a.
Test whether the proportions falling into each of the two response categories for males and females are same or not.
a.
Answer to Problem 21E
There is no convincing evidence that the proportions falling into each of the two response categories for males and females are not the same.
Explanation of Solution
Calculation:
The given table represents the classification of male and female students according to their response about whether they usually eat three meals a day or rarely eat three meals a day.
In order to test whether the proportions falling into each of the two response categories for males and females are same or not, the appropriate test is
The steps in chi-square test for homogeneity are as follows:
The null and alternative hypotheses are as follows:
Null hypothesis:
Alternative hypothesis:
Level of significance:
Assumptions:
- It is given that data are resulted from classifying each person in a random sample of male and female students at a particular college.
- The expected cell counts are calculated as shown below:
Usually eat 3 meals a day | Rarely eat 3 meals a day | Total | |
Male | 48 | ||
Female | 91 | ||
Total | 63 | 76 | 139 |
From the expected cell counts table, it is observed that all the expected counts are greater than 5.
Test statistic:
Chi-square statistic:
Degrees of freedom:
Here, the number of rows is 2 and the number of columns is 2. The degrees of freedom is calculated as follows:
From the calculations,
P-value:
From Appendix A: Table 8-Upper-Tail Areas for Chi-Square Distributions, the right-tailed area is greater than 0.100 for the chi-squared values that are less than 2.70.
Thus, P-value > 0.100.
Decision rule:
- If P-value is less than or equal to the level of significance, reject the null hypothesis.
- Otherwise, fail to reject the null hypothesis.
Conclusion:
Here, the level of significance is 0.05.
Here, P-value is greater than the level of significance.
That is,
Hence, fail to reject the null hypothesis.
Therefore, there is no convincing evidence that the proportions falling into each of the two response categories for males and females are not the same.
b.
Check whether the calculations and conclusions from Part (a) are consistent with the accompanying Minitab output.
b.
Explanation of Solution
From Part (a), the chi-square value obtained is
In the given MINITAB output, the chi-square value is 2.314 and the P-value is 0.128.
Since
Hence, the calculations and conclusions from Part (a) are consistent with the accompanying MINITAB output.
c.
Check whether the two-sample z test would lead to the same conclusion as in Part (a).
c.
Answer to Problem 21E
The two-sample z test leads to the same conclusion as in Part (a).
Explanation of Solution
From the given output, the test for two proportions is
Conclusion:
Here, the level of significance is 0.05.
Here, P-value is greater than the level of significance.
That is,
Hence, fail to reject the null hypothesis.
Therefore, there is no convincing evidence that the proportions falling in the two response categories are not the same for males and females.
Thus, the two-sample z test leads to the same conclusion as in Part (a).
d.
Compare the P-values from the tests in Parts (a) and (c).
d.
Explanation of Solution
The P-value obtained in Part (a) is greater than 0.100, which is not an exact P-value, but a range for the P-value.
The P-value obtained in Part (c) is 0.127.
Both the P-values are greater than 0.100.
This is not surprising as both the chi-square statistic and the z-statistic measure the probability of getting sample proportions that are far from the expected proportions under null hypothesis.
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Chapter 12 Solutions
Introduction to Statistics and Data Analysis
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