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Chapter 11: Problem 7

Determine the value of \(\chi^{2}\) for 13 degrees of freedom and a. 025 area in the left tail of the chi-square distribution curve b. \(.995\) area in the right tail of the chi-square distribution curve

Short Answer

Expert verified
The \(\chi^{2}\) value for 13 degrees of freedom and .025 area in the left tail is approximately 5.892, and the \(\chi^{2}\) value for .995 area in the right tail is approximately 32.852.

Step by step solution

01

Understanding the problem

We are given the degrees of freedom as 13 and the areas in the left and right tail of the chi-square distribution curve. The task is to find the \(\chi^{2}\) values corresponding to these areas.
02

Determine the \(\chi^{2}\) for .025 in the left tail

In a chi-square table, the probability or area provided is always the area to the right of the \(\chi^{2}\) value. Therefore, to find the \(\chi^{2}\) value for .025 area in the left tail, one must subtract .025 from 1 to find the equivalent area in the right tail. This results in .975. Now, using a \(\chi^{2}\) distribution table for 13 degrees of freedom, the \(\chi^{2}\) value that corresponds to the .975 area to the right is found.
03

Determine the \(\chi^{2}\) for .995 in the right tail

For the part b, we are already given that .995 is the area in the right tail. Thus, we just need to reference the \(\chi^{2}\) distribution table for 13 degrees of freedom to find the \(\chi^{2}\) corresponding to the .995 area to the right.

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Most popular questions from this chapter

A sample of 22 observations selected from a normally distributed population produced a sample variance of 18 . a. Write the null and alternative hypotheses to test whether the population variance is different from 14 . b. Using \(\alpha=.05\), find the critical values of \(\chi^{2}\). Show the rejection and nonrejection regions on a chi-square distribution curve. c. Find the value of the test statistic \(\chi^{2}\) d. Using the \(5 \%\) significance level, will you reject the null hypothesis stated in part a?

National Electronics Company buys parts from two subsidiaries. The quality control department at this company wanted to check if the distribution of good and defective parts is the same for the supplies of parts received from both subsidiaries. The quality control inspector selected a sample of 300 parts received from Subsidiary A and a sample of 400 parts received from Subsidiary \(B\). These parts were checked for being good or defective. The following table records the results of this investigation. \begin{tabular}{lcc} \hline & Subsidiary A & Subsidiary B \\ \hline Good & 284 & 381 \\ Defective & 16 & 19 \\ \hline \end{tabular} Using the \(5 \%\) significance level, test the null hypothesis that the distributions of good and defective parts are the same for both subsidiaries.

Of all students enrolled at a large undergraduate university, \(19 \%\) are seniors, \(23 \%\) are juniors, \(27 \%\) are sophomores, and \(31 \%\) are freshmen. A sample of 200 students taken from this university by the student senate to conduct a survey includes 50 seniors, 46 juniors, 55 sophomores, and 49 freshmen. Using the \(10 \%\) significance level, test the null hypothesis that this sample is a random sample. (Hint: This sample will be a random sample if it includes approximately \(19 \%\) seniors, \(23 \%\) juniors, \(27 \%\) sophomores, and \(31 \%\) freshmen.)

Find the value of \(\chi^{2}\) for 12 degrees of freedom and an area of \(.025\) in the right tail of the chi-square distribution curve.

Two drugs were administered to two groups of randomly assigned 60 and 40 patients, respectively, to cure the same disease. The following table gives information about the number of patients who were cured and not cured by each of the two drugs. \begin{tabular}{lcc} \hline & Cured & Not Cured \\ \hline Drug I & 44 & 16 \\ Drug II & 18 & 22 \\ \hline \end{tabular} Test at the \(1 \%\) significance level whether or not the two drugs are similar in curing and not curing the patients.
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