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The standard recommendation for automobile oil changes is once every 3000 miles. A local mechanic is interested in determining whether people who drive more expensive cars are more likely to follow the recommendation. Independent random samples of 45 customers who drive luxury cars and 40 customers who drive compact lower-price cars were selected. The average distance driven between oil changes was 3187 miles for the luxury car owners and 3214 miles for the compact lower-price cars. The sample standard deviations were \(42.40\) and \(50.70\) miles for the luxury and compact groups, respectively. Assume that the population distributions of the distances between oil changes have the same standard deviation for the two populations. a. Construct a \(95 \%\) confidence interval for the difference in the mean distances between oil changes for all luxury cars and all compact lower-price cars. b. Using a \(1 \%\) significance level, can you conclude that the mean distance between oil changes is less for all luxury cars than that for all compact lower-price cars?

Step by step solution

01

Calculate the Sample Means and Standard Deviations

Our samples provide us with the following data: \n mean of the luxury group (\( \bar{x}_1 \)) = 3187 miles, \n mean of compact group (\( \bar{x}_2 \)) = 3214 miles, \n standard deviation of the luxury group (\( s_1 \)) = 42.40 miles, \n standard deviation of the compact group (\( s_2 \)) = 50.70 miles.

02

Calculate the Standard Error of the Difference

Given that the two standard deviations should theoretically be the same, we take their average as our best estimate of the common standard deviation. So we calculate the pooled standard deviation \( s_p \) using the formula: \[ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \] where \( n_1 \) is the size of the first sample and \( n_2 \) is the size of the second sample. After calculating \( s_p \), we can calculate the standard error of the difference (\( SE \)) using the formula: \[ SE = s_p \sqrt{1/n_1 + 1/n_2} \]

03

Construct the Confidence Interval

The formula to create a 95% confidence interval for the difference in means is: \[ CI = (\bar{x}_2 - \bar{x}_1) \pm (t^* \cdot SE) \] where \( t^* \) is the critical t-values for a 95% confidence interval with \( n_1 + n_2 - 2 \) degrees of freedom.

04

Hypothesis Test

For part b, we will use a 1% significance level to test the hypothesis that the mean distance between oil changes for all luxury cars (\( \mu_1 \)) is less than that for all compact lower-price cars (\( \mu_2 \)). The null hypothesis will be \( H_0: \mu_1 = \mu_2 \) and the alternative hypothesis \( H_A: \mu_1 < \mu_2 \). For this, we calculate the test statistic using the formula: \[ t = \frac{(\bar{x}_1 - \bar{x}_2)}{SE} \] Then, we compare the t-statistic with the critical t-value from the t-distribution table at a 1% significance level with \( n_1 + n_2 - 2 \) degrees of freedom to see if we reject or do not reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a fundamental statistical method that allows us to make inferences or draw conclusions about populations based on sample data. In this context, the hypothetical question is whether the mean distance driven between oil changes for luxury cars is less than that for compact cars. We'll set up using two hypotheses:

• The null hypothesis (\( H_0 \)) posits that there is no difference, meaning the average distances between oil changes for luxury and compact cars are the same: \( H_0: \mu_1 = \mu_2 \).
• The alternative hypothesis (\( H_A \)) suggests that luxury car owners drive less distance between oil changes: \( H_A: \mu_1 < \mu_2 \).

To determine which hypothesis to support, we calculate a test statistic from our sample data and compare it against a critical value from statistical tables, determining whether we can reject \( H_0 \). This procedure helps decide if the observed data offers strong enough evidence against the null hypothesis.

Pooled Standard Deviation

When comparing two independent sample groups in statistics, an essential step is calculating a common measure of variability known as the pooled standard deviation. This is especially useful if we assume the two groups have equal variances, as it combines the variabilities of both sets.To find the pooled standard deviation, we use the formula:\[ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \]where \( n_1 \) and \( n_2 \) are the sample sizes, and \( s_1 \) and \( s_2 \) are the standard deviations of the samples. This pooled measure helps in calculating the standard error of the difference between the two means, a key step when constructing confidence intervals or conducting hypothesis tests. The pooled standard deviation provides an average of the variability within the datasets, furnishing a stable foundation for further analysis.

Significance Levels

Significance levels (commonly denoted as \( \alpha \)) in hypothesis testing define the probability threshold at which we accept that the observed data's outcome is due to chance. In other words, it sets the bar for how much risk we're willing to take in rejecting the null hypothesis incorrectly. In our example, we use a 1% significance level, translating to \( \alpha = 0.01 \). This means if the data yields results that would occur only 1% of the time under the null hypothesis, we deem the evidence strong enough to reject \( H_0 \). Choosing a significance level involves balancing Type I errors (rejecting a true null hypothesis) and Type II errors (failing to reject a false null hypothesis). Lower significance levels offer more stringent tests but increase the likelihood of concluding that no effect exists when there might be one. The 1% level used here is quite stringent, aiming for high confidence in the results.

T-Distribution

The t-distribution is a crucial component when dealing with small sample sizes or unknown population standard deviations. It’s especially helpful in inference, such as constructing confidence intervals or conducting hypothesis tests.In the exercise, we use it to derive critical values and test statistics. The t-distribution accounts for added variability and provides a more conservative estimate than the normal distribution, especially with smaller sample sizes.Its shape depends on the degrees of freedom (df), typically \( n_1 + n_2 - 2 \) when pooling samples. A higher df value creates a curve that closely resembles a normal distribution, while fewer degrees of freedom produce thicker tails, indicating greater uncertainty.By leveraging the t-distribution, we better manage uncertainties inherent in using sample statistics to infer population parameters, ensuring sound conclusions, especially in hypothesis testing and confidence interval construction.