Question

The top food snacks consumed by adults aged 18-54 are gum, chocolate candy, fresh fruit, potato chips, breath mints/candy, ice cream, nuts, cookies, bars, yogurt, and crackers. Out of a random sample of 26 men, 18 ranked fresh fruit in their top five snack choices. Out of a random sample of 33 women, 27 ranked fresh fruit in their top five snack choices. Is there a difference in the proportion of men and women who rank fresh fruit in their top five list of snacks?

(a-1) Choose the appropriate hypotheses. Assume ππ_M is the proportion of men and ππ_w is the proportion of women.

a. H₀: πMπM – πWπW = 0vs. H₁: πMπM – πWπW ≤ 0

b. H₀: πMπM – πWπW ≠ 0 vs. H₁: πMπM – πWπW = 0

c. H₀: πMπM – πWπW = 0vs. H₁: πMπM – πWπW ≥ 0

d. H₀: πMπM – πWπW = 0 vs. H₁: πMπM – πWπW ≠ 0

	a
	b
	c
	d

(a-2) State the decision rule for α = .02. (Round your answers to 3 decimal places. A negative value should be indicated by a minus sign.)

Reject the null hypotheses if  z_calc < or z_calc > .

(b) Calculate the sample proportions. (Round your answers to 4 decimal places.)

	Sample proportion
Men
Women

(c-1) Calculate the test statistic and find the p-value. (Do not round intermediate calculations. Round your answers to 3 decimal places. A negative value should be indicated by a minus sign.)

The value of the test statistic is  and the p-value is .

(c-2) What is your conclusion?

	The sample shows a significant difference in proportions.
	The sample does not show a significant difference in proportions.

(d) Is normality of p₁ – p₂ assured?

	No
	Yes

Answer 1

(a-1)

d. H0: πMπM – πWπW = 0 vs. H1: πMπM – πWπW ≠ 0

(a-2)

Rejection region:

If z < -2.33 or z > 2.33, reject H0

(b)

Sample proportions:

Men = 0.6923

Women = 0.8182

(c)

The test statistics is

z = -1.129

The p-value: 0.259

(c-2)

Since p-value is greater than 0.02 so we fail to reject the null hypothesis.

The sample does not show a significant difference in proportions.

(d)

Yes because number of successes and failures are both greater than 5 in both samples.