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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 the \(1 \%\) 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 difference of means and standard error

First, calculate the difference of sample means which is \( \bar{X}_1 - \bar{X}_2 \) = 3187 - 3214 = -27 miles. Then, calculate the standard error of the difference of means, which is \( SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \) = \sqrt{\frac{42.4^2}{45} + \frac{50.7^2}{40}} = 11.77 miles.

02

Construct a 95% confidence interval for the difference in means

For a 95% confidence interval, the critical z value is 1.96. The interval is given by \( (\bar{X}_1 - \bar{X}_2) \pm z(SE) \) = -27 \pm 1.96(11.77) = [-50.04, -3.96] miles.

03

Perform the hypothesis test

The null hypothesis is that the means are equal, or \( \mu_1 = \mu_2 \). The alternative hypothesis, however, is \( \mu_1 < \mu_2 \). Here we will use one-sided z-test. First, calculate the z-statistic: \( z = \frac{(\bar{X}_1 - \bar{X}_2) - 0}{SE} \) = -27 / 11.77 = -2.29. The p-value corresponding to this z value for a one-sided test is 0.011. Hence we reject the null hypothesis at the 1% level of significance because the p-value < 0.01. Thus, we can conclude that the mean distance between oil changes is less for all luxury cars than that for all compact lower-price cars.

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Key Concepts

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

Confidence Interval

A confidence interval provides a range of values that is believed to contain a parameter of the population, such as the mean. It's an important aspect of inferential statistics that helps us understand data variability. In the context of the exercise, constructing a 95% confidence interval means we're 95% confident that the true difference in the mean distances between oil changes for luxury and compact cars lies within this range. We calculated it as

• Lower bound: -50.04 miles
• Upper bound: -3.96 miles

This interval tells us that, on average, the distance driven between oil changes is less for luxury cars than compact cars, as the interval is all negative. It indicates that following the oil change recommendations differs between luxury and compact car groups.

Z-Test

The z-test is a statistical method used to determine whether there is a significant difference between sample and population means, or between two sample means. This test is applicable when the population variance is known, or the sample size is large, typically over 30.
A one-sided z-test was performed in this exercise to see if luxury cars get oil changes at shorter intervals than compact cars. With the calculated z-statistic of -2.29 and corresponding p-value being less than 0.01, we find evidence to reject the null hypothesis. This suggests a statistically significant difference in oil change behavior, corroborating the hypothesis that luxury car drivers tend to adhere more closely to the oil change recommendations.

Significance Level

The significance level, often denoted as alpha (\(\alpha\), defines the cutoff for making decisions about the null hypothesis. A common choice in hypothesis testing is a 5% significance level, but in this exercise, a stricter 1% level was used.
This lower alpha means we require stronger evidence against the null hypothesis to conclude that luxury car drivers, on average, change oil more frequently. When the p-value of 0.011 was found, it was below the 1% significance threshold, leading to the conclusion that there’s a statistically significant difference in oil change patterns between the groups.

Sample Statistics

Sample statistics involve data from a sample of the population to estimate population parameters. It includes values like sample mean and standard deviation.
In our exercise, the samples came from 45 luxury car drivers and 40 compact car drivers, with means of 3187 and 3214 miles respectively. The goal was to estimate the population mean difference using these samples, leading us to confidently assert differences in behavior around oil changes. Understanding and calculating these statistics help make real-world decisions based on the data without needing to survey the entire population.