## 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.

### Difference between Means

#### Learning Objectives

- State the assumptions for computing a confidence interval on the difference between means
- Compute a confidence interval on the difference between means
- Format data for computer analysis

It is much more common for a researcher to be interested in the difference between means than in the specific values of the means themselves. We take as an example the data from the "Animal Research" case study. In this experiment, students rated (on a 7-point scale) whether they thought animal research is wrong. The sample sizes, means, and variances are shown separately for males and females in Table 1.

**Table 1. **Means and Variances in Animal Research study.

Condition | n | Mean | Variance |
---|---|---|---|

Females | 17 | 5.353 | 2.743 |

Males | 17 | 3.882 | 2.985 |

As you can see, the females rated animal research as more wrong than did the males. This sample difference between the female mean of 5.35 and the male mean of 3.88 is 1.47. However, the gender difference in this particular sample is not very important. What is important is the difference in the population. The difference in sample means is used to estimate the difference in population means. The accuracy of the estimate is revealed by a confidence interval.

In order to construct a confidence interval, we are going to make three assumptions:

- The two populations have the same variance. This assumption is called the assumption of homogeneity of variance.
- The populations are normally distributed.
- Each value is sampled independently from each other value.

The consequences of violating these assumptions are discussed in a later section. For now, suffice it to say that small-to-moderate violations of assumptions 1 and 2 do not make much difference.

A confidence interval on the difference between means is computed using the following formula:

where is the difference between sample means, is the t for the desired level of confidence, and is the estimated standard error of the difference between sample means. The meanings of these terms will be made clearer as the calculations are demonstrated.

We continue to use the data from the "Animal Research" case study and will compute a confidence interval on the difference between the mean score of the females and the mean score of the males. For this calculation, we will assume that the variances in each of the two populations are equal.

The first step is to compute the estimate of the
standard error of the difference between means .
Recall from the relevant
section in the chapter on sampling distributions that the
formula for the standard error of the difference in means in the
population is:

In order to estimate this quantity, we estimate and use that estimate in place of . Since we are assuming the population variances are the same, we estimate this variance by averaging our two sample variances. Thus, our estimate of variance is computed using the following formula:

where MSE is our estimate of σ^{2}.
In this example,

Note that MSE stands for "mean square error" and is the mean squared deviation of each score from its group's mean.

Since (the number of scores in each condition) is 17,

The next step is to find the t to use for the confidence interval (). To calculate , we need to know the degrees of freedom. The degrees of freedom is the number of independent estimates of variance on which MSE is based. This is equal to where is the sample size of the first group and is the sample size of the second group. For this example, . When , it is conventional to use "" to refer to the sample size of each group. Therefore, the degrees of freedom is 16 + 16 = 32.

From either the above calculator or a table, you can find that the for a 95% confidence interval for is 2.037.

We now have all the components needed to compute the confidence interval. First, we know the difference between means:

We know the standard error of the difference between means is

and that the for the 95% confidence interval with is

Therefore, the 95% confidence interval is

We can write the confidence interval as:

where is the population mean for females and is the population mean for males. This analysis provides evidence that the mean for females is higher than the mean for males, and that the difference between means in the population is likely to be between 0.29 and 2.65.

#### Formatting data for Computer Analysis

Most computer programs that compute tests require your data to be in a specific form. Consider the data in Table 2.

**Table 2. **Example Data.

Group 1 | Group 2 |
---|---|

3 | 5 |

4 | 6 |

5 | 7 |

Here there are two groups, each with three observations. To format
these data for a computer program, you normally have to use two
variables: the first specifies the group the subject is in and the
second is the score itself. For the data in Table 2, the reformatted
data look as follows:

**Table 3. **Reformatted Data.

G | Y |
---|---|

1 | 3 |

1 | 4 |

1 | 5 |

2 | 5 |

2 | 6 |

2 | 7 |

To use Analysis Lab to do the calculations, you would copy the data and then

- Click the "Enter/Edit User Data" button. (You may be warned that for security reasons you must use the keyboard shortcut for pasting data).
- Paste your data.
- Click "Accept Data".
- Set the Dependent Variable to Y.
- Set the Grouping Variable to G.
- Click the t-test confidence interval button.

The 95% confidence interval on the difference between means extends from -4.267 to 0.267.

#### Computations for Unequal Sample Sizes (optional)

The calculations are somewhat more
complicated when the sample sizes are not equal. One consideration
is that MSE, the estimate of variance, counts the sample with
the larger sample size more than the sample with the smaller sample
size. Computationally this is done by computing the sum of squares
error (SSE) as follows:

where is the mean for group 1 and
is the mean for group 2. Consider
the following small example:

Table 4. Example Data.

Group 1 | Group 2 |
---|---|

3 | 2 |

4 | 4 |

5 |

and .

Then, MSE is computed by:

where the degrees of freedom () is computed as before:

.

.

The formula

is replaced by

where is the harmonic mean of the sample sizes and is computed
as follows:

and

.

for 3 and the 0.05 level = 3.182.

Therefore the 95% confidence interval is

We can write the confidence interval as: