Records of 40 used passenger cars and 40 used pickup trucks...

Records of 40 used passenger cars and 40 used pickup trucks (none used commercially) were randomly selected to investigate whether there was any difference in the mean time in years that they were kept by the original owner before being sold. For cars the mean was 5.3 years with standard deviation 2.2 years. For pickup trucks the mean was 7.1 years with standard deviation 3.0 years. a. Construct the $95 \%$ confidence interval for the difference in the means based on these data. b. Test the hypothesis that there is a difference in the means against the null hypothesis that there is no difference. Use the $1 \%$ level of significance. c. Compute the observed significance of the test in part (b).

Records of 40 used passenger cars and 40 used pickup trucks (none used commercially) were randomly
selected to investigate whether there was any difference in the mean time in years that they were kept by
the original owner before being sold. For cars the mean was 5.3 years with standard deviation 2.2 years. For
pickup trucks the mean was 7.1 years with standard deviation 3.0 years.
a. Construct the $95 \%$ confidence interval for the difference in the means based on these data.
b. Test the hypothesis that there is a difference in the means against the null hypothesis that there
is no difference. Use the $1 \%$ level of significance.
c. Compute the observed significance of the test in part (b).

Key Concepts

Standard Error of the Difference Between Means

The standard error quantifies the variability in the sampling distribution of the difference between two independent sample means. It is calculated by combining the individual sample variances (or standard deviations) and the sample sizes. This value is crucial for both constructing confidence intervals and performing hypothesis tests in two-sample scenarios.

Hypothesis Testing for Two Independent Means

Hypothesis testing in this context involves setting up a null hypothesis that there is no difference in the population means and an alternative hypothesis that there is a difference. The test statistic derived from the difference between sample means is compared to a critical value or used to compute a p-value, which determines whether the evidence is strong enough to reject the null hypothesis.

Two-Sample t-Test

This is a statistical method used to compare the means of two independent samples. In such tests, the test statistic is calculated by taking the difference between the sample means and then standardizing that difference by the estimated standard error. It uses the t-distribution to account for the sample size and variability when the population standard deviations are unknown.

Confidence Interval for Difference in Means

A confidence interval provides a range of plausible values for the difference between two population means. It is constructed by taking the difference of the sample means and adding and subtracting the margin of error, which is obtained from the critical value of the t-distribution multiplied by the standard error of the difference.

Significance Level and p-Value

The significance level is a predetermined threshold (e.g., 1% in this case) below which the probability of observing the test statistic, assuming the null hypothesis is true, is deemed too low, leading to its rejection. The p-value quantifies the exact probability of observing data as extreme as, or more extreme than, that observed under the null hypothesis. Comparing the p-value to the significance level helps in making a decision about the hypothesis test.

Transcript

00:01 The following is the solution to number 14, and this looks at used passenger cars versus pickup trucks, and we're looking at the difference in mean time used by the original owner before that either passenger car or pickup truck is being sold.

00:13 So here, see, this is the normal car.

00:16 So there were 40 of them.

00:17 The average lifespan or the average time that the original owner had it was 5 .3 years.

00:21 Standard deviation 2 .2, and then for the pickup trucks, still the same sample size.

00:26 And 7 .1 is the mean and 3 .0 is the standard deviation.

00:30 We're asked to find the 95 % confidence interval.

00:33 So i'm going to use the ti -84 just because it goes by a little bit more quickly.

00:36 If we go to stat and tests, we're going to go to this ninth option here, the two -sample -z interval.

00:41 The reason why we're using the z interval is because the sample sizes are large enough.

00:45 So as long as it's bigger than 40, then you can pretty much use the z or the t.

00:48 It doesn't really matter.

00:50 Now, for sigma -1 and sigma -2, those are your standard deviation.

00:52 So just think of those as s's for now.

00:55 And so you can just punch in your data there and then the c level is .95 because we're asked to find the 95 % confidence interval.

01:02 And then whenever we calculate, this gives us our answer there.

01:06 So notice zero is not contained in that interval.

01:08 So negative 2 .9 to negative .64...

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