## Confidence Intervals for the Mean

This section explains the need for confidence intervals and why a confidence interval is not the probability the interval contains the parameter. Then, it discusses how to compute a confidence interval on the mean when sigma is unknown and needs to be estimated. It also explains when to use t-distribution or a normal distribution. Next, it covers the difference between the shape of the t distribution and the normal distribution and how this difference is affected by degrees of freedom. Finally, it explains the procedure to compute a confidence interval on the difference between means.

### Confidence Interval on the Mean

#### Answers

- Because you know the standard deviation of the population, you can use the normal distribution. If you use the normal distribution calculator, you will find that 90% of the area is within 1.65 standard deviations of the mean.
- You have to estimate the standard deviation of the population, so you need to use the t distribution. Because your sample size is 11, your degrees of freedom are 11 - 1 = 10. Looking at the t distribution table, you can see that 95% of the distribution falls within 2.23 standard deviations of the mean.
- First, the standard error of the mean is 6.5/5 = 1.3. Second, you know the population standard deviation, so you can use the normal distribution. 95% of the distribution lies within 1.96 standard deviations of the mean. Lower limit: 38 - (1.3)(1.96) = 35.45; Upper limit: 38 + (1.3)(1.96) = 40.55
- In this case, the population standard deviation is unknown, so you need to use the t distribution (df = N-1 = 9-1 = 8). From the t distribution table, you see that 99% of the distribution lies within 3.355 standard deviations of the mean when t has 8 degrees of freedom. The standard error of the mean = 4/3 = 1.333. Lower limit: 49 - (1.333)(3.355) = 44.53; Upper limit: 49 + (1.333)(3.355) = 53.47
- Use a statistical program to compute the 95% CI. To compute it by hand, you need the sample mean and standard error and the t value for 19 df. 14.25 + (2.093)(1.784) = 17.984