Question

Test on the difference of two population proportions on page 406 (Optional). A 20-minutee consumer survey mailed to 500 adults aged 25-34 included a $5 Starbucks gift certificate. The same survey was mailed to 500 adults aged 25-34 without the gift certificate. There were 65 responses from the first group and 45 from the second group. Perform a two-tailed test comparing the response rates (proportions) at α=0.05.

Answer 1

There is a significant difference in response rates between the two groups.

Step-by-step explanation:

To compare response rates, conduct a two-proportion z-test at a 5% significance level (α=0.05). The formula for the z-test for two proportions is:

\[ z = \frac{(p_1 - p_2)}{\sqrt{p(1-p)(\frac{1}{n_1}+\frac{1}{n_2})}} \]

Where:

- \( p_1 \) and \( p_2 \) are the sample proportions (responses/total) for each group.

- \( n_1 \) and \( n_2 \) are the sample sizes.

- \( p \) is the combined sample proportion.

For Group 1 (with the gift certificate):

- Sample size (\( n_1 \)) = 500

- Responses (\( p_1 \)) = 65/500 = 0.13

For Group 2 (without the gift certificate):

- Sample size (\( n_2 \)) = 500

- Responses (\( p_2 \)) = 45/500 = 0.09

The combined sample proportion (\( p \)) is calculated as:

\[ p = \frac{65 + 45}{500 + 500} = \frac{110}{1000} = 0.11 \]

Now, calculate the standard error and the z-value using the formula. Afterward, check the z-value against the critical z-values for a two-tailed test at α=0.05 (±1.96).

Upon calculation, if the absolute value of the calculated z-value is greater than 1.96, it indicates a significant difference between the response rates of the two groups. Here, the z-value, when compared to the critical values, determines whether the difference in response rates is statistically significant or occurred by chance. Therefore, based on the comparison, it can be concluded whether the response rates between the two groups significantly differ or not at the specified significance level.