text:
# R/ChiSquare.R In distributions3: Probability Distributions as S3 Objects

#### Documented in cdf.ChiSquareChiSquarelog_pdf.ChiSquarepdf.ChiSquarequantile.ChiSquarerandom.ChiSquaresupport.ChiSquare

#' Create a Chi-Square distribution
#'
#' Chi-square distributions show up often in frequentist settings
#' as the sampling distribution of test statistics, especially
#' in maximum likelihood estimation settings.
#'
#' @param df Degrees of freedom. Must be positive.
#'
#' @return A ChiSquare object.
#' @export
#'
#' @family continuous distributions
#'
#' @section Transformations:
#'
#' A squared standard [Normal()] distribution is equivalent to a
#' \eqn{\chi^2_1} distribution with one degree of freedom. The
#' \eqn{\chi^2} distribution is a special case of the [Gamma()]
#' distribution with shape (TODO: check this) parameter equal
#' to a half. Sums of \eqn{\chi^2} distributions
#' are also distributed as \eqn{\chi^2} distributions, where the
#' degrees of freedom of the contributing distributions get summed.
#' The ratio of two \eqn{\chi^2} distributions is a [FisherF()]
#' distribution. The ratio of a [Normal()] and the square root
#' of a scaled [ChiSquare()] is a [StudentsT()] distribution.
#'
#' @details
#'
#'   We recommend reading this documentation on
#'   <https://alexpghayes.github.io/distributions3/>, where the math
#'   will render with additional detail and much greater clarity.
#'
#'   In the following, let \eqn{X} be a \eqn{\chi^2} random variable with
#'   df = \eqn{k}.
#'
#'   **Support**: \eqn{R^+}, the set of positive real numbers
#'
#'   **Mean**: \eqn{k}
#'
#'   **Variance**: \eqn{2k}
#'
#'   **Probability density function (p.d.f)**:
#'
#'   \deqn{
#'     f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2}
#'   }{
#'     f(x) = 1 / (2 \pi \sigma^2) exp(-(x - \mu)^2 / (2 \sigma^2))
#'   }
#'
#'   **Cumulative distribution function (c.d.f)**:
#'
#'   The cumulative distribution function has the form
#'
#'   \deqn{
#'     F(t) = \int_{-\infty}^t \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} dx
#'   }{
#'     F(t) = integral_{-\infty}^t 1 / (2 \pi \sigma^2) exp(-(x - \mu)^2 / (2 \sigma^2)) dx
#'   }
#'
#'   but this integral does not have a closed form solution and must be
#'   approximated numerically. The c.d.f. of a standard normal is sometimes
#'   called the "error function". The notation \eqn{\Phi(t)} also stands
#'   for the c.d.f. of a standard normal evaluated at \eqn{t}. Z-tables
#'   list the value of \eqn{\Phi(t)} for various \eqn{t}.
#'
#'   **Moment generating function (m.g.f)**:
#'
#'   \deqn{
#'     E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2}
#'   }{
#'     E(e^(tX)) = e^(\mu t + \sigma^2 t^2 / 2)
#'   }
#'
#' @examples
#'
#' set.seed(27)
#'
#' X <- ChiSquare(5)
#' X
#'
#' mean(X)
#' variance(X)
#' skewness(X)
#' kurtosis(X)
#'
#' random(X, 10)
#'
#' pdf(X, 2)
#' log_pdf(X, 2)
#'
#' cdf(X, 4)
#' quantile(X, 0.7)
#'
#' cdf(X, quantile(X, 0.7))
#' quantile(X, cdf(X, 7))
ChiSquare <- function(df) {
d <- list(df = df)
class(d) <- c("ChiSquare", "distribution")
d
}

#' @export
print.ChiSquare <- function(x, ...) {
cat(glue("Chi Square distribution (df = {x$df})"), "\n") } #' @export mean.ChiSquare <- function(x, ...) { ellipsis::check_dots_used() x$df
}

#' @export
variance.ChiSquare <- function(x, ...) x$df * 2 #' @export skewness.ChiSquare <- function(x, ...) sqrt(8 / x$df)

#' @export
kurtosis.ChiSquare <- function(x, ...) 12 / x$df #' Draw a random sample from a chi square distribution #' #' @inherit ChiSquare examples #' #' @param x A ChiSquare object created by a call to [ChiSquare()]. #' @param n The number of samples to draw. Defaults to 1L. #' @param ... Unused. Unevaluated arguments will generate a warning to #' catch mispellings or other possible errors. #' #' @return A numeric vector of length n. #' @export #' random.ChiSquare <- function(x, n = 1L, ...) { rchisq(n = n, df = x$df)
}

#' Evaluate the probability mass function of a chi square distribution
#'
#' @inherit ChiSquare examples
#'
#' @param d A ChiSquare object created by a call to [ChiSquare()].
#' @param x A vector of elements whose probabilities you would like to
#'   determine given the distribution d.
#' @param ... Unused. Unevaluated arguments will generate a warning to
#'   catch mispellings or other possible errors.
#'
#' @return A vector of probabilities, one for each element of x.
#' @export
#'
pdf.ChiSquare <- function(d, x, ...) {
dchisq(x = x, df = d$df) } #' @rdname pdf.ChiSquare #' @export #' log_pdf.ChiSquare <- function(d, x, ...) { dchisq(x = x, df = d$df, log = TRUE)
}

#' Evaluate the cumulative distribution function of a chi square distribution
#'
#' @inherit ChiSquare examples
#'
#' @param d A ChiSquare object created by a call to [ChiSquare()].
#' @param x A vector of elements whose cumulative probabilities you would
#'   like to determine given the distribution d.
#' @param ... Unused. Unevaluated arguments will generate a warning to
#'   catch mispellings or other possible errors.
#'
#' @return A vector of probabilities, one for each element of x.
#' @export
#'
cdf.ChiSquare <- function(d, x, ...) {
pchisq(q = x, df = d$df) } #' Determine quantiles of a chi square distribution #' #' quantile() is the inverse of cdf(). #' #' @inherit ChiSquare examples #' @inheritParams random.ChiSquare #' #' @param probs A vector of probabilities. #' @param ... Unused. Unevaluated arguments will generate a warning to #' catch mispellings or other possible errors. #' #' @return A vector of quantiles, one for each element of probs. #' @export #' quantile.ChiSquare <- function(x, probs, ...) { # TODO: in the documentation, more information on return and # how quantiles are calculated ellipsis::check_dots_used() qchisq(p = probs, df = x$df)
}

#' Return the support of the ChiSquare distribution
#'
#' @param d An ChiSquare object created by a call to [ChiSquare()].
#'
#' @return A vector of length 2 with the minimum and maximum value of the support.
#'
#' @export
support.ChiSquare <- function(d){
c(0, Inf)
}

## Try the distributions3 package in your browser

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format: plain


text:
# How much more time will it take to finish the filling the pool?

The question is:

An empty pool being filled with water at a constant rate takes 8 hours to fill to \frac{3}{5} of its capacity.How much more time will it take to finish the filling the pool? (Ans 5 hr 20min)

Now here is how I am solving it:

The entire time required to fill the pool would be $\frac{40}{3}$ hours or $13$ hours and $1$ min

So how much more time would be $(13-8)$ hours will be $5$ hours and $1$ min.Why am i getting the wrong answer. I would appreciate it if someone could tell me where I am going wrong

-
$40/3$ hours is 13 hours and $\bf 20$ minutes ($13$ hours and $1/3$ hours). – David Mitra Sep 14 '12 at 0:51


text:
## Colorado State University — Pueblo, Spring 2017 Math 156, Introduction to Statistics Course Schedule and Homework Assignments

Here is a link back to the course syllabus/policy page.

