Preliminary data analyses indicate that you can reasonably...

Preliminary data analyses indicate that you can reasonably apply the $t$ -interval procedure (Procedure 8.2 on page 346 ). According to Scarborough Research, more than $85 \%$ of working adults commute by car. Of all U.S. cities, Washington, D.C., and New York City have the longest commute times. A sample of 30 commuters in the Washington, D.C., area yielded the following commute times, in minutes. $$\begin{array}{llllll} \hline 24 & 28 & 31 & 29 & 54 & 28 \\ 27 & 38 & 24 & 14 & 46 & 38 \\ 31 & 16 & 21 & 11 & 21 & 15 \\ 30 & 29 & 17 & 23 & 27 & 18 \\ 29 & 44 & 19 & 35 & 34 & 38 \\ \hline \end{array}$$ a. Find a $90 \%$ confidence interval for the mean commute time of all commuters in Washington, D.C. (Note: $\bar{x}=27.97$ minutes and $s=10.04$ minutes.) b. Interpret your answer from part (a).

Preliminary data analyses indicate that you can reasonably apply the $t$ -interval procedure (Procedure 8.2 on page 346 ).
According to Scarborough Research, more than $85 \%$ of working adults commute by car. Of all U.S. cities, Washington, D.C., and New York City have the longest commute times. A sample of 30 commuters in the Washington, D.C., area yielded the following commute times, in minutes.
$$\begin{array}{llllll}
\hline 24 & 28 & 31 & 29 & 54 & 28 \\
27 & 38 & 24 & 14 & 46 & 38 \\
31 & 16 & 21 & 11 & 21 & 15 \\
30 & 29 & 17 & 23 & 27 & 18 \\
29 & 44 & 19 & 35 & 34 & 38 \\
\hline
\end{array}$$
a. Find a $90 \%$ confidence interval for the mean commute time of all commuters in Washington, D.C. (Note: $\bar{x}=27.97$ minutes and $s=10.04$ minutes.)
b. Interpret your answer from part (a).

Key Concepts

Sample Mean and Standard Deviation

The sample mean is the average of the observed values and is used as a point estimate of the population mean, while the sample standard deviation measures the dispersion or spread of the data around the mean. Both of these statistics are essential in constructing a confidence interval, as the sample mean centers the interval and the sample standard deviation contributes to calculating the standard error, which determines the interval's width.

Confidence Interval

A confidence interval is a range of values used to estimate a population parameter, such as the mean. It is constructed around a sample statistic and provides an interval that, with a given level of confidence (e.g., 90%), is believed to contain the true population parameter. This concept is central to inferential statistics, where the goal is to make conclusions about a population based on sample data.

t-Interval Procedure

The t-interval procedure is a method used to estimate the population mean when the sample size is small and the population standard deviation is unknown. It relies on the t-distribution, which adjusts for additional uncertainty in the variability estimate, making it appropriate for situations where the normal approximation may not be accurate. This procedure is especially applicable when the data are approximately normally distributed or when the sample size is sufficiently small.

t-Distribution

The t-distribution is a probability distribution that is similar to the normal distribution but has thicker tails, which provides for a higher probability of extreme values. This characteristic accounts for the extra variability introduced when estimating the population standard deviation from a small sample. The shape of the t-distribution depends on the degrees of freedom, and as the sample size increases, it converges to the normal distribution.

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