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A restaurant manager believes that a majority of customers who dine at his restaurant would rate the food as excellent. To investigate this belief, he sends an email survey to a random sample of 50 customers from the large list of customers who have signed up for his frequent diner club. To encourage the customers to complete the survey, he offers a $10 gift card upon completion of the survey. He receives 22 responses, of which 20 rated the food as excellent. He would like to know if these data provide convincing evidence that more than 50% of customers who dine at his restaurant would rate the food as excellent.

Are the conditions for inference met?

A. Yes, the conditions for inference are met.
B. No, the 10% condition is not met.
C. No, the Large Counts Condition is not met.
D. No, the randomness condition is not met.

Answer :

Answer:

D. No, the randomness condition is not met.

Step-by-step explanation:

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Rewritten by : Barada

The restaurant manager's case reflects a statistical hypothesis test regarding customer satisfaction. While the manager's survey meets the randomness condition and potentially the 10% condition, it fails the Large Counts Condition since the number of failures (non-excellent ratings) in the sample is less than 10.

The restaurant manager's inquiry about the food rating from his customers is an example of hypothesis testing in statistics. To determine whether the data provides convincing evidence that a majority would rate the food as excellent, the conditions for inference must be met. These include the randomness condition, the 10% condition, and the Large Counts Condition.

The randomness condition is typically met if the sample is random and representative of the population. In this case, the manager used a random sample from the frequent diner club, which could be considered a random selection if every member had an equal chance of being chosen. Therefore, we might assume that the randomness condition is met.

The Large Counts Condition, part of the requirements for the Central Limit Theorem to apply, stipulates that we must expect at least 10 successes (excellent ratings) and 10 failures (not excellent ratings) in the sample. Since 20 out of 22 responses rated the food as excellent, this condition is not met, because there are only 2 failures.

Therefore, the correct answer is: No, the Large Counts Condition is not met.