Width of a 95% confidence interval for the mean

In the previous page, it was demonstrated that an interval estimate calculated using

has a lower confidence level than 95%. In order to achieve a 95% confidence level, the interval must be widened. This can be done by replacing 1.96 by a slightly larger number. We will try

where k > 1.96.

Simulation

The diagram below initially shows a random sample of n = 30 values from a normal population with µ = 10 and σ = 2.

When k = 1.96, the confidence level is 94.0%. Use the slider to adjust the value k to give a 95% confidence level. (The arrow keys on the keyboard can be used for fine adjustment of k in increments of 0.01.)

The resulting 95% confidence interval is displayed in red behind the data. Click Another sample to obtain 95% confidence intervals for other data sets.

Use the pop-up menu to change the sample size to 5 and repeat. Observe that k must be increased in order to achieve a 95% confidence level.

The value of k that gives a 95% confidence level depends on the sample size.


The t-value

The value k that is needed to achieve a 95% confidence level depends on the sample size, n, and is denoted by tn-1. This t-value can be obtained from a table or graph or computer software. Small values of n require larger t-values. A 95% confidence interval for µ is therefore

For reasons that cannot be explained here, you must look up the t-value using the sample size minus one rather than the sample size itself. The value n - 1 is called the degrees of freedom of the constant.

Finding the t-value

You will often use a printed table or computer software to obtain the t-value for any problem. The diagram below shows how the t-value depends on the degrees of freedom (and therefore the sample size).

Drag the slider to change the degrees of freedom and read off the corresponding t-value. Note that