This schedule is will be changing very frequently, please check it at least every class day, and before starting work on any assignment (in case the content of the assignment has changed).

Below, we refer to the class text Lies, Damned Lies, or Statistics as LDLoS. The whole book is available here, although below there will be links to particular sections of the book that you are asked to read and in which you will find problems to do.

If you see the symbol below, it means that class was videoed and you can get a link by e-mailing me. Note that if you know ahead of time that you will miss a class, you should tell me and I will be sure to video that day for you.

Homework for a particular day is due that day, either in class or handed in at my office by 3pm.

#### Week 1

• :
• :
• Read LDLoS up to and including §1.3.2
• Some content we discussed, terms defined:
• graphs for categorical variables:
• bar charts
• pie charts
• graphs for a quantitative variables:
• a stem-and-leaf plot
• the most important such graph: a histogram
• Hand in BI1. This will be an unusual one: write down two or three things you were surprised about in the organization of this course. Be specific, mention types or numbers or formats of some assignments, etc.
• :
• Some content we discussed, terms defined:
• histograms
• variants: frequency and relative frequency
• bins [or classes] and their edge behavior
• shape: symmetric, skew, multimodal
• Quiz 1 today (on material covered on Monday and Wednesday this week)
• Hand in HW1: problems 1.{1, 2, 3} on this page of LDLoS.
• Hand in BI2 (on an idea from Wednesday's class or readings)
• Today [Friday] is the last day to add classes.

#### Week 4

• :
• Some content we discussed, terms defined:
• going over Quiz 3
• going over HW3
• independent and dependent variables [also called explanatory and response variables]
• deterministic and non-deterministic relationships between variables
• scatterplots; their shape, strength, and direction
• Hand in BI9 (on an idea from Friday's class or readings)
• Hand in ASE3. See above, here and here, for the required parts of an ASE (the second of those explanations above also has a list of a few sites you could go to in order to find materials for an ASE, although of course something you are interested in yourself would probably be much more fun). Please look for something which mentions variability, standard deviation, quartiles, and/or outliers, or which uses the five-number summary or boxplots
• :
• Read LDLoS §2.2 and §2.3
• Some content we discussed, terms defined:
• qualitative description of the association shown in a scatterplot
• shape,
• strength, and
• direction
• the [Pearson] correlation coefficient $r$ of bivariate data
• defining formula
• computing with electronic tools
• always satisfies $-1\le r\le 1$
• meaning of the size of $|r|$
• meaning of the sign of $r$
• independence of units
• Hand in BI10 (on an idea from Monday's class or readings)
• :
• Some content we discussed, terms defined:
• the idea of linear regression
• sketching the line of best fit
• one point the regression line should go through: $(\overline{x},\overline{y})$.
• the residuals of a regression line
• the least squares regression line [LSRL]: definition
• formulæ for the LSRL
• Hand in BI11 (on an idea from Wednesday's class or readings)
• Quiz 4 today [on material from last Friday to this Wednesday]

#### Week 5

• :
• Read LDLoS Chapter 3 Intro & §3.1 & §3.2
• Some content we discussed, terms defined:
• going over Quiz 4
• going over HW4
• Review of equations of lines — particularly their slope and $y$-intercept
• Repeating (from Friday) the basic definitions and facts for the LSRL
• Simpson's Paradox (which is in our textbook LDLoS here)
• Applications of the LSRL:
• interpolation
• interpretation in a problem context of the slope and $y$-intercept of an LSLR
• Hand in BI12 (on an idea from Friday's class or readings)
• :
• Some content we discussed, terms defined:
• bivariate outliers
• sensitivity of the correlation coefficient and LSRL to outliers
• correlation is not causation
• extrapolation
• Review for Test I. See this review sheet.
• Hand in BI13 (on an idea from Monday's class or readings)
• Hand in HW5: problems 3.{1, 2, 3} on this page of LDLoS.
• :
• Test I in class today. Make sure you are comfortable with the material outlined on this review sheet. Don't forget your calculator, or other favorite electronic device, if you use one!

#### Week 6

• :
• Yes, we do have class today, even though it is the federal holiday commemorating the birthday of George Washington (who was actually born on February 22nd, 1732), although this holiday is often casually referred to as "Presidents' Day."
• Test I post-mortem.
• Hand in ASE4. Please try to find something which mentions or uses scatterplots, correlation or the correlation coefficient, or maybe even [linear] regression
• :
• Read LDLoS Introduction to Part 2, Introduction to Chapter 4, and §4.1
• Some content we discussed, terms defined:
• the idea of randomness
• sample spaces, outcomes, events
• the idea of probability, a probability model
• complement of a subset [event], notation $E^c$, translation into English: not
• probability rule for complements
• Hand in BI14. This is a special one: please write a paragraph about how you think Test I went for you. Are you perfectly content with how it turned out? If not, what do you think was the cause of the trouble? And what can you do next time to make things better?
• Hand in Test I revisions, if you like.
• :
• Some content we discussed, terms defined:
• intersection of sets [events], notation $A\cap B$, translation into English: and
• union of sets [events], notation $A\cup B$, translation into English: or
• disjoint events, notation $\emptyset$ for the empty set
• finishing the definition of a probability model
• Venn diagrams — how to fill in numbers correctly
• Hand in BI15 (on an idea from Wednesday's class or readings)
• Hand in HW6: problems 4.{1, 2, 3} on this page of LDLoS. [NOTE: due to network problems, this assignment was not posted until late Thursday afternoon. Therefore, if you hand it in on Monday, that will not be considered late!]

#### Week 7

• :
• Hand in HW6 [details above], if you did not do so on Friday
• Hand in BI16 (on an idea from Friday's class or readings)
• ASE5 is not due today -- in fact, no ASE is due this week; if you did one before seeing this notice, save it and use it in the future.
• Some content we discussed, terms defined:
• more on the ideas of sample spaces, events, and probability models
• finite probability models, such as [fair] coins and [fair] dice
• :
• Some content we discussed, terms defined:
• mutually exclusive is a synonym of disjoint
• the general rule for computing $P(A\cup B)$, whether or not the events $A$ and $B$ are disjoint
• starting conditional probability, including its formal definition, computing examples, etc.
• two events being independent
• Hand in BI17 (on an idea from Monday's class or readings)
• :
• Some content we discussed, terms defined:
• more examples of turning statements in English into formulæ with events, and computing their probabilities.
• more discussion of the difference (they're completely different!) between disjoint and independent, and the different uses they have
• very start of the idea of a random variables [RVs]
• Hand in BI18 (on an idea from Wednesday's class or readings)
• Hand in HW7: problems 4.{4, 5, 6} starting on this page of LDLoS; if you want a little extra time, this could come in on Monday without any penalty
• Quiz 5 was handed out today — if you didn't pick up a copy in class, e-mail your instructor and you will get a PDF by return email.

