Question

1. Major television networks have never seemed to have issues showing commercials for beer and other alcoholic beverages. Even though adult viewers tend to enjoy the commercials, most adults seem to think that the commercials target teenagers and young adults (those under 21 years old). To study this belief, the networks conducted a joint poll of viewers and asked them if they felt that beer and other alcoholic beverage commercials targeted teenagers and young adults. The results of the survey are as follows

Network Advertising Survey Age Group Number Surveyed Number of "Yes" Responses 30 or Younger 1000 450 Older than 30 1000 655

a. Are the sample sizes large enough such that inferences about the differences between two population proportion can be made? If so, calculate a 99% confidence interval for the difference in the proportions of those older than 30 and those 30 or younger that believe alcoholic beverage commercials targeted teenagers and young adults. Interpret the interval.

b. Based on the data, can the networks conclude that the percentage of viewers who believe beer and alcoholic beverage commercials target teenagers and young adults is significantly higher in the over 30 age group than in the 30 or younger age group? Construct the 10 steps of hypothesis testing using α = 0.01 to answer the question.

Step 1 (Define the hypotheses to be tested in plain English)

Step 2 (Select the appropriate statistical measure, such as the population mean, proportion, or
              variance.)

Step 3 (Determine whether the alternative hypothesis should be one-sided or two-sided.)

Step 4 (State the hypotheses using the statistical measure found in Step 2)

Step 5 (Specify α, the level of the test.)

Step 6 (Select the appropriate test statistic based on the information at hand and the
              assumptions you willing to make.)

Step 7 (Determine the critical value of the test statistic.)

Step 8 (Collect sample data and compute the value of the test statistic.)

Step 9 (Make the decision.)

Step 10 (State the conclusion in terms of the original question.)

Answer 1

Solution:

Part a

Here, we have to find the 99% confidence interval for the difference between population proportions.

Confidence interval = (P1 – P2) ± Z*sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]

Where, P1 and P2 are sample proportions for first and second groups respectively.

We are given

X1 = 655, N1 = 1000, P1 = X1/N1 = 655/1000 = 0.655

X2 = 450, N2 = 1000, P2 = X2/N2 = 450/1000 = 0.450

Confidence level = 99%

Critical Z value = 2.3263 (by using z-table)

Confidence interval = (P1 – P2) ± Z*sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]

Confidence interval = (0.655 – 0.450) ± 2.5758*sqrt[(0.655*(1 – 0.655)/1000) + (0.450*(1 – 0.450)/1000)]

Confidence interval = 0.205 ± 2.5758*sqrt[(0.655*(1 – 0.655)/1000) + (0.450*(1 – 0.450)/1000)]

Confidence interval = 0.205 ± 2.5758* 0.0218

Confidence interval = 0.205 ± 0.0560

Lower limit = 0.205 - 0.0560 = 0.149

Upper limit = 0.205 + 0.0560 = 0.261

Confidence interval = (0.149, 0.261)

Part b

Here, we have to use two sample z test for population proportions. The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: The percentage of viewers who believe beer and alcoholic beverage commercials target teenagers and young adults is same in the over 30 age group and in the 30 or younger age group.

Alternative hypothesis: H_a: the percentage of viewers who believe beer and alcoholic beverage commercials target teenagers and young adults is significantly higher in the over 30 age group than in the 30 or younger age group.

H₀: p₁ = p₂ versus H_a: p₁ > p₂

This is a one tailed test. This is an upper tailed or right tailed test.

p₁ is the population proportion of the group ‘older than 30’ and p₂ is the population proportion of the group ’30 or younger’.

We are given

Level of significance = α = 0.01

For group 1, we are given

X1 = 655, N1 = 1000, P1 = X1/N1 = 655/1000 = 0.655

For group 2, we are given

X2 = 450, N2 = 1000, P2 = X2/N2 = 450/1000 = 0.450

Z = (P1 – P2) / sqrt[(P1*(1 – P1)/N1) + (P2*(1 – P2)/N2)]

Z = (0.655 – 0.450) / sqrt[(0.655*(1 – 0.655)/1000) + (0.450*(1 – 0.450)/1000)]

Z = (0.655 – 0.450) / 0.0218

Z = 0.205/0.0218

Z = 9.40367

P-value = 0.00 (by using z-table)

Upper critical value = 2.3263 (by using z-table)

P-value < α = 0.01

So, we reject the null hypothesis H₀

We reject the null hypothesis that the percentage of viewers who believe beer and alcoholic beverage commercials target teenagers and young adults is same in the over 30 age group and in the 30 or younger age group.

There is sufficient evidence to conclude that the percentage of viewers who believe beer and alcoholic beverage commercials target teenagers and young adults is significantly higher in the over 30 age group than in the 30 or younger age group.