91Ó°ÊÓ

Stats teachers' cars A random sample of \(\mathrm{AP}^{\mathbb{R}}\) Statistics teachers was asked to report the age (in years) and mileage of their primary vehicles. A scatterplot of the data is shown at top right. Computer output from a least-squares regression analysis of these data is shown below \((\mathrm{df}=19)\). Assume that the conditions for regression inference are met. $$ \begin{aligned} &\text { Variable coef } \quad \text { SE Coef t-ratio prob }\\\ &\begin{array}{llll} \text { Constant } & 7288.54 & 6591 & 1.11 & 0.2826 \end{array}\\\ &\begin{array}{lll} \text { Car age } & 11630.6 & 1249 \quad\quad&<0.0001\\\ \end{array}\\\ &S=19280 \quad \mathrm{R}-\mathrm{Sq}=82.0 \% \quad \mathrm{RSq}(\mathrm{adj})=81.1 \% \end{aligned} $$ (a) Verify that the \(95 \%\) confidence interval for the slope of the population regression line is \((9016.4,\) $$ 14,244.8) $$ (b) A national automotive group claims that the typical driver puts 15,000 miles per year on his or her main vehicle. We want to test whether \(\mathrm{AP}^{R}\) Statistics teachers are typical drivers. Explain why an appropriate pair of hypotheses for this test is \(H_{0}: \beta=15,000\) versus \(H_{a}: \beta \neq 15,000\) (c) Compute the test statistic and \(P\) -value for the test in part (b). What conclusion would you draw at the \(\alpha=0.05\) significance level? (d) Does the confidence interval in part (a) lead to the same conclusion as the test in part (c)? Explain.

Step by step solution

01

Calculate the Confidence Interval

The 95% confidence interval for the slope \( \beta \) is calculated using the formula: \[ CI = \text{estimate} \pm (t* \times SE) \] where \( t* \) is the critical value for \( df = 19 \) at the 95% confidence level. From statistical tables, \( t* \approx 2.093 \). Therefore, the confidence interval is \[ 11630.6 \pm (2.093 \times 1249) = (9016.4, 14244.8) \].

02

Formulating Hypotheses

We want to determine if the slope \( \beta \) differs from 15000, so the hypotheses to address this question are: \( H_0: \beta = 15,000 \) and \( H_a: \beta eq 15,000 \). These hypotheses allow us to test if the age of AP Statistics teachers' cars is typical, based on typical mileage.

03

Calculate Test Statistic

The test statistic for the hypothesis test is calculated using the formula: \[ t = \frac{\text{estimate} - \text{hypothesized value}}{SE} = \frac{11630.6 - 15000}{1249} = -2.695 \].

04

Determine the P-Value

Using a t-distribution table with \( df = 19 \), the p-value corresponding to \( t = -2.695 \) is obtained. Since we're conducting a two-tailed test, the p-value is double the one-tailed p-value. The p-value is approximately 0.014.

05

Draw a Conclusion

At \( \alpha = 0.05 \), a p-value of 0.014 is less than 0.05, leading us to reject the null hypothesis. We conclude that AP Statistics teachers do not have typical driving patterns concerning mileage.

06

Compare with the Confidence Interval

The confidence interval from part (a) was \((9016.4, 14244.8)\), which does not include 15,000. This indicates that typical driving mileage (15,000 miles/year) is not plausible for AP Statistics teachers, thus confirming the conclusion from the hypothesis test.

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Key Concepts

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

Least-Squares Regression

Least-squares regression is a method used to determine the line of best fit for a set of data. This line minimizes the sum of the squares of the differences between the observed values and the values predicted by the line. In simpler terms, imagine plotting several data points on a graph and drawing a straight line through them. The goal of least-squares regression is to make sure that this line is as close to all the points as possible.

• The formula for a least-squares regression line is typically expressed as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
• The slope \( m \) indicates the direction and steepness of the line. In the context of our exercise, it represents how much mileage increases with each year of car age.
• The accuracy of this line is measured by how much the data points stray from it, usually assessed by a statistic called \( R^2 \), which shows the proportion of variability in the data that is explained by the regression line.

Confidence Interval

A confidence interval provides a range of values that likely contains a population parameter with a certain level of confidence. For the least-squares regression in our exercise, we calculated a 95% confidence interval for the slope \( \beta \).

• This interval helps us understand the reliability of our estimate. It is given by \( CI = \text{estimate} \pm (t^* \times SE) \), where \( t^* \) is the t-distribution critical value, and \( SE \) is the standard error of the estimate.
• A 95% confidence interval means that we are 95% certain that the true parameter (in this case, the average mileage per year) lies within the interval range—in this example, between 9016.4 and 14244.8 miles per year.
• By considering the confidence interval, we can make inferences about whether our sample data is representative of the population, ensuring more informed decisions are made from statistical analyses.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. In the context of our regression analysis, hypothesis testing helps determine if a certain assumption about the population parameter (for instance, average mileage) is valid.

• The null hypothesis (\( H_0 \)) represents a standard belief or default position, like "the slope is equal to 15,000 miles per year."
• The alternative hypothesis (\( H_a \)) challenges the null, often suggesting a difference exists. It states the slope is "not equal to 15,000 miles per year" in our case.
• Hypothesis testing relies on the calculation of a test statistic, which helps determine how far the sample data deviates from what is expected under the null hypothesis.

T-distribution

The t-distribution is a probability distribution used in statistics when estimating population parameters and the sample size is small, or when the population standard deviation is unknown. It is similar to the standard normal distribution but has thicker tails, meaning it accounts for more variability.

• During regression analysis with hypothesis testing, the t-distribution helps estimate the confidence intervals and determine the p-value.
• It is characterized by the degrees of freedom (df), which is generally the sample size minus one (\( n-1 \)). In our exercise, \( df = 19 \), which is used in determining the critical value \( t^* \) and p-value.
• This distribution adjusts how confidently we can say something about a population based on our sample data, particularly useful when sample sizes aren't very large.

Significance Level

The significance level \( \alpha \) is a threshold used in hypothesis testing to determine when you should reject the null hypothesis. It represents the probability of making a Type I error, which occurs when you wrongly reject a true null hypothesis. Common values for \( \alpha \) are 0.05, 0.01, or 0.10.

• In our exercise, we used \( \alpha = 0.05 \). This means if the p-value calculated during our test is less than 0.05, we reject the null hypothesis in favor of the alternative hypothesis.
• The significance level is chosen based on how much error the analyst is willing to accept: lower \( \alpha \) values reduce the chance of a Type I error but increase Type II errors, where one fails to reject a false null hypothesis.
• It serves as a benchmark for deciding whether a result is statistically significant, providing guidance for making decisions based on statistical evidence.