#### Week 8

• :
• :
• Reread LDLoS §4.3.1, §4.3.2, & §4.3.3
• Some content we discussed, terms defined:
• mostly working through recent terms/ideas even more
• discussion of what makes an RV either discrete or continuous
• Hand in BI20 (on an idea from Monday's class or readings)
• :
• Read LDLoS §4.3.4 & §4.3.5
• Some content we discussed, terms defined:
• [probability] density functions for continuous RVs
• the uniform distribution on an interval $[x_{min},x_{max}]$
• the Normal distribution with mean $\mu$ and standard deviation $\sigma$
• the standard Normal distribution
• Hand in BI21 (on an idea from Wednesday's class or readings)
• Quiz 6 today [on material from last Friday to this Wednesday]
• No homework due today: it will instead be due on Monday

#### Week 9

• :
• Some content we discussed, terms defined:
• standardizing a non-standard Normal RV
• the 68-95-99.7 Rule for Normal RVs
• using electronic tools to compute Normal probabilities.
• Hand in BI22 (on an idea from Friday's class or readings)
• No ASE due today — in fact, none due this week. Although there will be one due after Spring Break, so if you think you will be unable to work at all over the break, it might make sense to start that ASE now — maybe just going as far as finding a source and, if you are worried about whether it will make a good ASE, asking your instructor.
• Hand in HW8: problems 4.{7, 8, 9, 10} starting on this page of LDLoS
• :
• Review for Test II. See this review sheet.
• Hand in HW9, which is just problem 4.11 on this page of LDLoS
• Hand in BI23 (on an idea from Monday's class or readings)
• :
• Test II in class today. Make sure you are comfortable with the material outlined on this review sheet. Don't forget your calculator, or other favorite electronic device, if you use one!
• Today [Friday] is the last day to withdraw (with a W) from classes.

#### Week 10

• Spring Break! No classes, of course.

#### Week 11

• :
• Test II post-mortem.
• ASE6 was postponed to Wednesday, to give you more time to concentrate on thinking through Test II and doing excellent re-dos.
• :
• Read LDLoS Introduction to Chapter 5, and §5.1
• Some content we discussed, terms defined:
• population parameter
• representative
• bias, against which our strongest weapon is randomness
• voluntary sample bias
• simple random sample [SRS]
• Hand in BI24. This is a special one: please write a paragraph about how you think Test II went for you. Are you perfectly content with how it turned out? If not, what do you think was the cause of the trouble? And what can you do next time to make things better? Do you think the ideas you talked about in BI14 for test improvement were effective -- were you actually able to use them?
• Hand in ASE6. Please try to find something which mentions or uses Normal distributions [which is sometimes called a "bell[-shaped] curve"], uniformly distributed, or maybe even some article which talks about the uses of randomness or random numbers in modern science or computers ‐ e.g., sometimes this is called "using Monte Carlo methods." For this ASE only, and only if you talk about one of these last few topics, you may write a brief summary of something which is mostly an educational source, and which therefore does not have live data, variables, population, etc. Contact your instructor if you have any concerns about what you you intend to do.
• :
• Some content we discussed, terms defined:
• sample statistic
• The Law of Large Numbers [LoLNs]
• general strategies in simple surveys:
1. get as big a sample as possible
• e.g., to take advantage of the LoLNs
2. pick a good statistic
• e.g., the LoLNs says that $\overline{x}$ is a good statistic for $\mu$
• also, make sure survey questions are not misleading
3. use randomness
• always aim for the [nearly impossible] SRS
• wording effects
• always, for heaven's sake, avoid voluntary samples
• observational studies vs experiments [and relationship to inferences about causality]
• control group
• confounding
• lurking variable
• placebo effect, placebo
The British doctor Ben Goldacre has written and spoken in very informative and entertaining ways about the placebo effect. His books are and some good talks of his you can find online are
• Hand in BI25 (on an idea from Wednesday's class or readings)
• Hand in Test II revisions, if you like.
• No Quiz or HW homework due today. But keep an eye out for what is due and what will be happening in class on Monday.

#### Week 12

• :
• Some content we discussed, terms defined:
• blind, double-blind
• finalizing discussion of the gold standard for studies involving human subjects: a randomized, [placebo-]controlled trial, [RCT]
• starting experimental ethics — some examples of what not to do:
• the Nazi doctors
• the Tuskegee and Guatemalan Syphilis Experiments
• the Stanford Prison Experiment
• the Milgram "Obedience to Authority" Experiment
• Hand in BI26 (on an idea from Friday's class or readings)
• :
• Some content we discussed, terms defined:
• One checklist of ethical guidelines would be
1. always get informed consent (Caution: this is less obvious than it seems!)
2. do no harm to the test subjects
3. always get independent oversight, which often consists of the IRB and/or the FDA
4. preserve test subjects' confidentiality by default and only break it with their prior permission.
• Hand in BI27 (on an idea from Monday's class or readings)
• :
• Reread LDLoS §5.3 and read LDLoS Introduction to Part 3, LDLoS Introduction to Chapter 6, and §6.1
• Some content we discussed, terms defined:
• the Central Limit Theorem [CLT]
• Hand in BI28 (on an idea from Wednesday's class or readings)
• Hand in HW10: problems 5.{1, 2, 3, 4} starting on this page of LDLoS
• Quiz 7 today [on any new material since Spring Break]

#### Week 13

• :
• Hand in ASE7, which is a special one: read this article and any other sources you find on the same subject which are useful (such as the research report in the Proceedings of the National Academy of Sciences to which there is a link in that first article), and then do as detailed an ASE as you can on this topic, containing all the usual parts. Also include a section in this ASE discussing the ethics of this study. Use the ethical criteria we discussed in class and which are in the readings from last week. As a consequence, this ASE will probably be a fair bit longer than usual.
• Some content we discussed, terms defined:
• general idea of a confidence interval for the mean of a quantitative RV
• Hand in BI29 (on an idea from Friday's class or readings)
• :
• Some content we discussed, terms defined:
• formulæ for confidence intervals for means
• the meaning of the confidence level of a confidence interval
• Hand in BI30 (on an idea from Monday's class or readings)
• :
• Quiz 8 [on material from last Friday to this Wednesday] handed out today, due Monday
• Some content we discussed, terms defined:
• a bit more on the CLT and CIs
• general idea of a hypothesis test for the mean of a quantitative RV
• Hand in HW11: problems 6.{1, 2, 3} starting on this page of LDLoS. Or, if you like, it may be handed in on Monday.
• Hand in BI31 (on an idea from Wednesday's class or readings)

#### Week 14

• :
• No ASE due today. Next week, though, after Test III....
• Quiz 8 due today.
• Hand in HW11 due today, if you didn't hand it in last Friday.
• Some content we discussed, terms defined:
• formulæ for hypothesis tests for means
• the meaning of the $p$-value of a hypothesis test
• Hand in BI32 (on an idea from Friday's class or readings)
• :
• Quiz 9 today [on material from last Friday to this Monday]
• Some content we discussed, terms defined:
• type I error
• caution for hypothesis testing, particularly about interpreting the $p$-value
• Hand in HW12: problems 6.{4, 5, 6} starting on this page of LDLoS
• Hand in BI33 (on an idea from Monday's class or readings)
• :
• Going over lots of recent homeworks, ASEs, and quizzes.
• Hand in BI34 (on an idea from Wednesday's class or readings)
• Review for Test III. See this review sheet.

#### Week 15

• :
• :
• Hand in ASE8. Three options for this one:
1. Find one about a confidence interval for a population mean or average (same thing). It can help to look for the phrase "margin of error," sometimes called "sampling error" in the press. Be careful not to get a source which is about a confidence interval for a percentage (like election data often is, for example), since that is not a CI for a mean (means are not percentages).
2. Find one which is about a hypothesis test, as we have done in this class. Look for "$p$-value", "null/alternative hypothesis," other other such term[s].
3. Otherwise, you can just do a "free-range" ASE: pick a topic that interests you, a nice article or webpage or whatever, which has a clear bit of statistical content, and write up an ASE as we've been doing all semester. [So be sure to clearly talk about the population, variable[s], parameter[s], sample, methods, etc.] If you go this route, please make sure you find a one with rather a lot of interesting statistical content for which you can show off all of your knowledge. Then make sure you do show it off!
• Test III post-mortem.
• :
• Hand in Test III revisions, if you like.
• Review for Final Exam. See this review sheet.
• Hand in BI35, a special one: what do you intend to do for the next few days to enable you to do the best you possibly can on the final exam for this class? Be specific!


text:
+0

# Percent

0
132
3

Pizzas are sized by diameter. What percent increase in area results if Chantel's pizza increases from an 8-inch pizza to a 12-inch pizza?

