A consumer group is interested in comparing operating costs...

A consumer group is interested in comparing operating costs for two different types of automobile engines. The group is able to find 15 owners whose cars have engine type $A$ and 15 who have engine type B. All 30 owners bought their cars at roughly the same time and all have kept good records for a certain 12 month period. In addition, owners were found that drove roughly the same mileage. The cost statistics are $y_{A}=\$ 87.00 / 1,000$ miles, $y_{B}=\$ 75.00 / 1,000$ miles, $s_{A}=\$ 5.99,$ and $S B-\$ 4.85 .$ Compute a $95 \%$ confidence interval to estimate $\mu_{A}-\mu_{B_{1}}$ the difference in the mean operating costs. Assume normality and equal variance.

A consumer group is interested in comparing operating costs for two different types of automobile engines. The group is able to find 15 owners whose cars have engine type $A$ and 15 who have engine type
B. All 30 owners bought their cars at roughly the same time and all have kept good records for a certain 12 month period. In addition, owners were found that drove roughly the same mileage. The cost statistics are $y_{A}=\$ 87.00 / 1,000$ miles, $y_{B}=\$ 75.00 / 1,000$ miles, $s_{A}=\$ 5.99,$ and $S B-\$ 4.85 .$ Compute a $95 \%$ confidence interval to estimate $\mu_{A}-\mu_{B_{1}}$ the difference in the mean operating costs. Assume normality and equal variance.

Key Concepts

Assumptions of Normality and Equal Variance

These assumptions are critical for the validity of the two-sample t-interval. Normality ensures the sample means follow a t-distribution, and the equal variance assumption allows for the pooling of variances from both samples, which simplifies the analysis and strengthens the inference.

Degrees of Freedom

Degrees of freedom in this context are used to determine the appropriate critical value from the t-distribution for the interval estimate. In a two-sample test with equal variances, it is typically calculated based on the total number of observations minus the number of groups.

Confidence Interval

A confidence interval estimates a range of values which is likely to contain a population parameter with a specified level of confidence. In this context, it is used to determine the plausible range for the difference between the true mean operating costs of the two engine types.

Two-Sample t-Interval

This statistical method is applied when comparing the means of two independent groups. It uses the t-distribution to calculate an interval estimate of the difference in the population means when the sample sizes are relatively small and the population variances are not known exactly.

Pooled Variance

When the assumption of equal variances is justified, the pooled variance method combines the individual sample variances into a single estimate. This improves the estimate of the common variance and hence the standard error used in constructing the confidence interval.

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