People are always interested in comparing themselves to othe

People are always interested in comparing themselves to others. If you are male, how does the length of your forearm, from elbow to wrist, compare to the average for the population of men? Suppose that the forearm lengths (in centimeters) for a randomly selected sample of 9 men has a sample mean length of 25.5 cm with a sample standard deviation of 1.52 cm.

(a) Find a 95% confidence interval for the average forearm length of men. Give your answer to 2 decimal places.

(b) Suppose that you assumed that the sample standard deviation was close enough to the population standard deviation and used 1.96 in your calculation of the confidence interval rather than the appropriate t value. What would be the consequence of incorrectly calculating your 95% confidence interval in this way? Choose all that apply.

The actual confidence level will be larger than 95%.

The actual confidence level will be smaller than 95%.

If we were to sample repeatedly and calculate a 95% confidence interval from each sample using 1.96 instead of the appropriate t value, then the percent of confidence intervals that capture the average forearm length of all men will be less than 95%.

The width of the confidence interval will be too large.

There is no problem with using 1.96 because the mean of all sample means is still the true average forearm length of men.

Solution

a)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    25.5          
t(alpha/2) = critical t for the confidence interval =    2.306004135          
s = sample standard deviation =    1.52          
n = sample size =    9          
df = n - 1 =    8          
Thus,              
Margin of Error E =    1.168375429          
Lower bound =    24.33162457          
Upper bound =    26.66837543          
              
Thus, the confidence interval is              
              
(   24.33162457   ,   26.66837543   ) [ANSWER]

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b)

As we accidentally calculated a wider confidence interval, then

CORRECT: The actual confidence level will be larger than 95%.

CORRECT: The width of the confidence interval will be too large.

[There are two answers]