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 45 male students and each person in a random sample of 92 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	25	20
Female	38	54

(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 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 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 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?

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.    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.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)

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	25	20	45
	20.69	24.31
	0.896	0.763
Female	38	54	92
	42.31	49.69
	0.438	0.373
Total	63	74	137
Chi-Sq = 2.471, DF = 1, P-Value = 0.116

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	25	45	0.555556
Female	38	92	0.413043
Difference = p(1) − p(2)
Test for difference = 0 (vs not = 0): Z = 1.57
P-Value = 0.116

(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 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 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 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.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.

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	25	20	45
Female	38	54	92
Total	63	74	137

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	20.69	24.31	45
Female	42.31	49.69	92
Total	63	74	137

Following table shows the calculations for chi square test statistics:

O	E	(O-E)^2/E
25	20.69	0.898
38	42.31	0.439
20	24.31	0.764
54	49.69	0.374
Total		2.475

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 = 0.1157

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 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.