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An insurance company wants to know if the average speed at which men drive cars is higher than that of women drivers. The company took a random sample of 27 cars driven by men on a highway and found the mean speed to be 72 miles per hour with a standard deviation of \(2.2\) miles per hour. Another sample of 18 cars driven by women on the same highway gave a mean speed of 68 miles per hour with a standard deviation of \(2.5\) miles per hour. Assume that the speeds at which all men and all women drive cars on this highway are both approximately normally distributed with unknown and unequal population standard deviations. a. Construct a 98\% confidence interval for the difference between the mean speeds of cars driven by all men and all women on this highway. b. Test at a \(1 \%\) significance level whether the mean speed of cars driven by all men drivers on this highway is higher than that of cars driven by all women drivers. c. Suppose that the sample standard deviations were \(1.9\) and \(3.4\) miles per hour, respectively. Redo parts a and b. Discuss any changes in the results.

Step by step solution

01

Setup the problem

First, it is important to clearly outline the parameters. For men, the variables are: sample size \(n_1 = 27\), sample mean \(x_1 = 72\) mph, and sample standard deviation \(s_1 = 2.2\) mph. For women: sample size \(n_2 = 18\), sample mean \(x_2 = 68\) mph, and sample standard deviation \(s_2 = 2.5\) mph.

02

Calculate the confidence interval

The formula for the confidence interval of the difference betweens is given by: \((x_1 - x_2) \pm Z_{\alpha/2}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\). Here \(Z_{\alpha/2}\) can be found from standard Z-table considering that we need a 98% confidence interval. Then use the gathered and calculated information to find the confidence interval.

03

Conduct the hypothesis testing

The null hypothesis \(H_o\) would be that there is no difference between the mean speeds of men and women while the alternative hypothesis \(H_1\) is that men drive faster. Therefore, \(H_o: \mu_1 - \mu_2 \leq 0\) and \(H_1: \mu_1 - \mu_2 > 0\). At a \(1\%\) level of significance, calculate the test-statistics to decide whether to reject \(H_o\).

04

Repeat the calculations with new standard deviations

Redo steps 2 and 3 with the changed standard deviations \(s_1 = 1.9\) mph and \(s_2 = 3.4\) mph and observe any changes in results. This tests the sensitivity of hypothesis testing results and confidence intervals to changes in sample standard deviations.

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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 gives us a range within which we expect the true difference in mean speeds to lie. It is based on the data we collected from the sample of men and women drivers. In this problem, we calculate the 98% confidence interval for the difference in mean speeds between male and female drivers. This means we are 98% confident that the true difference lies within this range.

To find this interval, we use the formula:

• \((x_1 - x_2) \pm Z_{\alpha/2}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)

Here, \(x_1\) and \(x_2\) are the sample means, and \(s_1\) and \(s_2\) are the sample standard deviations, while \(n_1\) and \(n_2\) are the sample sizes for men and women respectively.

The \(Z_{\alpha/2}\) value comes from the standard normal distribution table and corresponds to the chosen confidence level of 98%. This formula accounts for the variability in our samples and helps us estimate the range with confidence.

Significance Level

The significance level, often denoted by \(\alpha\), helps us determine the threshold at which we might reject the null hypothesis. In hypothesis testing, it represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. In our context, a 1% significance level implies a 1% risk of concluding that men drive faster when they don't.

• Using 1% makes our test very strict, reducing the likelihood of a false positive.
• We conclude there is enough evidence to state that men drive faster only if the test statistic falls beyond this threshold.

It's crucial to choose an appropriate significance level based on the context, as it directly influences the confidence in your conclusions.

Sample Mean

The sample mean gives a central value from our data. For hypothesis testing, we use the sample mean as our evidence point. In this case, 72 mph for men and 68 mph for women serve as estimates of the average speeds.

• It's calculated by adding up all the observed values and dividing by the number of observations.
• Sample mean acts as an indicator of the population mean, though it includes some variability inherent to the sample itself.

Sample means are crucial for calculating both the confidence intervals and hypothesis tests. They form the foundation of statistical inference in our evaluation.

Sample Standard Deviation

Sample standard deviation measures the amount of variation or dispersion in a set of values. In our example, the recorded standard deviations for men and women are 2.2 mph and 2.5 mph, respectively.

• It helps us understand how much individual speeds in each group differ from their group's average speed.
• The larger the standard deviation, the more spread out the speeds are.

When these values are used in hypothesis tests or confidence interval calculations, they influence our final estimates. Any change in these values, as discussed in the exercise, can significantly alter the results of our statistical analyses.