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The t-test is a statistical tool that helps us decide if there are significant differences between the means of two groups or between a sample mean and a known population mean. It is particularly useful when dealing with small sample sizes, like in this problem, where we have only 15 households.

To perform a t-test, we need some key components: the sample mean, the population mean, the sample standard deviation, and the sample size. These elements allow us to calculate the t-statistic, which is a kind of "distance" measure between the sample mean and the population mean.

It's important to check that the underlying data follows a normal distribution, which is an assumption when using the t-test. If this condition is met, the t-test can give us reliable results about the differences we are investigating.

The null hypothesis, often denoted as \(H_0\), is the assumption that there is no effect or difference. It is the default position that there is no change from the status quo. In the context of our problem, the null hypothesis states that the average miles driven per household has not changed, remaining at 18.59 thousand miles.

The purpose of hypothesis testing is to challenge the null hypothesis. We seek evidence against it, to potentially reject it, rather than to prove it right. If our test shows significant evidence against \(H_0\), we may reject it and consider the alternative hypothesis.

The alternative hypothesis, symbolized as \(H_a\), represents what we are testing against the null hypothesis. It suggests that there is a change, effect, or difference. In this scenario, the alternative hypothesis claims that people are driving less than before, meaning the population mean is less than 18.59 thousand miles.

This hypothesis is typically what the researcher wants to prove with the data. If evidence from our test is strong enough to reject the null hypothesis, the alternative hypothesis is supported. An alternative hypothesis can be one-tailed or two-tailed, depending on the direction of the predicted effect. Here, a one-tailed test is used because we are specifically interested in whether people are driving less.

The significance level, denoted as \(\alpha\), is a threshold we set to determine whether to reject the null hypothesis. It reflects the probability of rejecting a true null hypothesis, also known as making a Type I error. The significance level chosen for our problem is \(0.05\), or 5%.

By setting \(\alpha = 0.05\), we are allowing ourselves a 5% risk of concluding that people drive less, when in real terms they do not. The significance level helps us define the critical value or area in the statistical test, within which if our test statistic falls, we will reject the null hypothesis.

Choosing a significance level is a balance between risk and conservatism. A lower \(\alpha\) offers a more conservative test but increases the risk of not detecting a real effect, known as a Type II error.