Question

The data in the accompanying table is from a paper. Suppose that the data resulted from classifying each person in a random sample of 50 male students and each person in a random sample of 91 female students at a particular college according to their response to a question about whether they usually eat three meals a day or rarely eat three meals a day.

	Usually Eat 3 Meals a Day	Rarely Eat 3 Meals a Day
Male	26	24
Female	39	52

(a)

Is there evidence that the proportions falling into each of the two response categories are not the same for males and females? Use the

X²

statistic to test the relevant hypotheses with a significance level of 0.05.

State the null and alternative hypotheses.

H₀: The proportions falling into the two response categories are not the same for males and females.
H_a: The proportions falling into the two response categories are the same for males and females.H₀: The proportions falling into the two response categories are 0.5 for both males and females.
H_a: The proportions falling into the two response categories are not 0.5 for both males and females.    H₀: The proportions falling into the two response categories are the same for males and females.
H_a: The proportions falling into the two response categories are not the same for males and females.H₀: The proportions falling into the two response categories are not 0.5 for both males and females.
H_a: The proportions falling into the two response categories are 0.5 for both males and females.

Calculate the test statistic. (Round your answer to two decimal places.)

χ² =

What is the P-value for the test? (Use a statistical computer package to calculate the P-value. Round your answer to four decimal places.)

P-value =

What can you conclude?

Do not reject H₀. There is not convincing evidence that the proportions falling into the two response categories are not the same for males and females.Do not reject H₀. There is convincing evidence that the proportions falling into the two response categories are not the same for males and females.     Reject H₀. There is not convincing evidence that the proportions falling into the two response categories are not the same for males and females.Reject H₀. There is convincing evidence that the proportions falling into the two response categories are not the same for males and females.

(b)

Are your calculations and conclusions from part (a) consistent with the accompanying Minitab output?

Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
	Usually	Rarely	Total
Male	26	24	50
	23.05	26.95
	0.378	0.323
Female	39	52	91
	41.95	49.05
	0.207	0.177
Total	65	76	141
Chi-Sq = 1.086, DF = 1, P-Value = 0.297

YesNo    

(c)

Because the response variable in this exercise has only two categories (usually and rarely), we could have also answered the question posed in part (a) by carrying out a two-sample z test of

H₀: p₁ − p₂ = 0

versus

H_a: p₁ − p₂ ≠ 0,

where

p₁

is the proportion who usually eat three meals a day for males and

p₂

is the proportion who usually eat three meals a day for females. Minitab output from the two-sample z test is shown below. Using a significance level of 0.05, does the two-sample z test lead to the same conclusion as in part (a)?

Test for Two Proportions

Sample	X	N	Sample p
Male	26	50	0.520000
Female	39	91	0.428571
Difference = p(1) − p(2)
Test for difference = 0 (vs not = 0): Z = 1.04
P-Value = 0.297

(d)

How do the P-values from the tests in parts (a) and (c) compare? Does this surprise you? Explain?

The two P-values are very different. It is quite surprising that the P-values are this different, since both measure the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations.The two P-values are equal when rounded to three decimal places. It is surprising that the P-values are at least similar, since the P-value from the chi-squared test is measuring the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations and the z-test is measuring the probability of getting sample proportions closer to the expected proportions than what was observed, given that the proportions who usually eat three meals per day are the same for the two populations.    The two P-values are equal when rounded to three decimal places. It is not surprising that the P-values are at least similar, since both measure the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations.The two P-values are not equal when rounded to three decimal places. It is not surprising that the P-values are different, since the P-value from the chi-squared test is measuring the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations and the z-test is measuring the probability of getting sample proportions closer to the expected proportions than what was observed, given that the proportions who usually eat three meals per day are the same for the two populations.

Answer 1

(a)

H0: The proportions falling into the two response categories are the same for males and females.
Ha: The proportions falling into the two response categories are not the same for males and females.

Following table shows the row total and column total:

	Usually Eat 3 Meals a Day	Rarely Eat 3 Meals a Day	Total
Male	26	24	50
Female	39	52	91
Total	65	76	141

Expected frequencies will be calculated as follows:

Following table shows the expected frequencies:

	Usually Eat 3 Meals a Day	Rarely Eat 3 Meals a Day	Total
Male	23.05	26.95	50
Female	41.95	49.05	91
Total	65	76	141

Following table shows the calculations for chi square test statistics:

O	E	(O-E)^2/E
26	23.05	0.377548807
39	41.95	0.207449344
24	26.95	0.322912801
52	49.05	0.177420999
Total		1.085331952

The test statistics is:

Degree of freedom: df =( number of rows -1)*(number of columns-1) = (2-1)*(2-1)=1

The p-value is:

p-value = 0.298

Since p-value is greater than 0.05 so we fail to reject the null hypothesis.

Do not reject H0. There is not convincing evidence that the proportions falling into the two response categories are not the same for males and females.

(b)

yes

(c)

Yes. The p-value of test is 0.297 which is greater than 0.05 so we fail to reject the null hypothesis.

(d)

The two P-values are equal when rounded to three decimal places. It is not surprising that the P-values are at least similar, since both measure the probability of getting sample proportions at least as far from the expected proportions as what was observed, given that the proportions who usually eat three meals per day are the same for the two populations.