### Chapter 6

#### The Big Picture

In Chapter 5, we studied the distribution of statistics (such as the sample mean) when we knew the population from which the sample was drawn. In Chapter 6, we reverse the problem and study what we can say about a population parameter on the basis of data from a random sample. It is never sufficient to simply estimate a parameter with a number. We should also include a description of how much error we think there is in our estimation procedure. We do this with confidence intervals. The logic for constructing confidence intervals will be the same for all the examples we consider this semester. The logic is illustrated in the context of estimating a population mean with a 95% confidence interval.

#### The Logic of Confidence Intervals

1. The sampling distribution for is approximately normally distributed with a mean of and a standard deviation of .
2. For 95% of all possible samples, is within 1.96 of .
3. Thus, for 95% of all possible samples, is within 1.96 of .
4. Therefore, I am 95% confident that is within 1.96 of for the particular sample I have chosen.
Different contexts will call for using different formulas, but the basic structure of all confidence intervals we study this semester will be the same.
```(estimate) +/- (reliability coefficient)(standard error)
```
We will consider confidence intervals in the four situations from Chapter 5: the population mean from a single population, the difference between two population means from two independent samples, the population proportion from a single population, and the difference between two population proportions from two independent samples. For estimating population means when population standard deviations are unknown, the t distribution should be used to determine reliability coefficients.