Jul 5, 2022

#1
+2602
0

Original pizza: $$4 ^2 \pi = 16 \pi$$

New pizza: $$6^2 \pi = 36 \pi$$

So, $$36 \div 16 = 2.25 = 225\text{%}$$. But, recall that we are looking for the increase, so we subtract 100, to find that it is a $$\color{brown}\boxed{125 \text{%}}$$ increase.

Jul 5, 2022
#3
+1158
+7

my answer shows an easier way for the percentage step. but still. same answer!

nerdiest  Jul 5, 2022
#2
+1158
+9

An 8-inch pizza has a radius of 4 inches.

A 12-inch pizza has a radius of 6 inches.

Area of first pizza: $$\pi r^2$$$$= 16\pi$$

Area of second pizza: $$\pi r^2=36\pi$$

Calculation of percentage: $$\frac{36-16}{16} \times 100 = 125%$$%

Therefore, the area is increased by $$125$$%

Jul 5, 2022


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# Mimic mathjax behavior on undefined control sequences

I recently posted something on math.stackexchange.com, and was astonished by the behavior of the auto-renderer on undefined sequences. Instead of stopping the compilation right here, it would display in the rendered text the name of the sequence in red, and just carry on processing the rest of the document.

Needless to say, such a feature could prove incredibly useful for some usages of *TeX. One simple example is live-texing lecture notes, where a lot of spelling mistakes are likely to appear, but where one will prefer a successfully-compiled, badly-rendered document to a perfect, but half-compiled document.

The --nonstopmode option of pdflatex already offers some kind of such behavior, but then that's the opposite problem : the document is too well compiled, and the only way to find the errors is to either carefully proof-read the document, or to search through the log file.

Hence the question: is there a way to get latex to replace "Undefined control sequence" errors with simple warnings, and in that case print the name of the faulty macro verbatim (with its arguments, expanded or not) in red (for example)?

Or is it something too deeply buried in the core features of *TeX or mathjax, which would require to build a brand new engine?

• basically the answer to the question at the end is "yes" (unless luatex offers a hook to trap that primitive error message) – David Carlisle Jun 10 '13 at 11:00
• A quick scan through the LuaTeX manual gave me the following idea. The undefineds that you'd be interested in would all be directly from the input token stream so you could interrupt that using the process_input_buffer callback and test for a command token, then look it up and if it isn't there insert an appropriate token list instead. – Loop Space Jun 10 '13 at 11:06
• @AndrewStacey A command might be undefined when the buffer is populated, but get defined as part of the action of another command. Some environments define commands only as part of the \begin action. – egreg Oct 5 '13 at 21:56
• @DavidCarlisle Care to write an answer? – egreg Nov 2 '13 at 22:42


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### Home > CCAA8 > Chapter 2 Unit 2 > Lesson CCA: 2.1.3 > Problem2-32

2-32.

When Yoshi graphed the lines $y=2x+3$ and $y=2x−2$, she got the graph shown at right.

1. One of the lines above matches the equation $y=2x+3$, and the other matches $y=2x−2$. Which line matches which equation?

In the equation $y=mx+b,b$ is the $y$-intercept.
Match the $b$-value in each equation to the $y$-intercept on the graph.

2. Yoshi wants to add the line $y=2x+1$ to her graph. Predict where it would lie and sketch a graph to show its position. Justify your prediction.

The line has the same slope so it will be parallel to the other two lines.
Again, use the $y$-intercept to decide where the line will be on the graph.

It would lie between the lines $a$ and $b$ because it has $y=1$ as the intercept.

3. Where would the line $y=−2x+1$ lie? Again, justify your prediction and add the graph of this line to your graph from part (b).

The negative slope tells you that the line will travel downward. Once again, use the slope to decide where to put the line.

Use the graph in the eTool below to solve the problem.
Click the link at right for the full version of the eTool: 2-32 HW eTool (Desmos)


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# Question 3d0a0

Dec 8, 2015

$\text{28.08 g}$

#### Explanation:

As you know, potassium hydroxide, $\text{KOH}$, is a strong base and hydrochloric acid, $\text{HCl}$, is a strong acid, which means that they will both dissociate completely in aqueous solution to form hydroxide anions, ${\text{OH}}^{-}$, and hydrogen cations, ${\text{H}}^{+}$, respectively.

${\text{KOH"_text((aq]) -> "K"_text((aq])^(+) + "OH}}_{\textrm{\left(a q\right]}}^{-}$

${\text{HCl"_text((aq]) -> "H"_text((aq])^(+) + "Cl}}_{\textrm{\left(a q\right]}}^{-}$

The hydrogen cations will actually exist as hydronium ions, ${\text{H"_3"O}}^{+}$, in aqueous solution, but you can represent them as ${\text{H}}^{+}$ if you want.

These ions will then neutralize each other to form water. That is what your neutralization reaction is all about - how many hydroxide anions and how many hydrogen cations are present in solution.

The key to this problem is the mole ratio that exists between these two ions in the balanced chemical equation.

${\text{KOH"_text((aq]) + "HCl"_text((aq]) -> "KCl"_text((aq]) + "H"_2"O}}_{\textrm{\left(l\right]}}$

Potassium chloride, $\text{KCl}$, is a soluble salt, so it will exist as cations and anions in solution. The net ionic equation for this reaction will be

${\text{OH"_text((aq])^(-) + "H"_text((aq])^(+) -> "H"_2"O}}_{\textrm{\left(l\right]}}$

So, one mole of hydroxide anions will react with one mole of hydrogen cations to form one mole of water.

This tells you that you need equal numbers of moles of strong base and strong acid to reach complete neutralization.

Use hydrochloric acid's molar mass to determine how many moles you have in that $\text{18.25-g}$ sample

18.25 color(red)(cancel(color(black)("g"))) * "1 mole HCl"/(36.46 color(red)(cancel(color(black)("g")))) = "0.5005 moles HCl"

This means that in order to neutralize the acid, you need to have $0.5005$ moles of strong base.

Use potassium hydroxide's molar mass to determine how many grams would contain this many moles

0.5005 color(red)(cancel(color(black)("moles KOH"))) * "56.106 g"/(1color(red)(cancel(color(black)("mole KOH")))) = color(green)("28.08 g")#


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# Mechanics

calculus-based physics for scientists and engineers

## Dimensions

### Dimension of a Physical Quantity

We (the royal we, meaning physicists) have established seven quantities to serve as fundamental quantities. The dimension of these quantities is simply the name of the quantity! Since only three are used in mechanics, I've limited our discussion to those, which are shown below.

•   Quantity
• Length
• Mass
• Time
•   Dimension Name
• Length
• Mass
• Time
•   Dimension Symbol
• L
• M
• T

Other quantities don't get their own dimension name. Their dimension is written in terms of the fundamental dimensions.

example: The quantity of force has a dimension of Mass·Length/Time2, symbolically given by M·L/T2.

In the short video below, I describe what this means and the notation one uses to work with dimensions.

### Three Rules for Manipulating Dimensions

As you might have heard, in physics we do math with physical quantities! There are three rules to remember about dimensions when it comes to math.

1. You can only add quantities with the same dimension. This means that both qualities on either side of an equals sign have the same dimension.
2. Dimensions multiply like scalars.
3. Exponents and results of trig functions and logarithms are dimensionless. The arguments of trig functions are also dimensionless.

To learn the notation and how to work with dimensions, see the short video below.

### Dimensional Analysis

Dimensional analysis is a way of using the above rules to both determine the dimensions of physical quantities and check your work when you solve problems. The best way to learn this is to see and do examples. The video below walks you through it.

