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An automotive researcher wanted to estimate the difference in distance required to come to a complete stop while traveling 40 miles per hour on wet versus dry pavement. Because car type plays a role, the researcher used eight different cars with the same driver and tires. The braking distance (in feet) on both wet and dry pavement is shown in the table below. Construct a \(95 \%\) confidence interval for the mean difference in braking distance on wet versus dry pavement where the differences are computed as "wet minus dry." Interpret the interval. Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. $$ \begin{array}{lcccccccc} \text { Car } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \hline \text { Wet } & 106.9 & 100.9 & 108.8 & 111.8 & 105.0 & 105.6 & 110.6 & 107.9 \\ \hline \text { Dry } & 71.8 & 68.8 & 74.1 & 73.4 & 75.9 & 75.2 & 75.7 & 81.0 \\\ \hline \end{array} $$

Step by step solution

01

Calculate the Differences

Subtract each dry braking distance from the corresponding wet braking distance to find the differences. The differences are:1. 106.9 - 71.8 = 35.12. 100.9 - 68.8 = 32.13. 108.8 - 74.1 = 34.74. 111.8 - 73.4 = 38.45. 105.0 - 75.9 = 29.16. 105.6 - 75.2 = 30.47. 110.6 - 75.7 = 34.98. 107.9 - 81.0 = 26.9

02

Calculate the Mean Difference

To find the mean difference, add all the differences together and divide by the number of cars:\[\overline{d} = \frac{35.1 + 32.1 + 34.7 + 38.4 + 29.1 + 30.4 + 34.9 + 26.9}{8} = \frac{261.6}{8} = 32.7\]

03

Calculate the Standard Deviation of Differences

First, find the squared differences from the mean for each data point, then calculate the variance:\[\text{Variance} = \frac{\sum (d_i - \overline{d})^2}{n-1}\]Calculations for each car:1. (35.1 - 32.7)^2 = 5.762. (32.1 - 32.7)^2 = 0.363. (34.7 - 32.7)^2 = 4.004. (38.4 - 32.7)^2 = 32.495. (29.1 - 32.7)^2 = 12.966. (30.4 - 32.7)^2 = 5.297. (34.9 - 32.7)^2 = 4.848. (26.9 - 32.7)^2 = 33.64Variance: \[\text{Variance} = \frac{5.76 + 0.36 + 4.00 + 32.49 + 12.96 + 5.29 + 4.84 + 33.64}{7} = \frac{99.34}{7} 鈮� 14.19\]Standard Deviation: \[S_d = \sqrt{14.19} 鈮� 3.77\]

04

Calculate the Standard Error

The standard error (SE) is given by:\[SE = \frac{S_d}{\sqrt{n}} = \frac{3.77}{\sqrt{8}} = \frac{3.77}{2.83} 鈮� 1.33\]

05

Find the Critical t-Value

For a 95% confidence interval and 7 degrees of freedom (n-1), the critical t-value (from t-table or calculator) is approximately 2.365.

06

Calculate the Margin of Error

The margin of error (ME) is calculated by multiplying the critical t-value by the standard error:\[ME = t_{\alpha/2} \times SE = 2.365 \times 1.33 鈮� 3.14\]

07

Construct the Confidence Interval

The 95% confidence interval for the mean difference is given by:\[\overline{d} \pm ME = 32.7 \pm 3.14\]Therefore, the confidence interval is (29.56, 35.84).

08

Interpret the Interval

With 95% confidence, the mean difference in braking distances on wet versus dry pavement (computed as 'wet minus dry') is between 29.56 feet and 35.84 feet.

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Key Concepts

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

braking distance

Braking distance is the distance a vehicle travels after the brakes are applied and before it comes to a complete stop. It's crucial for ensuring safety on the roads.
Several factors can influence braking distance, including:

• Vehicle speed
• Brake condition
• Type of tires
• Road conditions

In this exercise, we're looking at the braking distances of eight different cars traveling at 40 miles per hour on both wet and dry pavement. By comparing these distances, we can understand how different road conditions impact safety.

wet versus dry pavement

When it comes to braking, road conditions make a huge difference. Wet pavement can significantly increase braking distances compared to dry pavement. This happens because:

• Less friction between the tires and the road
• Hydroplaning risk, where tires lose contact with the road due to a layer of water
• Longer reaction times

In our exercise, we're comparing the braking distances on wet and dry pavement for the same cars. This comparison helps us quantify how much additional distance is required to stop safely on a wet road.

mean difference calculation

To determine the average impact of wet versus dry pavement on braking distances, we calculate the 'mean difference.'
This involves:

• Subtracting each dry braking distance from the corresponding wet distance (wet - dry)
• Summing up all these differences
• Dividing the sum by the number of cars to get the mean

The mean difference helps us understand the average change in braking distance when going from dry to wet pavement.

standard deviation

Standard deviation measures how much the differences in our braking distances vary around the mean difference.
This involves:

• Calculating each difference from the mean
• Squaring these values
• Summing them up to find the variance
• Taking the square root of the variance to get the standard deviation

A higher standard deviation indicates more variability in our braking differences, while a lower standard deviation indicates they are closer to the mean.

critical t-value

In statistics, the critical t-value helps us establish a confidence interval.
It's a value derived from the Student's t-distribution and depends on:

• Desired confidence level (in this case, 95%)
• Degrees of freedom (number of cars minus one)

In our exercise, with 7 degrees of freedom (8 cars minus 1), and aiming for a 95% confidence interval, the critical t-value is approximately 2.365. This value is multiplied by the standard error to find the margin of error, which allows us to construct the confidence interval for the mean difference.