A car magazine is comparing the total repair costs incurred...

A car magazine is comparing the total repair costs incurred during the first three years on two sports cars, the T-999 and the XPY. Random samples of 45 T-999s and 51 XPYs are taken. All 96 cars are 3 years old and have similar mileages. The mean of repair costs for the 45 T-999 cars is $\$ 3300$ for the first 3 years. For the 51 XPY cars, this mean is $\$ 3850 .$ Assume that the standard deviations for the two populations are $\$ 800$ and $\$ 1000$, respectively. a. Construct a $99 \%$ confidence interval for the difference between the two population means. b. Using the $1 \%$ significance level, can you conclude that such mean repair costs are different for these two types of cars? c. What would your decision be in part $\mathrm{b}$ if the probability of making a Type I error were zero? Explain.

A car magazine is comparing the total repair costs incurred during the first three years on two sports cars, the T-999 and the XPY. Random samples of 45 T-999s and 51 XPYs are taken. All 96 cars are 3 years old and have similar mileages. The mean of repair costs for the 45 T-999 cars is $\$ 3300$ for the first 3 years. For the 51 XPY cars, this mean is $\$ 3850 .$ Assume that the standard deviations for the two populations are $\$ 800$ and $\$ 1000$, respectively.
a. Construct a $99 \%$ confidence interval for the difference between the two population means.
b. Using the $1 \%$ significance level, can you conclude that such mean repair costs are different for these two types of cars?
c. What would your decision be in part $\mathrm{b}$ if the probability of making a Type I error were zero? Explain.

Key Concepts

Significance Level

The significance level, denoted by alpha, is the threshold probability used in hypothesis testing to compare against the p-value. It represents the maximum acceptable risk of making a Type I error and is used to decide whether to reject the null hypothesis. In this context, a 1% significance level means there is a 1% risk of incorrectly rejecting the null hypothesis.

Two-Sample Z-Test

A two-sample z-test is used for comparing the means of two independent samples when the population standard deviations are known or the sample sizes are large enough for the central limit theorem to apply. It involves calculating a test statistic based on the difference between sample means, the known standard deviations, and the sample sizes, which is then compared to a critical value or p-value to decide on the null hypothesis.

Type I Error

A Type I error occurs when the null hypothesis is rejected even though it is actually true. It represents a false positive result in hypothesis testing. Discussing the probability of a Type I error, such as setting it to zero, highlights the balance between being cautious of false positives and the practical implications for testing, particularly when drawing conclusions from sample data.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is believed to contain the true population parameter (such as a mean difference) with a specified level of confidence. In the context of comparing two means, it provides an estimated range within which the true difference between the two population means is expected to lie.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. When comparing two group means, it involves setting up a null hypothesis (typically, no difference between means) and an alternative hypothesis (a specified difference exists), then using sample data to determine whether the null hypothesis can be rejected.

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