### Solution

1. If the position of a particle, $$x(t)$$ as a function of time, $$t$$ is given by $$x(t) = \alpha t^2 + \beta t^3 + C \sin (\omega t)$$ what are the dimensions of $$\alpha$$, $$\beta$$, $$C$$, and $$\omega$$?

### Solution

2. To keep an object moving in circular motion, a force must be applied to the object that is equal to some dimensionless constant times a product of the mass of the object, $$m$$, the radius of circular motion, $$r$$ (a length), and the speed of the object, $$v$$, each to some unknown power. We would write such an expression as: $$F = \alpha m^q r^p v^n$$ where $$\alpha$$ is the dimensionless constant. Find the unknown powers using dimensional analysis.

### Solution

1. According to Newton's universal law of gravity, the magnitude of the force of gravity a particle with mass, $$m_1$$ exerts on a mass, $$m_2$$ is given by $$F = \frac{Gm_1m_2}{r^2}$$ where $$r$$ is the distance between them and $$G$$ is a constant. What is the dimension of $$G$$?


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# Q&A - Ask Doubts and Get Answers

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Q

Q : 4    Number of persons in various age groups in a town is given in the following table.

 Age group (in years) $\small 1-14$ $\small 15-29$ $\small 30-44$ $\small 45-59$ $\small 60-74$ 75 and above Number of persons 2 lakhs 1 lakh 60 thousands 1 lakh 20 thousands 1 lakh 20 thousands 80 thousands 40 thousands

Draw a bar graph to represent the above information and answer the following questions. (take 1 unit length $\small =$ 20 thousands)

(a) Which two age groups have same population?

(b) All persons in the age group of 60 and above are called senior citizens. How many senior citizens are there in the town?

Following bar graph represents the above given informatio (a) The age groups 30 - 44 and  45 - 59 have the same population. (b) People in the age group 60 - 74 = 80,000       People whare 75 and above = 40,000       Total number of senior citizens in the town = 80,000 + 40,000 = 1,20,000

Q : 3     Following table shows the number of bicycles manufactured in a factory during the years 1998 to 2002. Illustrate this data using a bar graph. Choose a scale of your choice.

 Years Number of bicycles manufactured 1998 800 1999 600 2000 900 2001 1100 2002 1200

(a) In which year were the maximum number of bicycles manufactured?
(b) In which year were the minimum number of bicycles manufactured?

The following bar graph illustrates the above-given data: (a) In 2002 the maximum number of bicycles were manufactured. (b) In 1999 the minimum number of bicycles were manufactured.

Q : 2    The number of Mathematics books sold by a shopkeeper on six consecutive days is shown below :

 Days Sunday Monday Tuesday Wednesday Thursday Friday Number of books sold 65 40 30 50 20 70

Draw a bar graph to represent the above information choosing the scale of your choice.

Using the above given information the followiing bar graph has been drawn The scale is 1 unit = 10 books.

Q : 1   A survey of 120 school students was done to find which activity they prefer to do in their free time.

 Preferred activity Number of students Playing 45 Reading story books 30 Watching TV 20 Listening to music 10 Painting 15

Draw a bar graph to illustrate the above data taking scale of 1 unit length $\small =$ 5 students.  Which activity is preferred by most of the students other than playing?

Reading storybooks is preferred by most of the students other than playing.

Q : 3    Observe this bar graph which shows the marks obtained by Aziz in half-yearly examination in different subjects. Answer the given questions.

(a) What information does the bar graph give?

(b) Name the subject in which Aziz scored maximum marks.

(c) Name the subject in which he has scored minimum marks.

(d) State the name of the subjects and marks obtained in each of them.

(a) This bar graph shows the marks obtained by Aziz in half-yearly examination in different subjects. (b) Aziz scored maximum marks in Hindi. (c) Aziz scored minimum marks in Social Science. (d) ) Hindi – 80, English – 60, Mathematics – 70, Science – 50 and Social Studies – 40.

Q : 2    Observe this bar graph which is showing the sale of shirts in a ready made shop from Monday to Saturday.

Now answer the following questions :

(a) What information does the above bar graph give?

(b) What is the scale chosen on the horizontal line representing number of shirts?

(c) On which day were the maximum number of shirts sold? How many shirts were sold on that           day?

(d) On which day were the minimum number of shirts sold?

(e) How many shirts were sold on Thursday?

(a) The bar graph is showing the sale of shirts in a ready-made shop from Monday to Saturday. (b) 1 unit = 5 shirts. (c) On Saturday the maximum number of shirts sold. 60 shirts were sold on Saturday. (d) On Tuesday the minimum number of shirts were sold. (e) 35 shirts were sold on Thursday.

Q: 1     The bar graph given alongside shows the amount of wheat purchased by government during the year $1998-2002$. Read the bar graph and write down your observations. In which year was

(a) the wheat production maximum?

(b) the wheat production minimum?

(a) In 2002 the wheat production maximum. (b) In 1998 the wheat production minimum.

Q : 2    Total number of students of a school in different years is shown in the following table

 Years Number of students 1996 400 1998 535 2000 472 2002 600 2004 623

B. Prepare another pictograph of students using any other symbol each representing 50 students. Which pictograph do you find more informative?

B. The second pictograph is more informative

Q : 2     Total number of students of a school in different years is shown in the following table

 Years Number of students 1996 400 1998 535 2000 427 2002 600 2004 623

A.     Prepare a pictograph of students using one symbol  to represent 100 students and answer the following questions:

(a) How many symbols represent total number of students in the year 2002?

(b) How many symbols represent total number of students for the year 1998?

A.  (a) 6 symbols represent the total number of students in the year 2002. (b)      5 and 7/20 symbols , i.e 5 complete and 1 incomplete represent the total number of students for the year 1998.

Q : 7    Total number of animals in five villages are as follows :

Village A: 80    Village B: 120      Village C: 90   Village D: 40     Village E: 60

Prepare a pictograph of these animals using one symbol $\small \otimes$ to represent 10 animals and answer the following questions :

(a) How many symbols represent animals of village E?

(b) Which village has the maximum number of animals?

(c) Which village has more animals: village A or village C?

Villages Number of Animals         = Ten Animals     A           B               C            D       E        (a) 6 symbols represent animals of village E. (b) Village B has the maximum number of animals. (c) Village C has more animals.

Q : 7    In a village six fruit merchants sold the following number of fruit baskets in a particular            season :

Observe this pictograph and answer the following questions :

(a) Which merchant sold the maximum number of baskets?

(b) How many fruit baskets were sold by Anwar?

(c) The merchants who have sold 600 or more number of baskets are planning to buy a godown for the next season. Can you name them?

(a) Martin has sold the maximum number of baskets. (b) Number of fruit baskets were sold by Anwar = (c) Anwar, Martin and Ranjit Singh have sold more than 600 baskets.

Q : 6    The sale of electric bulbs on different days of a week is shown below :

Observe the pictograph and answer the following questions :

(a) How many bulbs were sold on Friday?

(b) On which day were the maximum number of bulbs sold?

(c) On which of the days same number of bulbs were sold?

(d) On which of the days minimum number of bulbs were sold?

(e) If one big carton can hold 9 bulbs. How many cartons were needed in the given week?

(a) The number of bulbs sold on Friday = . (b) On Sunday the maximum number of bulbs were sold. (c) On both Wednesday and saturday, 8 bulbs were sold. (d) On Wednesday and saturday, the minimum number of bulbs (8) were sold. (e) Total number of bulbs sold in the weekend =                      Number of cartons required = 9 + 1 = 10 ( we cannot have fractional number of boxes)

Q: 5    The number of girl students in each class of a co-educational middle school is depicted by the pictograph :

Observe this pictograph and answer the following questions :

(a) Which class has the minimum number of girl students?

