Section 6.2: Confidence Interval for a Population Mean

Key Concepts

The sampling distribution of is approximately normal with mean and standard deviation by the central limit theorem for sufficiently large samples. The logic of constructing confidence intervals for from sample data depends on understanding the sampling distribution of .

An example at the end of the section demonstrates the type of exercise you should be able to solve.

Interpretation

Using the general formula
```(estimate) +/- (reliability coefficient)(standard error)
```
constructing a confidence interval is simply plugging into the formula To construct a C% confidence interval, the multiplier z* is chosen from the standard normal distribution so that the area between -z* and z* is C%. Common choices are

```Confidence Level | z*
------------------------------------
90%             |  1.645
95%             |  1.960
99%             |  2.576
```
A 95% confidence interval expressed as 4.5 +/- 2.3 can be interpreted as

"I am 95% confident that the population mean is within 2.3 of the sample mean 4.5" or "I am 95% confident that the population mean is between 2.2 and 6.8".

Nonnormal Populations

Even if the population is not normal, the central limit theorem says that the shape of the sampling distribution will be approximately normal for sufficiently large samples. For most practical situations in biology, samples of size 25 or 30 are sufficiently large for confidence intervals based on the normal distribution to be valid.

If your sample size is small and noticeably nonnormal, with extreme outliers, or strong skewness apparent in histograms, you should not use the formula in this section for constructing confidence intervals.

Example

In studies from recent years, the birthweights of infants born in Boston had population standard deviations of 20.6 ounces. 40 infants were randomly sampled from recent births in this population, and the mean of their birth weights was 114.0 ounces. Give a 95% confidence interval for the population mean.

The sample size is large enough that it is reasonable to conclude that the shape of the sampling distribution of is approximately normal. From previous studies, it is reasonable to conclude that the standard error of this distribution is = 3.26.

Plugging into the formula gives an answer 114.0 +/- 6.4, where we used 1.96 for z.

This can interpreted as "we are 95% confident that the unknown mean weight of all infants in Boston is between 107.6 and 120.4 ounces".