Crash Test Results for "Mini" Cars The Insurance Institute...

Crash Test Results for "Mini" Cars The Insurance Institute for Highway Safety (IIHS) routinely conducts crash tests on vehicles to determine the cost of repairs. The following data represcnt the vehicle repair costs for 2009 ]model mini- and micro-cars resulting from front-full and rear-full crash tests at 6 miles per hour. Treat these data as a simple random sample of 14 low-impact crashes Construct and interpret a $90 \%$ confidcnce interval for the mean repair cost of a low-impact bumper crash on a mini- or micro-car. Use the normal probability plot and boxplot to assist in verifying the model requirements.

Crash Test Results for "Mini" Cars The Insurance Institute for Highway Safety (IIHS) routinely conducts crash tests on vehicles to determine the cost of repairs. The following data represcnt the vehicle repair costs for 2009 ]model mini- and micro-cars resulting from front-full and rear-full crash tests at 6 miles per hour. Treat these data as a simple random sample of 14 low-impact crashes Construct and interpret a $90 \%$ confidcnce
interval for the mean repair cost of a low-impact bumper crash on a mini- or micro-car. Use the normal probability plot and boxplot to assist in verifying the model requirements.

Key Concepts

Model Assumptions and Diagnostic Plots

Diagnostic tools like the normal probability plot and boxplot are used to assess whether the underlying assumptions of the statistical model (for example, the normality of the population distribution) are satisfied. These plots help identify potential outliers and deviations from normality, which can affect the reliability and interpretation of the confidence interval.

Simple Random Sample

A simple random sample ensures that every individual or measurement in the population has an equal chance of being selected, which minimizes bias. This assumption is fundamental in statistical inference, as it underpins the validity of the confidence interval and other estimation procedures.

Confidence Interval

A confidence interval provides a range of plausible values for a population parameter, such as the mean repair cost in this context. It is computed so that, if the same process were repeated many times, a specified percentage (90% in this case) of these intervals would contain the true mean value. This concept is key in inferential statistics as it allows us to make probabilistic statements about unknown parameters based on sample data.

Student’s t-Distribution

When the population standard deviation is unknown and the sample size is small, the t-distribution is used instead of the normal distribution for constructing confidence intervals. This distribution accounts for the extra variability inherent in estimating the population standard deviation, making it particularly useful for small samples like the one in this example.

Transcript

00:01 Let's look at an example where we're asked to construct and interpret a 90 % confidence interval for the population mean repair cost of slow impact accidents on tiny cars.

00:12 Now, with this one, data was actually given to us instead of our statistics.

00:19 So when we get to the point of working with the confidence interval, we'll refer to the graphing calculator where i've entered the data.

00:27 If you are not allowed to use a graph and calculator and you're doing this process with a just scientific calculator, you would go through the steps with that data to calculate the sample mean and sample standard deviation.

00:40 Now, one of the first things we need to do is that when we look at the data values we have, we were just given 14 data values.

00:51 So that's a small sample size.

00:52 And we weren't told at all in the problem that we had a normal distribution of the individual values.

01:02 With a small sample size, any sample size that's less than 30, we need to somehow verify that we have a normal distribution of the individual numbers so that we can use this process that actually is based on the sampling distribution of the sample means being normally distributed.

01:20 And they will be normally distributed if my population of my individual values is normally distributed.

01:28 If i don't have the population of my individual values being normally distributed, then i need to have a sample size of 30 or more.

01:38 And the central limit theorem states that even if you don't have that for your individual values population, if you have a sample size of 30 or more, that the distribution of your sample means will be, approximately normal and you still can consider.

01:54 But with 14, that's less than 30.

01:57 So what we do is we look at a probability plot.

02:02 And once we look at the probability plot and many times in these introductory courses, the probability plots will be provided to you.

02:11 But look at the probability plot and just note that all of the data values are in the bounds in the probability plot.

02:19 And then also when you look at the box pot, you want to make sure there are no outliers.

02:24 And so no outliers in the box pot.

02:28 And note that all the data values are in the bounds when you're looking at a probability bot.

02:33 And those things are verified through the pictures and diagrams that were presented.

02:40 Now let's go ahead then and find our 90 % confidence interval for the population mean.

02:49 Now remember, once we have it and i want to find the confidence interval, our lower bound is our point estimate, x bar, minus t sub alpha over two times s over the square root of n.

03:02 That's the lower bound of your confidence interval for the population mean, and x bar plus that t sub alpha over two times s over the square root of n for our upper bound.

03:18 Now, looking over at the calculator, if you push stat and edit and enter, i have preloaded the data values from earth that are associated with this particular question instead of writing them all out individually and having you watch me enter them into the calculator, i just went ahead and already did that.

03:44 On your side, you push your stat and edit and enter, you just put the first number in, push enter, put the second number in, push enter, et cetera.

03:55 Now, from here, if i push the stat, calculate one var stat, my data values are in list one.

04:06 I do not have a frequency list.

04:08 They didn't give me a table that said my data values happen more than just once each.

04:14 And then i calculate.

04:17 And notice in the report, the very top number is giving me that.

04:22 X bar is about $2 ,139 .93.

04:28 And 93 cents.

04:34 And my sample standard deviation is about $1 ,114.

04:44 So i have that from the calculator.

04:47 So that gives me for my x bar and my s...