Question

A recent report from the National Center for Health Statistics (NCHS) states that the distribution of weights for men in the United States aged 35 to 45 is well approximated by a normal distribution with mean 176.4 pounds and standard deviation 31.3 pounds. Let X be the sample mean weight of a simple random sample of 16 men chosen from this population.

(a) Find the interval containing the central 80% of the X distribution. Interpret your answer in words.

(b) Repeat part (a) using the sample size of 64 instead of 16.

c) The length of an interval is the difference between its upper endpoint and its lower endpoint. Compute and compare the lengths of the intervals from parts (a) and (b).

(d) Suppose we use a sample mean to estimate the actual mean for the normal distribution of X, the weight of a man in the United States aged 35 to 45. Which size sample, 16 or 64, is more likely to give a closer estimate to the actual mean of the distribution (176.4 lbs)? Explain you answer, referring to any work above.

Answer 1

(a) The 80% confidence interval is between 166.372 and 186.428.

176.400	mean 1
31.300	std. dev.
7.825	std. error
16	n
166.372	confidence interval 80.% lower
186.428	confidence interval 80.% upper
10.028	margin of error

(b) The 80% confidence interval is between 171.386 and 181.414.

176.400	mean 1
31.300	std. dev.
3.913	std. error
64	n
171.386	confidence interval 80.% lower
181.414	confidence interval 80.% upper
5.014	margin of error

(c) The confidence interval from part (a) is wider than part (b).

(d) As the sample size increases, the standard error decreases. Therefore, we can conclude that the sample size, 64, is more likely to give a closer estimate to the actual mean of the distribution (176.4 lbs).

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