(b) Is the number of girls in Class VI less than the number of girls in Class V?

(c) How many girls are there in Class VII?

(a) Class VIII has the minimum number of girls.       Number of girls in class VIII  =  (b) Number of girls in class VI =        Number of girls in class V =        Therefore the number of girls in Class VI is not less than the number of girls in Class V. (c) Number of girls in class VII =

Q: 4   Following pictograph shows the number of tractors in five villages.

Observe the pictograph and answer the following questions.

(i) Which village has the minimum number of tractors?

(ii) Which village has the maximum number of tractors?

(iii) How many more tractors village C has as compared to village B.

(iv) What is the total number of tractors in all the five villages?

(a) Village D has only three tractors which is the minimum. (b) Village C has eight tractors which is the maximum. (c) Village C has eight tractors          Village B has five tractors      Village C has three tractors more than village B. (d) The total number of tractors in all the villages = 6 + 5 + 8 + 3 + 6 = 28.

Q : 3    Catherine threw a dice 40 times and noted the number appearing each time as shown below :

 1 3 5 6 6 3 5 4 1 6 2 5 3 4 6 1 5 5 6 1 1 2 2 3 5 2 4 5 5 6 5 1 6 2 3 5 2 4 1 5

Make a table and enter the data using tally marks. Find the number that appeared.

(a) The minimum number of times

(b) The maximum number of times                                                                                (

c) Find those numbers that appear an equal number of times.

Number shown  by the dice Tally Marks Number of Times 1 7 2 6 3 5 4 4 5 11 6 7 Total   40 (a) The number that appeared the minimum number of times is 4. ( 4 times) (b) The number that appeared the maximum number of times is 5. ( 11 times) (c) 1 and 6 both appeared 7 times.

Q : 2      Following is the choice of sweets of 30 students of Class VI.

(a) Arrange the names of sweets in a table using tally marks.

(b) Which sweet is preferred by most of the students?

(a) Sweets Tally Marks Number of Students Ladoo 11 Barfi 3 Jalebi 7 Rasgulla 9 Total   30 (b) Ladoo is preffered by the maximum number of students, 11 students prefer.

Q : 1      In a Mathematics test, the following marks were obtained by 40 students. Arrange these marks in a table using tally marks.

 8 1 3 7 6 5 5 4 4 2 4 9 5 3 7 1 6 5 2 7 7 3 8 4 2 8 9 5 8 6 7 4 5 6 9 6 4 4 6 6

(a) Find how many students obtained marks equal to or more than 7.

(b) How many students obtained marks below 4?

Marks Obtained Tally Marks Number of  Students 1     2 2      3 3     3 4 7 5 6 6 7 7 5 8   4 9   3 Total   40   (a) Number of  students who obtained marks equal to or more than 7 = 5 + 4 + 3 = 12 (b) Number of  students who students obtained marks below 4 = 3 + 3 + 2 = 8

6. Find the probabilities of the events given in Question 2.

2. When a die is thrown, list the outcomes of an event of getting

(i) (a) a prime number

(b) not a prime number.

(ii) (a) a number greater than 5

(b) a number not greater than 5.

Total outcomes = 6 6.(i) (a) prime numbers = 2,3,5 = 3 numbers           probability of prime number                        (b) non prime numbers = 1,4,6 = 3 numbers          probability of non prime number                  (ii) (a) number greater than 5 = 6  = 1 number only     probability of getting number greater than 5                      (b) number not greater than 5 = 1,2,3,4,5 = 5...

5. If you have a spinning wheel with $\inline 3$ green sectors, $\inline 1$ blue sector and $\inline 1$ red sector, what is the probability of getting a green sector? What is the probability of getting a non blue sector?

Total sectors = 5 the probability of getting a green sector                      Non blue sector = 5-1 = 4           probability of getting a non blue sector

4. Numbers 1 to 10 are written on ten separate slips (one number on one slip), kept in a box and mixed well. One slip is chosen from the box without looking into it. What is the probability of .

(i) getting a number $\inline 6$?

(ii) getting a number less than $\inline 6$ ?

(iii) getting a number greater than $\inline 6$ ?

(iv) getting a 1-digit number?

(i) the probability of getting a number (ii) the numbers less than 6 are: 1,2,3,4,5=5 numbers.     the probability of  getting a number less than   (iii) the numbers greater than 6 are: 7,8,9,10=4 numbers.       the probability of  getting a number greater than      (iv) 1 digit numbers = 1,2,3,4,5,6,7,8,9 = 9 numbers      the probability of  getting a 1 digit  number
Exams
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• ### On the photon or meson formation in $J/\psi$ decays into $p {\overline{p}}$(1802.00763)

Feb. 2, 2018 hep-ph, hep-ex, nucl-th
The measurements of the $J/\psi \rightarrow \gamma p \overline{p}$ decays by the BES Collaboration indicate an enhancement at the $p \overline{p}$ threshold which, however, is present neither in the J/$\psi$ decays into $\omega p \overline{p}$ nor into $\pi p \overline{p}$. Here, two processes for describing the decays $J/\psi \rightarrow \mathcal{B} p\overline{p}$ where $\mathcal{B} = \gamma ,\omega, \phi,\pi$ are presented. The first one, applied not only to the radiative decay to reproduce the threshold peak but also to the $\omega p \overline{p}$ decay channel to improve the description of the spectrum, postulates a direct emission of the boson before the baryon pair is formed. The second process assumes that the boson $\mathcal{B}$ is emitted from the baryon pair following the $J/\psi$ decay and includes for the decays into $\gamma p \overline{p}$ a final state nucleon-antinucleon interaction based on the Paris $N \overline{N}$ potential. The reproduction of the $p \overline{p}$ distribution in the $J/\psi \rightarrow \omega p \overline{p}$ decays needs a final state interaction involving a $N(2050)\ 3/2^-$ resonance. The photon- and meson-emission rates are reproduced in a semi-quantitative way.
• ### Search for Deeply Bound Kaonic Nuclear States with AMADEUS(1712.09908)

Dec. 28, 2017 nucl-ex
We briefly report on the search for Deeply Bound Kaonic Nuclear States with AMADEUS in the Sigma0 p channel following K- absorption on 12C and outline future perspectives for this work.
• ### Antikaon Interactions with Nucleons and Nuclei - AMADEUS At Da$\Phi$ne(1704.06562)

April 21, 2017 nucl-ex
The aim of AMADEUS is to provide unprecedented experimental information on K$^-$ absorption in light nuclear targets, to face major open problems in hadron nuclear physics in the strangeness sector, namely the nature of the $\Lambda$(1405), strongly related to the possible existence of kaonic nuclear clusters, kaons and hyperon scattering cross sections on nucleons and nuclei. These issues are fundamental for a better understanding of the non-perturbative QCD in the strangeness sector. AMADEUS step 0 deals with the analysis of the 2004-2005 KLOE collected data. The interactions of the negative kaons produced by the DA$\Phi$NE collider (a unique source of monochromatic low-momentum kaons) with the materials of the KLOE detector, used as active targets, provide samples of K$^-$ absorptions on H, ${}^4$He, ${}^{9}$Be and ${}^{12}$C, both at-rest and in-flight. A second step deals with the data from the implementation in the central region of the KLOE detector of a pure graphite target, providing a high statistic sample of K$^- \, {}^{12}$C nuclear captures at rest. For the future a new setup, with various dedicated gaseous and solid targets, is under preparation.
• ### K$^-$ absorption on two nucleons and ppK$^-$ bound state search in the $\Sigma^0$p final state(1511.04496)

