Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 14: Problem 5

Use the information and Table 5 in Appendix I to find the value of \(\chi^{2}\) with area \(\alpha\) to its right. $$ \alpha=.05, d f=5 $$

Short Answer

Expert verified
Answer: The chi-squared statistic with a right-tail area of 0.05 and 5 degrees of freedom is 11.07.

Step by step solution

01

Identify the given values

We're given the area to the right of the chi-squared value, \(\alpha = 0.05\), and the degrees of freedom, \(df = 5\).
02

Locate the values in Table 5

Using Appendix I's Table 5, locate the row corresponding to our degrees of freedom, \(df = 5\). Then, locate the column corresponding to our area, \(\alpha = 0.05\).
03

Find the chi-squared value

The value at the intersection of the row and column we identified in Step 2 is the chi-squared value we're looking for with our given area and degrees of freedom. In this case, the chi-squared value (\(\chi^2\)) is 11.07.
04

Write down the answer

The value of \(\chi^2\) with a right-tail area of \(\alpha=0.05\) and \(df=5\) is \(\chi^2 = 11.07\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Degrees of Freedom
Understanding the concept of degrees of freedom (df) is essential in the context of statistical analysis. In a chi-squared distribution, the degrees of freedom are the number of values in the final calculation of a statistic that are free to vary. Imagine you have a set of numbers. If you were to calculate their mean, you'd use all of the numbers in your calculation. However, if you then wanted to adjust the numbers to have a particular mean, one of your values would be dependent on the others. This dependency reduces the degrees of freedom by one.

For example, if we have 5 independent observations in a chi-squared test, our degrees of freedom would be 5. These degrees of freedom play a pivotal role in the shape of the chi-squared distribution curve and impact the critical values needed to determine statistical significance. The higher the degrees of freedom, the closer the distribution will resemble a normal distribution.
Right-Tail Area
When dealing with the chi-squared distribution, the 'right-tail area' refers to the probability of observing a value at least as extreme as the test statistic. It is the area under the curve of the distribution to the right of a specified chi-squared value. In our textbook exercise, the right-tail area, denoted by \(\alpha\), is the level of significance, which in this case is 0.05. This signifies that there is a 5% probability of observing a chi-squared value as extreme or more extreme by chance alone if the null hypothesis is true.

This concept is crucial in hypothesis testing. By comparing the computed or observed chi-squared value to the critical value from chi-squared distribution tables, we can decide whether to reject the null hypothesis or not. The smaller the right-tail area, the more extreme the test statistic needs to be for us to reject the null hypothesis.
Statistical Tables
Statistical tables, such as the chi-squared table, are invaluable tools in probability and statistics. They provide critical values for various distributions, which are necessary for carrying out hypothesis tests. A chi-squared table typically has rows labeled with degrees of freedom and columns labeled with the right-tail area, or significance levels. The intersection of a row and column provides the chi-squared value that serves as a threshold for decision-making in hypothesis testing.

To navigate these tables, you first find the row corresponding to your degrees of freedom. Then, move across to the column that represents your right-tail area or significance level. The value where the row and column intersect is the critical value. For example, in our exercise, the critical chi-squared value for 5 degrees of freedom and a significance level of 0.05 is found to be 11.07. If the calculated chi-squared statistic exceeds this value, it suggests that the null hypothesis can be rejected at this level of significance.
Probability and Statistics
The field of probability and statistics underpins much of the decision-making in science, finance, and quality control processes. Probability is the study of chance and is used to predict the likelihood of future events occurring, while statistics helps us to analyze historical data and draw conclusions.

In the realm of hypothesis testing, we use probability to determine how unlikely our sample results are, assuming that the null hypothesis is true. If the probability is low enough (less than our chosen significance level \(\alpha\)), we conclude that the observed data is statistically significant. For instance, an \(\alpha\) of 0.05 means there is a 95% confidence level that the results are not due to random chance. By merging probability with statistical methods, such as the chi-squared test, we are equipped to test hypotheses and make informed decisions based on statistical evidence.

