/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 True or false, Part I. Determine... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 37

True or false, Part I. Determine if the statements below are true or false. For each false statement, suggest an alternative wording to make it a true statement. (a) The chi-square distribution, just like the normal distribution, has two parameters, mean and standard deviation. (b) The chi-square distribution is always right skewed, regardless of the value of the degrees of freedom parameter. (c) The chi-square statistic is always positive. (d) As the degrees of freedom increases, the shape of the chi-square distribution becomes more skewed.

Short Answer

Expert verified
Statements (b) and (c) are true. False statements are (a) and (d).

Step by step solution

01

Analyze Statement (a)

The chi-square distribution does not have parameters like mean and standard deviation as the normal distribution does. It is instead described by a single parameter, which is the degrees of freedom. Hence, statement (a) is false.
02

Correct Statement (a)

An alternative true statement would be: "The chi-square distribution has one parameter, which is the degrees of freedom."
03

Analyze Statement (b)

The statement is true. The chi-square distribution is indeed right-skewed, and this skewness is present regardless of the degrees of freedom, although it decreases as the degrees of freedom increase.
04

Analyze Statement (c)

The chi-square statistic is a measure of the sum of squared standard normal variables. Since squaring any real number results in a non-negative value, the chi-square statistic is always positive. Thus, statement (c) is true.
05

Analyze Statement (d)

This statement is false. As the degrees of freedom increase, the chi-square distribution becomes less skewed (i.e., more symmetrical), not more skewed.
06

Correct Statement (d)

An alternative true statement would be: "As the degrees of freedom increase, the shape of the chi-square distribution becomes more symmetrical."

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degrees of freedom
In statistical terms, "degrees of freedom" refers to the number of independent values or quantities that can vary in the analysis of a data set. It serves as a key parameter in various statistical calculations and tests. For a chi-square distribution, the degrees of freedom define its shape and are calculated based on the number of categories in the data minus one.
This singular parameter plays an instrumental role in determining the characteristics and behavior of the chi-square distribution. The more degrees of freedom, the closer the chi-square distribution resembles a normal distribution.
If you ever encounter a chi-square test, remember that the degrees of freedom are what allow your test to adapt to the size and nature of your data, providing flexibility and robustness in statistical testing.
Statistical skewness
Skewness in a statistical distribution refers to the degree of asymmetry observed within the data set. When we talk about a chi-square distribution, it is famous for being right-skewed. This means that the distribution has a longer tail on the right side compared to the left.
Right skewness implies that a distribution has more low-value data points and fewer high-value data points. Interestingly, the skewness in the chi-square distribution decreases as the degrees of freedom increase, resulting in a distribution that appears more symmetrical and less skewed with higher degrees of freedom.
Understanding skewness is valuable, especially when interpreting results from a chi-square test, as it might influence perception or decisions based on how values tend to be distributed along its curve.
Chi-square statistic
The chi-square statistic is an important component of the chi-square test, which is commonly used in hypothesis testing. It quantifies the difference between observed and expected frequencies in one or more categories of a contingency table.
This statistic is calculated by summing the squared differences between observed and expected values, divided by the expected values.
Since it involves squaring these differences, the chi-square statistic is always non-negative. The chi-square statistic gives us an idea of how much observed data diverges from what we would expect if the null hypothesis were true.
In practice, if the chi-square statistic is high, it suggests that the observed data strongly deviates from the null hypothesis, leading to a potential rejection of the null in favor of the alternative hypothesis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sleep deprivation, CA vs. OR, Part I. According to a report on sleep deprivation by the Centers for Disease Control and Prevention, the proportion of California residents who reported insufficient rest or sleep during each of the preceding 30 days is \(8.0 \%\), while this proportion is \(8.8 \%\) for Oregon residents. These data are based on simple random samples of 11,545 California and 4,691 Oregon residents. Calculate a \(95 \%\) confidence interval for the difference between the proportions of Californians and Oregonians who are sleep deprived and interpret it in context of the data. \(^{44}\)

