The following data set represents the time (in minutes) for a random sample of phone calls made by employees at a company:

(b) Find the sample standard deviation.

(c) Use the t-distribution to construct a 90% confidence interval for the population mean and interpret the results. Assume the population of the data set is normally distributed.

(d) Repeat part (c) assuming that_{}. Compare results.

**Solution: **

**(b) **The sample standard deviation is computed as:

_{}

Also, using Excel we find that

_{}

**(c) **The 90% confidence interval for the population mean is given by:

_{}

where_{}corresponds to the tow-tailed cutoff point of t-distribution, for_{}, and 9 degrees of freedom, which means that

_{}

This means that

_{}

This means that there’s a 90% chance that the interval (4.463051, 8.603616) contains the actual population mean_{}.

**(d) **Now we assume that the population standard deviation is known, and is equal to_{}. The 90% confidence interval is this case is equal to:

_{}

_{}

This means that there’s a 90% chance that the interval (4.712649, 8.354011) contains the actual population mean_{}. This means we have a better estimate for_{}, which makes sense, given that we have more information (the population variance is known).

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