A nutritionist wants to determine how much time nationally people spend eating and drinking Suppose for a random sample of 1087 people age 15 or older, the mean amount of time spent eating or drinking per day is 1.33 hours with a standard deviation of 0.58 hour. Complete parts (a) through (d) below. Click the icon to view the table of critical t-values. (a) A histogram of time spent eating and drinking each day is skewed right. Use this result to explain why a large sample size is needed to construct a confidence interval for the mean time spent eating and drinking each day Since the distribution of time spent eating and drinking each day is highly skewed right, a large sample size is needed to minimize the margin of error to ensure only the peak of the sampling distribution is captured in the confidence interval. Since the distribution of time spent eating and drinking each day is highly skewed right, a large sample size is needed to ensure that it contains at least $5 \%$ of the population Since the distribution of time spent eating and drinking each day is highly skewed right, a large sample size is necessary to ensure that the distribution of the sample mean is approximately normal Since the distribution of time spent eating and drinking each day is highly skewed right, a large sample size is needed to maximize the margin of error to ensure that both tails are accounted for in the confidence interval (b) There are more than 200 million people nationally age 15 or older Explain why this, along with the fact that the data were obtained using a random sample, satisfies the requirements for constructing a confidence interval A. The sample size is less than $10 \%$ of the population B. The sample size is greater than $10 \%$ of the population. C. The sample size is less than $5 \%$ of the population. D. The sample size is greater than $5 \%$ of the population
Solution
Step 1 ：A large sample size is necessary to ensure that the distribution of the sample mean is approximately normal.
Step 2 ：The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution.
Step 3 ：Therefore, a large sample size is needed to construct a confidence interval that is based on the assumption of a normal distribution.
Step 4 ：The fact that there are more than 200 million people nationally age 15 or older satisfies the requirement for constructing a confidence interval because the sample size of 1087 is greater than 5% of the population.
Step 5 ：This ensures that the sample is representative of the population and allows for generalization of the results to the entire population.
Step 6 ：Therefore, the answer is D. The sample size is greater than 5% of the population.
