A nutritionist wants to determine how much time nationally people spend eating a

Question

A nutritionist wants to determine how much time nationally people spend eating and drinking. Suppose for a random sample of 994 people age 15 or older, the mean amount of time spent eating or drinking per day is 1.77 hours with a standard deviation of 0.63 hour. Complete parts (a) through (d) below. 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. The distribution of the sample mean will never be approximately normal. 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. The distribution of the sample mean will always be approximately normal. 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. In 2010, there were over 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. The sample size is greater than 5% of the population. The sample size is less than 10% of the population. The sample size is less than 5% of the population. The sample size is greater than 10% of the population. Determine and interpret a 99% confidence interval for the mean amount of time Americans age 15 or older spend eating and drinking each day. Select the correct choice below and fill in the answer boxes, if applicable, in your choice. (Type integers or decimals rounded to three decimal places as needed. Use ascending order.) The nutritionist is 99% confident that the amount of time spent eating or drinking per day for any individual is between and hours. The nutritionist is 99% confident that the mean amount of time spent eating or drinking per day is between and hours. There is a 99% probability that the mean amount of time spent eating or drinking per day is between and hours. The requirements for constructing a confidence interval are not satisfied. Could the interval be used to estimate the mean amount of time a 9-year-old spends eating and drinking each day? Explain. No, the interval is about individual time spent eating or drinking per day and cannot be used to find the mean time spent eating or drinking per day for specific age. Yes; the interval is about the mean amount of time spent eating or drinking per day for people people age 15 or older and can be used to find the mean amount of time spent eating or drinking per day for 9-year-olds. No; the interval is about people age 15 or older. The mean amount of time spent eating or drinking per day for 9-year-olds may differ. Yes; the interval is about individual time spent eating or drinking per day and can be used to find the mean amount of time a 9-year-old spends eating and drinking each day. A confidence interval could not be constructed in part (c).

Explanation / Answer

(a) - The central limit theorem states that the sampling distribution of the mean of any independent,random variable will be normal or nearly normal, if the sample size is large enough. Here large is usually 30 or more. In practice, if population is not normally distributed then sample size needs to be even larger. So in this case d seems best choice out of 4

(b) Here sample size is 994 out of 2 million population. Hence sample size is less than 5% of population.

(c) Since we know sample mean, so we use t-test

COnfidence interval - 99%

Standard error = s/sqrt(n) = 0.63/sqrt(994) = 0.01998

alpha = 1-confidence interval/100 = 0.01, critical probability = 1-(0.01/2) = 0.995

degree of freedom = 994-1 = 993

t value for (993,0.995) = 2.581

Hence Margin of error = 2.581*0.01998 = 0.052

Hence time spent eating/drinking per day is 1.77 hours+-0.052

So answer is A with values 1.77-0.52 to 1.77+0.52 hours per day.

(d) This data provided as per this Q is only for people with age 15 or older. Since age wise data is not available, so calculating interval for 9 years age would not be possible. Hence answer should be - C

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