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When we talk about a confidence interval, we refer to a range that estimates the difference between two population means. This range tells us how confident we are that the true difference falls within it. In our exercise, we're looking at the repair costs for two car models. To construct the confidence interval, we calculate the standard error, a simple measure of how accurately our sample represents the population. The formula for the standard error involves the standard deviations and sample sizes of the two groups. After finding this, we look up the critical value from the Z-table, based on our desired confidence level—99% in this case. The confidence interval is then built around the difference in sample means, adding and subtracting the product of the critical value and standard error. This interval gives us a range where the true difference in repair costs is likely to fall 99% of the time if we repeated this study with new samples.

Hypothesis testing is a method used to make decisions about population parameters. We start with a null hypothesis (\(H_0\)), which suggests no effect or no difference. For our exercise, \(H_0: \mu_1 - \mu_2 = 0\), means there's no difference in the average repair costs of the two cars. The alternative hypothesis (\(H_1\)) suggests that there is a difference. We use the Z-test to determine which hypothesis the data supports. This involves calculating a Z statistic using the difference in sample means and dividing it by the standard error. Once we have the Z statistic, we check its corresponding probability, known as the p-value. If the p-value is less than the chosen significance level (\(\alpha = 0.01\)), it suggests that the observed data would be very unlikely under the null hypothesis, leading us to reject \(H_0\).

Type I error is a crucial idea in statistics. It's when we falsely reject a true null hypothesis. Imagine if we conclude that the two cars have different repair costs when, in fact, they don't—that's a Type I error. In testing, this error is controlled by setting a significance level (\(\alpha\)), typically at 0.05 or 0.01, like in our exercise. This significance level is the probability of making a Type I error, meaning that 1% of the time, we might incorrectly reject the null hypothesis if our \(\alpha\) is 0.01. If the probability of making a Type I error were zero, it would mean we never make a false claim of difference. However, this is practically impossible as it would require perfect knowledge of the populations.

A Z-test is a statistical test used to determine differences between sample means. It's applicable when we know the population standard deviations, as in our exercise. To perform a Z-test, we first state our null and alternative hypotheses. Then, we calculate the Z statistic, which measures how far away our sample statistic is from the null hypothesis value—in standard errors. The formula for the Z statistic is \(Z = \frac{(\overline{x_1} - \overline{x_2})}{SE}\), where \(SE\) is the standard error. Next, we compare the calculated Z statistic against the critical Z value from a Z-table. If the calculated Z statistic is greater than the critical value or the p-value is less than our significance level, we reject the null hypothesis. This tells us that there is significant evidence to suggest a difference in mean repair costs between the two car models.