The following data represent the repair cost for a...

The following data represent the repair cost for a low-impact collision in a simple random sample of mini- and micro-vehicles (such as the Chevrolet Aveo or Mini Cooper). $$ \begin{array}{lrlrr} \hline \$ 3148 & \$ 1758 & \$ 1071 & \$ 3345 & \$ 743 \\ \hline \$ 2057 & \$ 663 & \$ 2637 & \$ 773 & \$ 1370 \\ \hline \end{array} $$ (a) Draw a normal probability plot to determine if it is reasonable to conclude the data come from a population that is normally distributed. (b) Draw boxplot to check for outliers. (c) Construct and interpret a $95 \%$ confidence interval for the population mean cost of repair. (d) Suppose you obtain a simple random sample of size $n=10$ of a Mini Cooper that was in a low-impact collision and determine the cost of repair. Do you think a $95 \%$ confidence interval would be wider or narrower? Explain.

The following data represent the repair cost for a low-impact collision in a simple random sample of mini- and micro-vehicles (such as the Chevrolet Aveo or Mini Cooper).
$$
\begin{array}{lrlrr}
\hline \$ 3148 & \$ 1758 & \$ 1071 & \$ 3345 & \$ 743 \\
\hline \$ 2057 & \$ 663 & \$ 2637 & \$ 773 & \$ 1370 \\
\hline
\end{array}
$$
(a) Draw a normal probability plot to determine if it is reasonable to conclude the data come from a population that is normally distributed.
(b) Draw boxplot to check for outliers.
(c) Construct and interpret a $95 \%$ confidence interval for the population mean cost of repair.
(d) Suppose you obtain a simple random sample of size $n=10$ of a Mini Cooper that was in a low-impact collision and determine the cost of repair. Do you think a $95 \%$ confidence interval would be wider or narrower? Explain.

Key Concepts

Sample Size Effects on the Confidence Interval

The width of a confidence interval is influenced by the sample size, among other factors. A larger sample size generally results in a smaller standard error, leading to a narrower confidence interval. Conversely, a smaller sample size yields a larger standard error, which in turn produces a wider confidence interval, indicating less precision in the estimation of the population mean.

Confidence Interval

A confidence interval for the mean is a range of values, derived from sample statistics, which is likely to contain the population mean with a specified level of confidence, in this case 95%. It is constructed based on the sample mean, standard error, and the appropriate critical value from the t-distribution or z-distribution, depending on sample size and whether the population standard deviation is known.

Normal Probability Plot

A normal probability plot, also known as a Q-Q plot, is a graphical technique for assessing whether a set of data approximately follows a normal distribution. It does this by plotting the quantiles of the data against the quantiles of the normal distribution and evaluating how closely the points fall along a straight line. If the points lie approximately along a straight line, this suggests that the data come from a normally distributed population.

Boxplot

A boxplot is a graphical summary that displays the distribution of data based on a five-number summary: minimum, first quartile, median, third quartile, and maximum. It is particularly useful for identifying potential outliers and understanding the central tendency and spread of the data. Outliers are typically shown as points that fall outside the 'whiskers' of the boxplot.

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