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A 20-minute consumer survey that included a $5 Starbucks gift card was emailed to 500 adults aged 25-34, resulting in 65 responses. The same survey, without the gift card, was also emailed to 500 adults, resulting in 45 responses. We would like to know if the inclusion of a Starbucks gift card led to any change in the survey response rate. Which of the following hypotheses should be used?

A. Null Hypothesis: \(\mu_1\) (average number of responses with Starbucks gift card) \(\leq\) \(\mu_0\) (average number of responses without Starbucks gift card)
Alternative Hypothesis: \(\mu_1\) (average number of responses with Starbucks gift card) \(>\) \(\mu_0\) (average number of responses without Starbucks gift card)

B. Null Hypothesis: \(\mu_1\) (average number of responses with Starbucks gift card) = \(\mu_0\) (average number of responses without Starbucks gift card)
Alternative Hypothesis: \(\mu_1\) (average number of responses with Starbucks gift card) \(\neq\) \(\mu_0\) (average number of responses without Starbucks gift card) (\(\neq\) means "not equal to")

C. Null Hypothesis: \(\pi_1\) (rate with Starbucks gift card) = \(\pi_2\) (rate without Starbucks gift card)
Alternative Hypothesis: \(\pi_1\) (rate with Starbucks gift card) \(\neq\) \(\pi_2\) (rate without Starbucks gift card) (\(\neq\) means "not equal to")

D. Null Hypothesis: \(\pi_1\) (rate with Starbucks gift card) - \(\pi_2\) (rate without Starbucks gift card) \(\leq 0\)
Alternative Hypothesis: \(\pi_1\) (rate with Starbucks gift card) - \(\pi_2\) (rate without Starbucks gift card) \(> 0\)

Answer :

Answer:

D. Null Hypothesis: pi1 (rate with Starbucks gift card) - pi2 (rate without Starbucks gift card)<=0 Alternative Hypothesis: pi1 (rate with Starbucks gift card) - pi2 (rate without Starbucks gift card) > 0

Step-by-step explanation:

The test described above test the difference in proportion :

With Starbucks gift card :

x1 = 65 n1 = 500

Phat 1 = x1 / n1

Phat 1 = 65 /500 = 0.13

Phat1

Without Starbucks gift card :

x2 = 45 n2 = 500

Phat1 = x2 / n2

Phat 2 = 45 / 500 = 0.09

To test the difference in proportion :

Phat 1 > Phat 2 (alternative)

Null takes the stance that there is no difference in proportion

Phat1 - phat2 <=0

Alternative claims that there is difference on proportion such that phat 1 is greater than phat 2

Phat 1 - phat > 0

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Rewritten by : Barada

Final answer:

In this scenario, we want to determine if there is a difference in survey response rates with and without the Starbucks gift card. Therefore, we need to use a hypothesis test to compare two population proportions. The appropriate hypotheses for this situation are: Null Hypothesis: The population proportions are equal. (pi1 = pi2) Alternative Hypothesis: The population proportions are not equal. (pi1 ≠ pi2)

Explanation:

In this scenario, we want to determine if there is a difference in survey response rates with and without the Starbucks gift card. Therefore, we need to use a hypothesis test to compare two population proportions. The appropriate hypotheses for this situation are:



Null Hypothesis: The population proportions are equal. (pi1 = pi2)



Alternative Hypothesis: The population proportions are not equal. (pi1 ≠ pi2)



This corresponds to option C.

Learn more about Hypothesis Testing here:

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