Nov. 14, 2015 nucl-ex
We report the measurement of K$^-$ absorption processes in the $\Sigma^0$p final state and the first exclusive measurement of the two nucleon absorption (2NA) with the KLOE detector. The 2NA process without further interactions is found to be 12\% of the sum of all other contributing processes, including absorption on three and more nucleons or 2NA followed by final state interactions with the residual nucleons. We also determine the possible contribution of the ppK$^-$ bound state to the $\Sigma^0$p final state. A yield of ppK$^- /\mathrm{K^-_{stop}}$ is found to be $(0.044 \pm 0.009\, stat ^{+ 0.004} _{- 0.005} \,syst) \cdot 10^{-2}$ but its statistical significance based on an F-test is only 1$\sigma$.
• ### Antinucleon-nucleus interaction near threshold from the Paris $\bar NN$ potential(1506.06965)

Sept. 1, 2015 hep-ph, nucl-th
A general algorithm for handling the energy dependence of hadron-nucleon amplitudes in the nuclear medium, consistently with their density dependence, has been recently applied to antikaons, eta mesons and pions interacting with nuclei. Here we apply this approach to antiprotons below threshold, analyzing experimental results for antiprotonic atoms across the periodic table. It is also applied to antiproton and antineutron interaction with nuclei up to 400 MeV/c, comparing with elastic scattering and annihilation cross sections. The underlying $\bar pN$ scattering amplitudes are derived from the Paris $\bar NN$ potential, including modifications in the medium. Emphasis is placed on the role of the $P$-wave amplitudes with respect to the repulsive $S$-wave amplitudes.
• ### Baryonium, a common ground for atomic and high energy physics(1503.03319)

March 11, 2015 hep-ph, nucl-th
Indications of the existence of quasi-bound states in the N-Nbar system are presented. Measurements by BES discovered a broad enhancement close to the p-pbar threshold in the S wave, isospin 0 state formed in radiative decays of J/psi. Another enhancement located about 50 MeV below the threshold was found in mesonic decays of J/psi. In terms of the Paris potential model it was shown that these are likely to represent the same state. Antiprotonic atomic data provide some support for this interpretation and indicate the existence of another fairly narrow quasi-bound state in a P wave.
• ### Probing the nuclide 180W for neutrinoless double-electron capture exploration(1111.6377)

Nov. 28, 2011 nucl-ex
The mass difference of the nuclides 180W and 180Hf has been measured with the Penning-trap mass spectrometer SHIPTRAP to investigate 180W as a possible candidate for the search for neutrinoless doubleelectron capture. The Q-value was measured to 143.20(27)keV. This value in combination with the calculations of the atomic electron wave functions and other parameters results in a half-life of the 0+ \rightarrow 0+ ground-state to ground-state double-electron capture transition of approximately 5\cdot10E27 years/<m_ee[eV]>^2.
• ### K-Mesic Nuclei Versus eta -Mesic Nuclei(1011.0866)

Nov. 3, 2010 nucl-th
The nuclear states of \bar K and \eta are bound by a similar mechanism - the excitations of nucleons to \Lambda (1405) and N*(1535) resonant states. The observed large differences in binding energies are understood in terms of separation of the involved energies and the resonance positions. The other experimental findings: broad \bar K-mesic and narrow \eta -mesic states are more difficult to understand. A phenomenological model for \eta -N interactions is used to explain the suppression of the \eta absorption in light nuclei.
• ### On the structure of the X(1835) baryonium(0904.2163)

Nov. 19, 2009 hep-ph, hep-ex, nucl-th
The measurement by the BES collaboration of J/psi -> gamma p pbar decays indicates an enhancement at the p-pbar threshold. In another experiment BES finds a peak in the invariant mass of pi-mesons produced in the possibly related decay J/psi -> gamma pi+ pi- eta'. Using a semi-phenomenological potential model which describes all the N-Nbar scattering data, we show that the explanation of both effects may be given by a broad quasi-bound state in the spin and isospin singlet S wave. The structure of the observed peak is due to an interference of this quasi-bound state with a background amplitude and depends on the annihilation mechanism.
• ### Paris nucleon-antinucleon potential constrained by recent antiprotonic-atom data and antineutron-proton total cross sections(0807.4454)

May 11, 2009 hep-ph, nucl-th
We report on an updated Paris nucleon-antinucleon optical potential. The long- and intermediate-range real parts are obtained by G-parity transformation of the Paris nucleon-nucleon potential based on a theoretical dispersion-relation treatment of the correlated and uncorrelated two-pion exchange. The short-range imaginary potential parametrization results from the calculation of the nucleon-antinucleon annihilation box diagram into two mesons with a nucleon-antinucleon intermediate state in the crossed channel. The parametrized real and imaginary short range parts are determined by fitting not only the existing experimental data included in the 1999 version of the Paris nucleon-antinucleon potential, but also the recent antiprotonic-hydrogen data and antineutron-proton total cross sections. The description of these new observables is improved. Only this readjusted potential generates an isospin zero 1S0, 52 MeV broad quasibound state at 4.8 MeV below the threshold. Recent BES data on J/psi decays could support the existence of such a state.
• ### Variational calculations for K-few-nucleon systems(0808.3329)

Aug. 25, 2008 hep-ph, nucl-th
Deeply bound KNN, KNNN and KNNNN states are discussed. The effective force exerted by the K meson on the nucleons is calculated with static nucleons. Next the binding energies are obtained by solving the Schrodinger equation or by variational calculations. The dominant attraction comes from the S-wave Lambda(1405) and an additional contribution is due to Sigma(1385). The latter state is formed at the nuclear peripheries and absorbs a sizable piece of the binding energy. It also generates new branches of quasi-bound states. The lowest binding energies based on a phenomenological KN input fall into the 40-80 MeV range for KNN, 90-150 MeV for KNNN and 120-220 MeV for K-alpha systems. The uncertainties are due to unknown KN interactions in the distant subthreshold energy region.
• ### Neutron density distributions from antiprotonic 208Pb and 209Bi atoms(nucl-ex/0702016)

Feb. 26, 2007 nucl-ex
The X-ray cascade from antiprotonic atoms was studied for 208Pb and 209Bi. Widths and shifts of the levels due to the strong interaction were determined. Using modern antiproton-nucleus optical potentials the neutron densities in the nuclear periphery were deduced. Assuming two parameter Fermi distributions (2pF) describing the proton and neutron densities the neutron rms radii were deduced for both nuclei. The difference of neutron and proton rms radii /\r_np equal to 0.16 +-(0.02)_{stat} +- (0.04)_{syst} fm for 208Pb and 0.14 +- (0.04)_{stat} +- (0.04)_{syst} fm for 209Bi were determined and the assigned systematic errors are discussed. The /\r_np values and the deduced shapes of the neutron distributions are compared with mean field model calculations.
• ### Nuclear surface studies with antiprotonic atom X-rays(nucl-th/0702029)

Feb. 8, 2007 nucl-th
The recent and older level shifts and widths in pbar atoms are analyzed. The results are fitted by an antiproton-nucleus optical potential with two basic complex strength parameters. These parameters are related to average S and P wave scattering parameters in the subthreshold energy region. A fair consistency of the X-ray data for all Z values, stopping pbar data and the Nbar-N scattering data has been achieved. The determination of neutron density profiles at the nuclear surface is undertaken, and the determination of the neutron R_{rms} radii is attempted. Uncertainties due to the input data and the procedure are discussed.
• ### Antiproton-Proton Channels in J/psi Decays(hep-ph/0501112)