Public option, Part 1. A Washington Post article from 2009 reported that "support for a government-run health-care plan to compete with private insurers has rebounded from its summertime lows and wins clear majority support from the public." More specifically, the article says "seven in 10 Democrats back the plan, while almost nine in 10 Republicans oppose it. Independents divide 52 percent against, 42 percent in favor of the legislation." \((6 \%\) responded with "other".) There were were 819 Democrats, 566 Republicans and 783 Independents surveyed. (a) A political pundit on TV claims that a majority of Independents oppose the health care public option plan. Do these data provide strong evidence to support this statement? (b) Would you expect a confidence interval for the proportion of Independents who oppose the public option plan to include \(0.5 ?\) Explain.

Full body scan, Part I. A news article reports that "Americans have differing views on two potentially inconvenient and invasive practices that airports could implement to uncover potential terrorist attacks." This news piece was based on a survey conducted among a random sample of 1,137 adults nationwide, interviewed by telephone November \(7-10,2010,\) where one of the questions on the survey was "Some airports are now using 'full-body' digital x-ray machines to clectronically screen passengers in airport security lines. Do you think these new x-ray machines should or should not be used at airports?" Below is a summary of responses based on party affiliation. \begin{tabular}{llccc} & & \multicolumn{3}{c} { Party Affilation } \\ \cline { 3 - 5 } & & Republican & Democrat & Independent \\ \cline { 2 - 5 } Answer & Should & 264 & 299 & 351 \\ & Should not & 38 & 55 & 77 \\ & Don't know/No answer & 16 & 15 & 22 \\ \cline { 2 - 5 } & Total & 318 & 369 & 450 \end{tabular} (a) Conduct an appropriate hypothesis test evaluating whether there is a difference in the proportion of Republicans and Democrats who think the full- body scans should be applied in airports. Assume that all relevant conditions are met. (b) The conclusion of the test in part (a) may be incorrect, meaning a testing error was made. If an error was made, was it a Type I or a Type II error? Explain.

True or false, Part. II. Determine if the statements below are true or false. For each false statement, suggest an alternative wording to make it a true statement. (a) As the degrees of freedom increases, the mean of the chi-square distribution increases. (b) If you found \(X^{2}=10\) with \(d f=5\) you would fail to reject \(H_{0}\) at the \(5 \%\) significance level. (c) When finding the p-value of a chi-square test, we always shade the tail areas in both tails. (d) As the degrees of freedom increases, the variability of the chi-square distribution decreases.

Shipping holiday gifts. A December 2010 survey asked 500 randomly sampled Los Angeles residents which shipping carrier they prefer to use for shipping holiday gifts. The table below shows the distribution of responses by age group as well as the expected counts for each cell (shown in parentheses). Method \begin{tabular}{l|cc|cc|cc|c} & \multicolumn{6}{|c|} { Age } & \\ \cline { 2 - 5 } & \multicolumn{2}{|c|} {\(18-34\)} & \multicolumn{2}{|c|} {\(35-54\)} & \multicolumn{2}{|c|} {\(55+\)} & Total \\ \hline USPS & 72 & \((\mathrm{~B} 1)\) & 97 & (102) & 76 & (62) & 245 \\ UPS & 52 & (83) & 76 & \((6 \mathrm{~b})\) & 34 & (41) & 162 \\ FedEx & 31 & (21) & 24 & (27) & 9 & (16) & 64 \\ Something else & 7 & (5) & 6 & (7) & 3 & (4) & 16 \\ Not sure & 3 & (5) & 6 & (5) & 4 & (3) & 13 \\ \hline Total & \multicolumn{2}{|c|} {165} & \multicolumn{2}{|c|} {209} & \multicolumn{2}{|c|} {126} & 500 \end{tabular} (a) State the null and alternative hypotheses for testing for independence of age and preferred shipping method for holiday gifts among Los Angeles residents. (b) Are the conditions for inference using a chi-square test satisfied?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.