A tire manufacturer tested the braking performance of one of...

A tire manufacturer tested the braking performance of one of its tire models on a test track. The company tried the tires on 10 different cars, recording the stopping distance for each car on both wet and dry pavement. Results are shown in the table. a) Write a $95 \%$ confidence interval for the mean dry pavement stopping distance. Be sure to check the appropriate assumptions and conditions, and explain what your interval means. b) Write a $95 \%$ confidence interval for the mean increase in stopping distance on wet pavement. Be sure to check the appropriate assumptions and conditions, and explain what your interval means.

A tire manufacturer tested the braking performance of one of its tire models on a test track. The company tried the tires on 10 different cars, recording the stopping distance for each car on both wet and dry pavement. Results are shown in the table. a) Write a $95 \%$ confidence interval for the mean dry pavement stopping distance. Be sure to check the appropriate assumptions and conditions, and explain what your interval means.
b) Write a $95 \%$ confidence interval for the mean increase in stopping distance on wet pavement. Be sure to check the appropriate assumptions and conditions, and explain what your interval means.

Key Concepts

Paired Samples (Matched-Pairs) Design

A paired samples design involves collecting two related measurements on the same subjects, allowing for comparison of two conditions while controlling for variability among subjects. This design is particularly useful when analyzing the difference between two treatments or conditions, as it uses the within-subject differences, which often results in increased statistical power.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is believed to contain the true value of a population parameter with a specified level of confidence. It conveys both an estimate of the parameter and the uncertainty associated with that estimate, allowing statisticians to infer about the population from the sample data.

t-Distribution

The t-distribution is used in constructing confidence intervals when the sample size is small and the population standard deviation is unknown. It accounts for additional uncertainty due to estimating the standard deviation from the sample, and its shape depends on the number of degrees of freedom, becoming more like the normal distribution as the sample size increases.

Assumptions and Conditions for Inference

Statistical inference using confidence intervals, especially with small samples, relies on assumptions such as independence of observations, roughly normal distribution of the data, and appropriate use of the t-distribution when the population standard deviation is unknown. These assumptions ensure the validity and reliability of the confidence interval estimates.

Transcript

00:01 For this problem, we're given a table with a bunch of different cars in it.

00:06 And for each car, the stopping distance was tested on dry pavement and wet pavement, and the stopping distance was recorded in here.

00:15 We want two things.

00:17 First, we want a 95 % confidence interval for the mean of the dry stopping distance.

00:22 That's just this column here.

00:25 And then we want a 95 % confidence interval for the increase from dry pavement to wet pavement in the stopping distance.

00:33 For both of these, we need to check our assumptions and then interpret our interval that we find.

00:38 So let's start with the first one.

00:40 What i've done over here on the calculator is already put the two lists into my calculator.

00:47 So these are, or list one is all the dry pavement stopping distances.

00:51 List two is the wet pavement stopping distances.

00:55 And remember, these are, like for example, the first row is the same car, measured first on dry, then on wet pavement.

01:02 As you can see in all of them, the wet pavement stopping distance is longer, which makes sense.

01:07 If you think about a wet road, it's going to take longer for your car to stop when you break.

01:12 But right now, we just want to look at the dry pavement data.

01:16 And we want to find a 95 % confidence interval for the mean.

01:20 So what that means is we're going to do a test, right? we're going to stat over the test.

01:26 And we just want a t interval because we are using one list.

01:33 It's not a two sample t test.

01:36 That would be if we were looking at both the wet and dry.

01:40 But we just want list one, confidence interval of 95%, and then we're going to calculate.

01:48 All right, so here's the interval.

01:50 I'll write it over here.

01:53 It's from 131 .79 to 145 .61 .61 .61 .61 .61 .61 .61.

02:03 You'll notice we're also given the mean for our data set.

02:07 And the reason we turn that into an interval or a confidence interval is because we want to generalize it to all cars.

02:16 So if we're generalizing to all cars, we know that it's not going to always be exactly 138 .7.

02:22 But we're 95 % confident it's going to be between these two stopping distance amounts.

02:30 So let's just double check our assumptions.

02:32 First, that it was random.

02:33 So if these cars were randomly chosen, we're just going to assume that, then we're good.

02:39 Is it independent? well, one car's stopping distance isn't going to affect another car's stopping distance, right? so we know that each car is independent of the other cars.

02:49 Nearly normal.

02:50 We can check this in our calculator.

02:52 All we have to do is second y equals or stat plot.

02:56 We go to plot one.

02:59 I already have it on a histogram, list one, frequency one, colored.

03:04 Doesn't quite matter.

03:06 So for this one, in order to see it, you need to go to zoom, down to zoom stat.

03:12 And here is our graph.

03:14 Now you're going to notice this is not nearly normal...