Dec. 16, 2005 hep-ph
The recent measurements by the BES Collaboration of J/psi decays into a photon and a proton-antiproton pair indicate a strong enhancement at the proton-antiproton threshold not observed in the decays into a neutral pion and a proton-antiproton pair. Is this enhancement due to a proton-antiproton quasi-bound state or a baryonium? A natural explanation follows from a traditional model of proton-antiproton interactions based on G-parity transformation. The observed proton-antiproton structure is due to a strong attraction in the 1S0 state, and possibly to a near-threshold quasi-bound state in the 11S0 wave.
• ### The antinucleon-nucleon quasi-bound states: J/psi and atomic evidence(hep-ph/0508064)

Aug. 5, 2005 hep-ph
The measurements of J/psi decays into photon, proton and antiproton show a strong enhancement at the proton-antiproton threshold not seen in the decays into neutral pion, proton and antiproton. What is the nature of this enhancement? A natural interpretation can be performed in terms of a classical model of nucleon-antinucleon interactions based on G-parity transformation. The observed proton-antiproton structure is the consequence of the strong attraction in the singlet S-wave state related predominantly to pion exchanges. Similar attractions generate near threshold: an isospin-zero virtual (or quasi-bound) state in singlet S-wave, an isospin-one quasi-bound state in triplet P-wave with total angular momentum one and an isospin-zero resonance in triplet P-wave with total angular momentum zero. These P-wave structures find support in the antiproton-atomic data.
• ### The physical interest in kaonic- and antiprotonic-deuterium atoms(nucl-th/0508013)

Aug. 5, 2005 nucl-th
Exotic deuterium and helium are discussed. The S, P and D levels of antiprotonic and kaonic atoms are calculated. Absorptive, subthreshold antiproton-nucleon amplitudes are extracted from experimental data and compared to model calculations. The existence of a quasi-bound state in the antiproton-nucleon system is indicated. In the kaonic atoms some effects of the Sigma(1385) resonance are evaluated.
• ### eta-Nucleon Scattering Length and Effective Range uncertainties(nucl-th/0411024)

June 28, 2005 nucl-th
The coupled eta-N, pi-N, gamma-N, 2pi-N system is described by a K-matrix method. The parameters in this model are adjusted to get an optimal fit to pi-N-->pi-N, pi-N-->eta-N, gamma-N-->pi-N and gamma-N-->eta-N data in an energy range of about 100 MeV or so each side of the eta-threshold. Compared with our earlier analysis, we now utilize recent Crystal Ball data. However, the outcome confirms our previous result that the eta-nucleon scattering length (a) is large with a value of 0.91(6)+i0.27(2) fm.
• ### On the nuclear states of K^- mesons(nucl-th/0501019)

Jan. 8, 2005 hep-ph, nucl-th
The KNNN bound states recently discovered at KEK are studied. It is shown that the Lambda(1405) and Sigma(1385) resonant states coupled to the KN system may generate an attraction strong enough to form such bound states.
• ### Search for and study of eta-mesic nuclei in pA-collisions at the JINR LHE nuclotron(nucl-ex/0412036)

Dec. 23, 2004 nucl-ex
An approved experiment at the internal proton beam of the JINR nuclotron on a search for eta-mesic nuclei in the reaction pA --> np + eta(A-1) --> np + pi-p + X is briefly presented.
• ### A phenomenological determination of the pion-nucleon scattering lengths from pionic hydrogen(hep-ph/0411230)

Nov. 17, 2004 hep-ph
A model independent expression for the electromagnetic corrections to a phenomenological hadronic pion-nucleon scattering length, extracted from pionic hydrogen, is obtained. In a non-relativistic approach and using an extended charge distribution, these corrections are derived up to terms of order (alpha)**2 log(alpha) in the limit of a short-range hadronic interaction. We infer a charged pion-proton scattering length of 0.0870(5) in units of inverse pion mass, which gives for the charged pion-proton-neutron coupling, through the GMO relation, a value of 14.04(17).
• ### Antiproton-proton resonant like channels in J/Psi decays into photon, proton and antiproton(hep-ph/0411218)

Nov. 16, 2004 hep-ph
The BES collaboration has recently observed a strong enhancement close to the proton-antiproton threshold in the J/Psi decays into photon, proton and antiproton. Such a structure can be explained by a traditional nucleon-antinucleon model. The near threshold 1S0 bound state and/or the well-established 3P0 resonant state found in this nucleon-antinucleon interaction can adequately describe the BES data.
• ### A phenomenological negatively charged pion-proton scattering length from pionic hydrogen(hep-ph/0310134)

May 5, 2004 hep-ph, nucl-th
We derive a closed, model independent, expression for the electromagnetic correction factor to a phenomenological hadronic scattering length extracted from a hydrogenic atom. It is obtained in a non-relativistic approach and in the limit of a short ranged hadronic interaction to terms of order ((alpha)**2)(log(alpha)) using an extended charge distribution. A hadronic negatively charged pion-proton scattering length of 0.0870(5), in units of inverse charged pion-mass, is deduced, leading to a pion-nucleon coupling constant from the GMO relation equals to 14.04(17).
• ### On eta-pi mixing close to the eta-Helium threshold(nucl-th/0308057)

Aug. 21, 2003 nucl-th
A K-matrix formalism is used to relate the amplitudes for the three reactions pd --> 3He eta, pi- 3H --> 3He eta and pd --> 3H pi+. Free parameters are fitted to the available experimental data and an extrapolation below the eta 3He threshold is made to see the origin of the eta 3He threshold enhancement. The existence of a virtual -- and not a quasi-bound -- state finds support in the data. The K-matrix permits a discussion of eta-pi mixing. A mixing parameter of 0.010(5), i.e. a mixing angle theta =0.6(3) degrees, is extracted from a best fit to the very recent pd --> 3He pi0 reaction data.
• ### Electromagnetic corrections to $\pi ^-$p scattering length from pionic hydrogen(hep-ph/0211433)

Nov. 27, 2002 hep-ph
We derive a closed, model space independent, expression for the electromagnetic correction factor $\delta$ to the scattering length $a$ extracted from a hydrogenic atom with an extended charge to order $\alpha ^2$ and $a^3$ in the limit of a short ranged hadronic interaction.
• ### Final state interactions in the eta-d system(nucl-th/0104053)

April 17, 2001 hep-ph, nucl-th
The eta-deuteron scattering amplitude is calculated with a recent K-matrix model of eta-N interactions. The existence of a narrow virtual state in the eta-deuteron system is inferred. This state couples strongly to the eta-d channel but its effect on final state interactions in the pn --> eta-d reaction is shown to be rather weak. Fairly large eta-N scattering lengths are allowed by the experimental data.


text:
At the time she was hired as a server at the Grumney Family Restaurant, Beth Brigden was told, %u201CYou can average $82 a day in tips.%u201D Assume the population of daily tips is normally distributed with a standard deviation of$3.26. Over the first 44 days she was employed at the restaurant, the mean daily amount of her tips was $84.61. At the .02 significance level, can Ms. Brigden conclude that her daily tips average more than$82?

(a) State the null hypothesis and the alternate hypothesis.

 H0:%u03BC %u2265 82 ; H1: %u03BC < 82 H0:%u03BC >82 ; H1: %u03BC = 82 H0:%u03BC = 82 ; H1: %u03BC %u2260 82 H0:%u03BC %u2264 82 ; H1: %u03BC > 82

(b) State the decision rule.

 Reject H1 if z < 2.05 Reject H1 if z > 2.05 Reject H0 if z < 2.05 Reject H0 if z > 2.05

 (c) Compute the value of the test statistic. (Round your answer to 2 decimal places.)

 Value of the test statistic

(d) What is your decision regarding H0 ?

 Do not reject H0 Reject H0

 (e) What is the p-value? (Round your answer to 4 decimal places.)

 p-value