Single-sample, Large Sample Confidence Interval for a Mean

Sometimes you are confronted with a deceptively simple-looking problem:

Construct the indicated confidence interval for the population mean, µ.

c = 0.90, x-bar = 16.2, s = 5.0, n = 75.

To solve it you need to carefully inspect the data you are given. You should notice two things. You are not given the population standard deviation, σ, and the n is > 30.

Depending upon the author of your stats book (and your instructor), you will choose either the normal distribution or the t-distribution to solve it. Some authors (e.g. Larson) say if you do not know sigma, use the t-distribution. Other authors say if n > 30, it is OK to use the normal distribution with s being approximately equal to σ. (McClave, Benson, & Sincich, 2014, p 303)

For these latter authors, the sample size, n, is used as a discriminator between small samples (n <30) and large samples (n >30).

We will solve it both ways and compare, but you should find out which way your textbook author leans because it might be important on a quiz.

Here is the Excel solution for the normal (z) distribution:

Here is the Excel solution for the t-distribution:

You can see the two intervals are close but different enough to cause you to miss the question even if they ask you to round to just one decimal, e.g. 14.9 vs 14.8 on the lower limits.

So, know the preference of your instructor/author. If you are in doubt, I would fall back on the rule of thumb that if n > 30, use the normal distribution. Both Larson & Farber, 4^th, and McClave et al, 12^th, use the “small vs large” concept.

Here are the StatCrunch solutions. Use the command sequence Stat > T Stats [ or Z Stats] > One Sample > With Summary. Enter the data, select the radio button next to Confidence Interval for µ, enter the confidence level, c, and click Compute!