A nutritionist wants to determine how much time nationally people spend eating a
ID: 3208492 • Letter: A
Question
A nutritionist wants to determine how much time nationally people spend eating and drinking. Suppose for a random sample of 1082 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 0.61 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. The distribution of the sample mean will never 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. 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.
D. The distribution of the sample mean will always be approximately normal.
(b) 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.
A. The sample size is less than 10% of the population.
B. The sample size is less than 5% of the population.
C. The sample size is greater than 5% of the population.
D. The sample size is greater than 10% of the population.
(c) 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.)
A. The nutritionist is 99% confident that the amount of time spent eating or drinking per day for any individual is between____ and ____ hours.
B. The nutritionist is 99% confident that the mean amount of time spent eating or drinking per day is between _____ and ____ hours.
C. There is a 99% probability that the mean amount of time spent eating or drinking per day is between ____ and____ hours.
D. The requirements for constructing a confidence interval are not satisfied.
(d) Could the interval be used to estimate the mean amount of time a 9-year-old spends eating and drinking each day? Explain.
A. 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.
B. 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.
C. 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.
D. 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.
E. A confidence interval could not be constructed in part (c).
Explanation / Answer
(A) we know that according to central limit theorem, the random variables may follow any distribution ,the only requirements are - They should be independent and identical , Sample size should be large . If these are followed by a set of random variables, then according to LAW OF LARGE NUMBERS, their average will follow approximately normal distribution. Here daily data of time spent eating and drinking is right skewed( not symmetrical as in normal distribution). But this doesn't matter if want to get the result that mean amount of the sample followsnormal distribution, the only requirement is sample size is large.
Hence correct answer is C.
(B) Random sample successfully satisfies the requirements for constructing a confidence interval. First, it is sufficiently large needed so that sample mean follows normal distribution .Second, Sample size is less than 5%of the population , If it would be more than 5% ,then a correction needs to be made to the standard error of the means by using finite population correction factor. Third, sample is random and thus free from any sampling bias which may invalidate the results in general.
Hence answer is B
(C) Interpretation of a confidence interval is that- If we have constructed 90% confidence interval, it is not the case that there is 90%chance that population mean falls between this confidence interval. Correct one is " we are confident or we would expect that true mean of population would lie in this confidence interval in 90% of the cases".
Hence answer is B
(D) I personally think that it would not be a wise step to use confidence interval of time spent eating and drinking by 15 year age or older to find the mean of time spent eating and drinking by 9 year old . Reason being that both meanand standard error may differ in the two populations.In the latter population ,children may be quicker than the older ones and may spend less time in eating and drinking.
Hence correct anser is A.
TY!
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