91Ó°ÊÓ

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.

Step by step solution

01

Organize the Data

Start by listing the given repair costs: 3148, 1758, 1071, 3345, 743, 2057, 663, 2637, 773, 1370.

02

Normal Probability Plot

To create a normal probability plot, use statistical software or graph paper. Plot the observed data on the vertical axis and the expected z-scores (corresponding to a standard normal distribution) on the horizontal axis. If the points approximately form a straight line, it suggests that the data may be normally distributed.

03

Draw Boxplot

Calculate the five-number summary of the data (minimum, Q1, median, Q3, maximum). Use these values to draw a boxplot. Identify potential outliers if any data points lie beyond 1.5 times the interquartile range (IQR) from the quartiles.

04

Calculate 95% Confidence Interval

First, compute the sample mean \(\bar{x}\) and sample standard deviation (s). Use the formula for the 95% confidence interval for the population mean:

05

Evaluate Sample Size Impact

With the new sample size \(n=10\), the 95% confidence interval would be narrower or wider based on sample variance and degrees of freedom. Generally, increasing sample size decreases standard error, leading to a narrower confidence interval.

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Key Concepts

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

normal probability plot

A normal probability plot helps us check if a set of data follows a normal distribution. To create this plot, we start by organizing the data in ascending order. We then plot the observed data points against the expected z-scores from a standard normal distribution.
When the points line up roughly in a straight line, it suggests that the data are normally distributed. However, significant deviations from the straight line indicate departures from normality.
This step is crucial because many statistical analyses assume normally distributed data. For our given repair costs, a good normal probability plot can be easily created using statistical software like R or Python.

boxplot

A boxplot is a graphical representation that shows the distribution of a dataset and highlights potential outliers. It's built using the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
To draw a boxplot for our repair costs:

• Calculate the five-number summary of the data.
• Draw a box from Q1 to Q3 with a line at the median.
• Extend whiskers from the box to the minimum and maximum values.

Data points that fall far from the whiskers are potential outliers. Checking for outliers is essential as they can heavily influence measures like the mean and standard deviation.

confidence interval

A confidence interval provides a range of values that likely contain the population mean. For a 95% confidence interval:

• Calculate the sample mean \(\bar{x}\) and sample standard deviation (s).
• Use the formula: \[\bar{x} \pm t_{n-1} \cdot \frac{s}{\sqrt{n}}\] where \( t_{n-1} \) is the t-value from the t-distribution with n-1 degrees of freedom.

This formula assumes a normally distributed population. The confidence interval for our repair costs helps us understand the likely range for the average repair cost across all vehicles. It's useful in making informed economic decisions.

sample size effect

The size of the sample (n) affects the width of the confidence interval. A larger sample generally reduces the standard error, which is \( \frac{s}{\sqrt{n}} \), leading to a narrower interval. Conversely, a smaller sample size increases the standard error, resulting in a wider confidence interval.
This relationship shows the importance of drawing sufficiently large samples to make precise estimates. For example, if we had a larger sample size than n=10, the 95% confidence interval for the repair costs would be narrower, increasing the precision of our estimates.