Determine why a large sample size is necessary when the histogram of time spent eating and drinking each day is skewed right:
A nutritionist wants to determine how much time nationally people spend eating and drinking. Suppose for a random sample of 943 people age 15 or older, the mean amount of time spent eating or drinking per day is 1.56 hours with a standard deviation of 0.71 hour. Complete parts (a) through (d) below.
(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.
A. Since the distribution of time spent eating and drinking each day is not normally distributed (skewed right), the sample must be large so that the distribution of the sample mean will be approximately normal.
B. Since the distribution of time spent eating and drinking each day is normally distributed, the sample must be large so that the distribution of the sample mean will be approximately normal.
C. The distribution of the sample mean will always be approximately normal.
D. The distribution of the sample mean will never be approximately normal.
(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 greater than $10 \%$ of the population.
B. The sample size is less than $5 \%$ of the population.
C. The sample size is less than $10 \%$ of the population.
D. The sample size is greater than $5 \%$ of the population.
Answer
\(\boxed{\text{A. Since the distribution of time spent eating and drinking each day is not normally distributed (skewed right), the sample must be large so that the distribution of the sample mean will be approximately normal.}}\)
Step 1 ：The question is asking why a large sample size is necessary when the histogram of time spent eating and drinking each day is skewed right. This is a theoretical question and does not require any calculations.
Step 2 ：The Central Limit Theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.
Step 3 ：This will hold true regardless of the shape of the population distribution. Therefore, when the population distribution is skewed right, a larger sample size is needed to ensure that the sample mean distribution is approximately normal.
Step 4 ：This is important for constructing a confidence interval for the mean time spent eating and drinking each day.
Step 5 ：Therefore, the answer is A. Since the distribution of time spent eating and drinking each day is not normally distributed (skewed right), the sample must be large so that the distribution of the sample mean will be approximately normal.
Step 6 ：\(\boxed{\text{A. Since the distribution of time spent eating and drinking each day is not normally distributed (skewed right), the sample must be large so that the distribution of the sample mean will be approximately normal.}}\)
