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The t-distribution is crucial when you're dealing with smaller sample sizes and unknown population standard deviations. Unlike the normal distribution, which is symmetric and bell-shaped, the t-distribution has thicker tails. These tails account for the increased variability and uncertainty that comes with small samples. As the sample size increases, the t-distribution approaches the standard normal distribution.
For confidence intervals involving a sample size less than 30, like in our tire test example, the t-distribution is the tool of choice. You use it to find the critical t-value, which depends on the degrees of freedom. The degrees of freedom is generally the sample size minus one (n - 1). In our case, with 25 samples, we have 24 degrees of freedom. This t-value from the t-distribution table is then used to construct the confidence interval around the sample mean.

The standard error of the mean (SEM) helps estimate the precision of the sample mean as an estimate of the population mean. Think of SEM as a measure of how far the sample mean of your data is likely to be from the true population mean.
The formula for SEM is:

• \[ \text{SEM} = \frac{s}{\sqrt{n}} \]where \( s \) is the sample standard deviation, and \( n \) is the sample size.

In the tire stopping distance test, the sample standard deviation was 8 feet and the sample size was 25, resulting in a SEM of 1.6 feet. This SEM is then used with the t-value to calculate the confidence interval. A smaller SEM indicates more precise estimates of the population mean.

The sample mean, represented by \( \bar{x} \), is an average of the sample data collected. It is a good estimator of the population mean when the data is normally distributed. To calculate the sample mean, sum all observations and divide by the number of observations.
In our example, the stopping distances of the tires were recorded, resulting in a sample mean of 173 feet. This figure is at the center of our confidence interval calculation.
The sample mean is particularly important because it serves as the basis for constructing the confidence interval. By understanding how close the sample mean is to the expected population mean, you can refine the expected range of outcomes (confidence interval) for similar tests.