Twenty people in one large metropolitan area were asked to...

Twenty people in one large metropolitan area were asked to record the time (in minutes) that it takes them to drive to work. These times were as follows: $\begin{array}{llllllllll}30 & 42 & 35 & 40 & 45 & 22 & 32 & 15 & 41 & 45 \\ 28 & 32 & 45 & 27 & 47 & 50 & 30 & 25 & 46 & 25\end{array}$ a. Calculate the standard error. b. Find $t_{v, \alpha / 2}$ for a $95 \%$ confidence interval for the true population mean. c. Calculate the width for a $95 \%$ confidence interval for the population mean time spent driving to work.

Twenty people in one large metropolitan area were asked to record the time (in minutes) that it takes them to drive to work. These times were as follows:
$\begin{array}{llllllllll}30 & 42 & 35 & 40 & 45 & 22 & 32 & 15 & 41 & 45 \\ 28 & 32 & 45 & 27 & 47 & 50 & 30 & 25 & 46 & 25\end{array}$
a. Calculate the standard error.
b. Find $t_{v, \alpha / 2}$ for a $95 \%$ confidence interval for the true population mean.
c. Calculate the width for a $95 \%$ confidence interval for the population mean time spent driving to work.

Key Concepts

Standard Error

The standard error is a measure of the variability of a sample statistic, typically the sample mean, from the true population parameter. It is calculated by taking the sample standard deviation and dividing it by the square root of the sample size. This concept is essential in inferential statistics as it helps to gauge the reliability of an estimate and is used to construct confidence intervals around the sample mean.

t-Distribution

The t-distribution is a probability distribution used in estimating population parameters when the sample size is relatively small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the additional uncertainty in the estimate of the standard deviation. This distribution is key for constructing confidence intervals and hypothesis tests when working with small samples.

Confidence Interval

A confidence interval is a range of values, derived from the sample statistics, that is believed to contain the true population parameter with a specified level of confidence, such as 95%. It is constructed using the sample mean, the standard error, and the critical value from the appropriate probability distribution (like the t-distribution for small samples). The width of the confidence interval reflects the precision of the estimate—the narrower the interval, the more precise the estimate.

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