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According to Exercise \(10.27\), 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 normally distributed with 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

- Calculate the difference in sample means

To start this problem, subtract the sample mean of the women's speeds from the sample mean of the men's speeds. The equation is: \[\bar{X}_{M} - \bar{X}_{W} = 72 - 68 = 4 \]

02

- Compute the standard error of the difference

The standard error of the difference can be given by the formula \[SE = \sqrt{(\frac{s^2_M}{n_M}) + (\frac{s^2_W}{n_W})}\]where \(s_M\) and \(s_W\) denote the standard deviations of men and women respectively, and \(n_M\) and \(n_W\) represent the sizes of the samples of the men and the women respectively.Using the given data:\[SE = \sqrt{(\frac{2.2^2}{27}) + (\frac{2.5^2}{18})} \]

03

- Construct the 98% confidence interval for the difference between the means

The confidence interval for the difference \(D\) between two means is calculated by the formula\[D \pm z \times SE\]where \(D\) is the difference in sample means calculated in Step 1 and \(z\) is the z-value that corresponds to the desired confidence level (2.33 for a 98% confidence level). This gives:\[4 \pm 2.33 \times SE \]

04

- Perform a Hypothesis Test

To test whether the mean speed of cars driven by all men drivers on the highway is higher than that of cars driven by all women drivers, a one-tailed test can be performed. A test statistic for this test can be calculated as:\[T = \frac{\bar{X}_M - \bar{X}_W}{SE}\]which will be compared against a critical value from the t distribution table with \((n_M + n_W - 2)\) degrees of freedom at a 1% level of significance. If \(T > T_{critical}\), then it can be concluded that the mean speed of cars driven by men is significantly higher.

05

- Redo parts a and b with different standard deviations

Lastly, repeat the steps above using the new standard deviations. Early measures should be recalculated and the resultant confidence interval and hypothesis test results should be compared to the earlier results. The impact of any changes should be discussed.

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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 key concept in statistics that helps us decide if there is enough evidence to support a specific claim about a population. In this exercise, we want to check if men drive faster than women on the highway. We begin with a null hypothesis, which represents the statement we aim to challenge. Here, the null hypothesis might say that men and women drive at the same average speed on the highway. The alternative hypothesis claims the opposite – that men drive faster than women.

To test these hypotheses, we use a statistical test to determine whether the data supports rejecting the null hypothesis in favor of the alternative. We rely on a test statistic calculated from our sample data, which we'll compare against a critical value. If our test statistic is greater than the critical value, we reject the null hypothesis with high confidence and conclude that men drive faster.

Standard Error

The standard error is an important concept in understanding data variability. It measures the accuracy of a sample mean by indicating how much it can be expected to vary from the population mean. In this exercise, before forming our conclusions, we compute the standard error of the difference between the men's and the women's driving speeds.

Using the formula \[SE = \sqrt{(\frac{s^2_M}{n_M}) + (\frac{s^2_W}{n_W})}\]where

• \(s_M\) and \(s_W\) are the sample standard deviations,
• \(n_M\) and \(n_W\) are the sample sizes.

this calculation provides us with a measure of how much these sample means could differ from their actual population means just by random chance. The smaller the standard error, the more reliable the sample mean.

Sample Mean

The sample mean is the average value of a sample, offering an estimate of the population mean. In our scenario, we have two sample means: one calculated for men's driving speeds and one for women's. These sample means help provide insight into overall tendency in the larger population of men and women drivers.

The sample mean formula is\[\bar{X} = \frac{\sum X_i}{n}\]where

• \(X_i\) are the individual values in the sample,
• \(n\) is the number of observations.

In practice, this easily computed statistic serves as a basic step in many statistical procedures, including confidence intervals and hypothesis tests.

Z-Value

Z-value, often found in procedures involving the normal distribution, helps establish how many standard deviations a sample mean is from the population mean under the null hypothesis. For confidence intervals and hypothesis tests, z-values are crucial because they set the boundary for deciding whether results are statistically significant.

In this exercise, the z-value corresponds to a high confidence level (98%), which might be 2.33. Using this z-value, we can compute confidence intervals or assess hypothesis test results by comparing it to calculated test statistics.

A z-value translates real-world data into quantifiable results that state how unlikely a sample finding would be if the null hypothesis were true. This makes it an essential component of many inferential statistics procedures.