Constructing a Confidence Interval for a Population Mean or Population Mean Difference

t-distributions

When you replace $\sigma$ with $s$, the standardized value $\frac{\bar{x}-\mu }{s/\sqrt{n}}$ no longer follows the standard normal curve. It follows a t-distribution, also called Student's t-distribution.

The t-distributions are a family of symmetric, bell-shaped, standardized curves centered at 0. Each one is identified by its degrees of freedom (df), which depend on the sample size. For one sample, $\mathrm{df}=n-1$. Compared with the standard normal curve, a t-distribution has a lower, narrower peak and fatter tails, so more of its area sits far from 0. With few degrees of freedom the difference is large. As the degrees of freedom increase, the t-distribution looks more and more like the standard normal curve, and the critical values close in on the familiar $z^{\ast }$.

The one-sample t-interval

To estimate a population mean $\mu$ when $\sigma$ is unknown, use the one-sample t-interval for a population mean:

The point estimate is the sample mean $\bar{x}$. The standard error, $\mathrm{SE}=s/\sqrt{n}$, estimates how far sample means typically land from $\mu$. It is the $\sigma /\sqrt{n}$ formula from Topic 4.1, with $s$ in place of $\sigma$. The margin of error is $t^{\ast }\times \mathrm{SE}$, where $t^{\ast }$ is the critical value for the central C% of the t-distribution with $\mathrm{df}=n-1$. On a TI-84, invT(0.975, df) gives $t^{\ast }$ for 95% confidence; the printed t table gives the same values.

State the parameter in context: the mean of the response variable for the population you sampled. "The mean caffeine content of all 16-ounce cold brews sold by the chain" names a parameter. "The mean" does not.

Three conditions

A one-sample t-interval for a population mean requires three conditions:

• Randomization condition: the data come from a random sample or a randomized experiment.
• 10% condition: when sampling without replacement, $n\le 0.10N$.
• Sample data condition: the population distribution is approximately normal, or $n\ge 30$, or, if $n<30$, the sample data are free from strong skewness and outliers.

The third one is new. With proportions you counted successes and failures. With means you look at the data. For a small sample, graph it with a dotplot, boxplot or histogram and look for strong skew or outliers. Mild unevenness is fine, because small samples from normal populations rarely look perfectly symmetric. An outlier or a long tail means the condition is not met, and the interval may not deserve the confidence level printed next to it.

Paired data: one sample of differences

Sometimes each value in one set is matched with exactly one value in the other, such as the same student measured before and after a program, or twins split between two treatments. This is a matched pairs design, and the two sets of values are dependent. Do not treat them as two separate samples. Subtract within each pair to get one sample of differences, and use the one-sample t-interval for a population mean difference:

Here ${\bar{x}}_d$ and $s_d$ are the mean and standard deviation of the differences, $n$ is the number of pairs, and $\mathrm{df}=n-1$. The parameter is ${\mu }_d$, the population mean difference, and you must say which way you subtracted: after minus before is a different parameter from before minus after. The three conditions are checked on the differences, so for the sample data condition you graph the differences, not the two original lists.