A Guide to Business Statistics by David M. McEvoy

Author:David M. McEvoy
Language: eng
Format: epub, pdf
ISBN: 9781119138372
Publisher: Wiley
Published: 2018-02-24T00:00:00+00:00

7.1.3 Confidence Intervals Using the -Distribution

While it is often considered acceptable to use a critical value to construct the confidence interval for a mean, even when the population standard deviation is unknown, there is a more conservative approach. The reason to be more conservative is that we used the sample of 75 to estimate two statistics, the mean and the standard deviation. If for some reason, we knew the value for the population standard deviation, then we could safely rely on the -distribution. Since we do not know the value for the population standard deviation (which is almost always the case), we must rely on our estimate of $5000. To deal with this added uncertainty about the sampling distribution, we use a more conservative distribution than . We use what is called the Student -distribution or simply the -distribution.

The -distribution, like the normal distribution, is symmetric and mound-shaped. In fact, when sample sizes are large, the -distribution and the -distribution are effectively identical and therefore it does not matter which distribution you use. The -distribution only becomes influential, and therefore important, with smaller sample sizes. With small sample sizes (0 to 30, for example), the -distribution looks like a normal distribution that has been stretched out a bit (see Figure 7.2 for a visual comparison). Imagine someone pulling both tails of the distribution slightly. The tails become a bit “fatter” than those in the -distribution and the bell shape looks flatter. In practical terms, this means that for any given level of confidence, critical -values will be bigger than critical -values when sample sizes are small. And when critical values are bigger, confidence intervals are wider. And a wider interval is a more conservative interval.3

Question: When do we use the -distribution instead of the -distribution when forming a confidence interval around a mean?

Answer: When we do not know the population standard deviation and the sampling distribution can be approximated as normal.

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