For another test of the tires in Exercise 31 , a car made...

For another test of the tires in Exercise 31 , a car made repeated stops from 60 miles per hour. The test was run on both dry and wet pavement, with results as shown in the table. (Note that actual braking distance, which takes into account the driver's reaction time, is much longer, typically nearly 300 feet at $60 \mathrm{mph} !$ ) a) Write a $95 \%$ confidence interval for the mean dry pavement stopping distance. Be sure to check the appropriate assumptions and conditions, and explain what your interval means. b) Write a $95 \%$ confidence interval for the mean increase in stopping distance on wet pavement. Be sure to check the appropriate assumptions and conditions, and explain what your interval means. $$\begin{array}{|c|c|} \hline {\text { Stopping Distance (ft) }} \\\begin{array}{c} \text { Dry } \\\text { Pavement }\end{array} & \begin{array}{c}\text { Wet } \\\text { Pavement } \end{array} \\\hline 145 & 211 \\152 & 191 \\141 & 220 \\143 & 207 \\131 & 198 \\ 148 & 208 \\126 & 206 \\140 & 177 \\135 & 183 \\ 133 & 223 \\\hline\end{array}$$

For another test of the tires in Exercise 31 , a car made repeated stops from 60 miles per hour. The test was run on both dry and wet pavement, with results as shown in the table. (Note that actual braking distance, which takes into account the driver's reaction time, is much longer, typically nearly 300 feet at $60 \mathrm{mph} !$ )
a) Write a $95 \%$ confidence interval for the mean dry pavement stopping distance. Be sure to check the appropriate assumptions and conditions, and explain what your interval means.
b) Write a $95 \%$ confidence interval for the mean increase in stopping distance on wet pavement. Be sure to check the appropriate assumptions and conditions, and explain what your interval means.
$$\begin{array}{|c|c|}
\hline {\text { Stopping Distance (ft) }} \\\begin{array}{c}
\text { Dry } \\\text { Pavement }\end{array} & \begin{array}{c}\text { Wet } \\\text { Pavement }
\end{array} \\\hline 145 & 211 \\152 & 191 \\141 & 220 \\143 & 207 \\131 & 198 \\
148 & 208 \\126 & 206 \\140 & 177 \\135 & 183 \\
133 & 223 \\\hline\end{array}$$

Key Concepts

Interpretation of Confidence Intervals

The interpretation of a confidence interval involves understanding that it is computed from sample data and, if the sampling process were repeated many times, a specified percentage (such as 95%) of the calculated intervals would contain the true population parameter. It does not imply that there is a 95% probability that any one computed interval contains the parameter, but rather that the method is reliable in the long run.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. It provides an estimate along with a level of certainty (e.g., 95%) that the interval includes the parameter, reflecting the precision and reliability of the estimate.

t-Distribution

The t-distribution is used in situations where the sample size is small and the population standard deviation is unknown. It accounts for extra uncertainty by having heavier tails than the normal distribution, and is pivotal in constructing confidence intervals for the mean under these conditions.

Assumptions for Inference

Key assumptions for constructing valid confidence intervals include random sampling, independence of observations, and the approximate normality of the underlying distribution. For small sample sizes, checking the normality assumption is particularly important to ensure the validity of the interval estimation.

Paired Data Analysis

Paired data analysis involves comparing two related measurements taken on the same unit or subject, such as stopping distances on dry versus wet pavement. This method controls for between-subject variability by focusing on the difference between paired observations, which is then used to construct a confidence interval for the mean difference.

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