Question

researchers at harris interactive wondered if there was a difference between males and females in regard to whether they typically buy name- brand or store- brand products. they asked a random sample of males and females the following question. "for each of the following types of products, please indicate whether you typically buy name- brand products or store- brand products?" among the 1104 males surveyed, 343 indicated they buy name- brand over-the-counter drugs; among the 1172 females surveyed, 295 indicated they buy name- brand over-the-counter drugs. does the evidence suggests a lower proportion of females by name- brand over-the-counter drugs?

(a) explain why this study can be analyzed using the methods for conducting a hypothesis test regarding two independent portions.

(b) what are the null and alternative hypotheses?

(c) describe the sampling distribution of Pfemale- Pale draw a normal model with the area representing the P-value shaded for this hypothesis test.

(d) determine the P-value based on the model from part (c).

(e) interpret the P-value.

(f) based on the p-value, what does the sample evidence suggest? that is, what is the conclusion of the hypothesis test? Assume an alpha=0.05 level of significance.

Answer 1

Solution:

(a) explain why this study can be analyzed using the methods for conducting a hypothesis test regarding two independent portions.

Here, we have to use z test for difference between two population proportions, because we are given two samples of males and females; and their corresponding proportions regarding they buy name- brand over-the-counter drugs. Both proportions are independent.

(b) what are the null and alternative hypotheses?

Null hypothesis: H₀: There is no statistically significant difference between the two population proportions of the male and female by name- brand over-the-counter drugs.

Alternative hypothesis: H_a: The population proportion of the female by name- brand over-the-counter drugs is lower than the population proportion of the male by name- brand over-the-counter drugs.

(c) describe the sampling distribution of Pfemale- Pale draw a normal model with the area representing the P-value shaded for this hypothesis test.

The sampling distribution of the difference between the proportions of females and male will follows an approximately normal distribution with mean (P_female – P_male). The standard deviation for this sampling distribution is given as below:

Let P₁ = P_female and P₂ = P_male

Mean = (P₁ – P₂)

Standard deviation = sqrt[(P₁*(1 – P₁)/n₁) + (P₂*(1 – P₂)/n₂)]

Where,

X1 = 295

X2 = 343

N1 = 1172

N2 = 1104

First sample proportion = P₁ = X1/N1 = 295/1172 = 0.251706485

Second sample proportion = P₂ = X2/N2 = 343/1104 = 0.310688406

Mean = (P₁ – P₂)

Mean = (0.251706485 - 0.310688406)

Mean = -0.05898

Standard deviation = sqrt[(P₁*(1 – P₁)/n₁) + (P₂*(1 – P₂)/n₂)]

Standard deviation = sqrt((0.251706485*(1 - 0.251706485)/1172) + (0.310688406*(1 - 0.310688406)/1104))

Standard deviation = 0.018833

(d) determine the P-value based on the model from part (c).

Test statistic formula is given as below:

Z = (P₁ – P₂) / sqrt[(P₁*(1 – P₁)/n₁) + (P₂*(1 – P₂)/n₂)]

Z = -0.05898/0.018833

Z = -3.13174

P-value = 0.0009

(by using z-table)

(e) interpret the P-value.

There is a 0.0009 estimated probability or significance level of the rejection of the null hypothesis.

(f) based on the p-value, what does the sample evidence suggest? that is, what is the conclusion of the hypothesis test? Assume an alpha=0.05 level of significance.

We have

P-value = 0.0009

α = 0.05

P-value < α

So, we reject the null hypothesis

There is a sufficient evidence to conclude that the population proportion of the female by name- brand over-the-counter drugs is lower than the population proportion of the male by name- brand over-the-counter drugs.