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

Not all ethnic groups have the same mix of blood types and \(\mathrm{Rh}\) factors. For example, Latino-Americans have a high number of Os while Asians have a high number of Bs. \({ }^{12}\) A tabulation of blood types including Rh factors for 300 people in each of these ethnic groups is given below. $$ \begin{array}{lllllllll} \hline \text { Type } & \text { O+ } & \text { O? } & \text { A+ } & \text { A- } & \text { B+ } & \text { B- } & \text { AB+ } & \text { AB- } \\ \hline \text { Latino- } & & & & & & & & \\ \text { American } & 161 & 10 & 88 & 6 & 21 & 5 & 6 & 3 \\ \text { Asian } & 115 & 4 & 79 & 4 & 72 & 3 & 19 & 4 \end{array} $$ Do these data provide evidence to conclude that the proportions of people in the various blood groups differ for these two ethnic groups? Use \(\alpha=.01\)

Suppose you wish to test the null hypothesis that three binomial parameters \(p_{A}, p_{B},\) and \(p_{c}\) are equal versus the alternative hypothesis that at least two of the parameters differ. Independent random samples of 100 observations were selected from each of the populations. Use the information in the table to answer the questions in Exercises \(5-7 .\) $$ \begin{array}{lrrrr} \hline & {\text { Population }} & \\ & \text { A } & \text { B } & \text { C } & \text { Total } \\ \hline \text { Successes } & 24 & 19 & 33 & 76 \\ \text { Failures } & 76 & 81 & 67 & 224 \\ \hline \text { Total } & 100 & 100 & 100 & 300 \end{array} $$ Calculate the test statistic and find the approximate \(p\) -value for the test in Exercise 5.

According to Americans, access to healthcare and the cost of healthcare remain the most urgent health problems. However, a recent Gallup poll \(^{11}\) shows that concern about substance abuse jumped from \(3 \%\) to \(14 \%\) in 2017 . Based on samples of size 200 for each year, the data that follow reflect the results of that poll. $$ \begin{array}{lrr} \hline \text { Concern } & 2016 & 2017 \\ \hline \text { Access } & 40 & 48 \\ \text { Cost } & 54 & 32 \\ \text { Substance abuse } & 6 & 28 \\ \text { Cancer } & 24 & 22 \\ \text { Obesity } & 16 & 14 \\ \text { Other } & 60 & 56 \\ \hline \text { Total } & 200 & 200 \\ \hline \end{array} $$ a. Calculate the proportions in each of the categories for 2016 and 2017 . Test for a significant change in proportions for the healthcare concerns listed from 2016 to 2017 using \(\alpha=.05\) b. How would you summarize the results of the analysis in part a? Can you conclude that the change in the proportion of adults whose concern was substance abuse is significant? Why or why not?

The "sandwich generation" refers to middle-aged Americans who are either providing support for an aging parent while raising child under 18 or supporting a child over 18 . The information that follows summarizes the results of a poll \(^{8}\) regarding support for a parent 65 years or older by demographic group. $$ \begin{array}{lrrr} \hline \text { Provide Financial } & & & \\ \text { Support } & \text { Yes } & \text { No } & \text { Total } \\ \hline \text { Hispanics } & 134 & 66 & 200 \\ \text { Blacks } & 86 & 114 & 200 \\ \text { Whites } & 49 & 151 & 200 \\ \hline \end{array} $$ a. Find the proportion of individuals providing financial support for their parents for each demographic group. b. Are there significant differences among the proportions providing financial support for their parents for these demographic groups of Americans? Use \(\alpha=.01\). Are your results consistent with the observed proportions in part a?

Find the appropriate degrees of freedom for the chisquare test of independence. $$\text { three rows and five columns }$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.

Sign-up for free