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Chapter 12: Problem 1

From the given information in each case below, state what you know about the \(P\) -value for a chi-square test and give the conclusion for a significance level of \(\alpha=.01\). a. \(X^{2}=7.5, \mathrm{df}=2\) d. \(X^{2}=21.3, \mathrm{df}=4\) b. \(X^{2}=13.0, \mathrm{df}=6\) e. \(X^{2}=5.0, \mathrm{df}=3\) c. \(X^{2}=18.0, \mathrm{df}=9\)

Short Answer

Expert verified
The null hypothesis is not rejected for a, b, e, and c where P-value is greater than 0.01. However, for d, the null hypothesis is rejected as the P-value is less than 0.01.

Step by step solution

01

Calculation for item a

Considering a chi-square statistic \(X^{2}=7.5\) with degrees of freedom (df) equal to 2. To get the P-value, check the value of the chi-square statistic in a chi-square distribution table. The P-value is greater than 0.01. Therefore, do not reject the null hypothesis.
02

Calculation for item d

Next, for \(X^{2}=21.3\), df = 4, again, look in a chi-square distribution table. The P-value is less than 0.01. Therefore, reject the null hypothesis.
03

Calculation for item b

For \(X^{2}=13.0\), df = 6, checking in a chi-square distribution table, the P-value is greater than 0.01. Hence, do not reject the null hypothesis.
04

Calculation for item e

Upon checking in a chi-square distribution table for \(X^{2}=5.0\), df = 3, the P-value is greater than 0.01. Therefore, do not reject the null hypothesis.
05

Calculation for item c

Finally, for \(X^{2}=18.0\), df = 9, when checked in a chi-square distribution table, the P-value turns out to be greater than 0.01. Hence, do not reject the null hypothesis.

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Most popular questions from this chapter

A particular state university system has six campuses. On each campus, a random sample of students will be selected, and each student will be categorized with respect to political philosophy as liberal, moderate, or conservative. The null hypothesis of interest is that the proportion of students falling in these three categories is the same at all six campuses. a. On how many degrees of freedom will the resulting \(X^{2}\) test be based? b. How does your answer in Part (a) change if there are seven campuses rather than six? c. How does your answer in Part (a) change if there are four rather than three categories for political philosophy?

A particular paperback book is published in a choice of four different covers. A certain bookstore keeps copies of each cover on its racks. To test the hypothesis that sales are equally divided among the four choices, a random sample of 100 purchases is identified. a. If the resulting \(X^{2}\) value were \(6.4\), what conclusion would you reach when using a test with significance level .05? b. What conclusion would be appropriate at significance level \(.01\) if \(X^{2}=15.3 ?\) c. If there were six different covers rather than just four, what would you conclude if \(X^{2}=13.7\) and a test with \(\alpha=.05\) was used?

A random sample of 1000 registered voters in a certain county is selected, and each voter is categorized with respect to both educational level (four categories) and preferred candidate in an upcoming election for county supervisor (five possibilities). The hypothesis of interest is that educational level and preferred candidate are independent factors. a. If \(X^{2}=7.2\), what would you conclude at significance level .10? b. If there were only four candidates vying for election, what would you conclude if \(X^{2}=14.5\) and \(\alpha=.05\) ?

Packages of mixed nuts made by a certain company contain four types of nuts. The percentages of nuts of Types \(1,2,3\), and 4 are supposed to be \(40 \%, 30 \%, 20 \%\), and \(10 \%\), respectively. A random sample of nuts is selected, and each one is categorized by type. a. If the sample size is 200 and the resulting test statistic value is \(X^{2}=19.0\), what conclusion would be appropriate for a significance level of .001? b. If the random sample had consisted of only 40 nuts. would you use the chi- square test here? Explain your reasoning.

The article "Victim's Race Affects Killer's Sentence, Study Finds" (New York Times, April 20, 2001) included the following information: The table was based on a study of \(a l l\) homicide cases in North Carolina for the period 1993 to 1997 where it was possible that the person convicted of murder could receive the death penalty. Explain why it would not be necessary (or appropriate) to use a chi-square test to determine if there was an association between the defendant-victim race combination and whether or not the convicted murderer receive a death sentence for convicted murders in North Carolina during 1993 to \(1997 .\)
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