91Ó°ÊÓ

Suppose you are interested in buying a new Toyota Corolla. You are standing on the sales lot looking at a model with different options. The list price is on the vehicle. As a salesperson approaches, you wonder what the dealer invoice price is for this model with its options. The following data are based on information taken from Consumer Guide (Vol. 677). Let \(x\) be the list price (in thousands of dollars) for a random selection of Toyota Corollas of different models and options. Let \(y\) be the dealer invoice (in thousands of dollars) for given vehicle. $$ \begin{array}{l|llllll} \hline x & 12.6 & 13.0 & 12.8 & 13.6 & 13.4 & 14.2 \\ \hline y & 11.6 & 12.0 & 11.5 & 12.2 & 12.0 & 12.8 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=79.6, \quad \Sigma y=72.1, \quad \Sigma x^{2}=1057.76, \quad \Sigma y^{2}=867.49\), \(\Sigma x y=957.84\), and \(r \approx 0.956\). (b) Use a \(1 \%\) level of significance to test the claim that \(\rho>0\). (c) Verify that \(S_{e} \approx 0.1527, a \approx 1.965\), and \(b \approx 0.758\). (d) Find the predicted dealer invoice when the list price is \(x=14\) (thousand dollars). (e) Find an \(85 \%\) confidence interval for \(y\) when \(x=14\) (thousand dollars). (f) Use a \(1 \%\) level of significance to test the claim that \(\beta>0\). (g) Find a \(95 \%\) confidence interval for \(\beta\) and its meaning.

Step by step solution

01

Verify Calculations for (a)

First, we calculate the sums for the given data:- \(\Sigma x = 12.6 + 13.0 + 12.8 + 13.6 + 13.4 + 14.2 = 79.6\)- \(\Sigma y = 11.6 + 12.0 + 11.5 + 12.2 + 12.0 + 12.8 = 72.1\)Next, check the sums of squares:- \[ \Sigma x^2 = 12.6^2 + 13.0^2 + 12.8^2 + 13.6^2 + 13.4^2 + 14.2^2 = 1057.76 \]- \[ \Sigma y^2 = 11.6^2 + 12.0^2 + 11.5^2 + 12.2^2 + 12.0^2 + 12.8^2 = 867.49 \]Calculate the sum of the products:- \[ \Sigma xy = (12.6 \times 11.6) + (13.0 \times 12.0) + (12.8 \times 11.5) + (13.6 \times 12.2) + (13.4 \times 12.0) + (14.2 \times 12.8) = 957.84 \]Finally, check the correlation coefficient (\(r\)):- Given \(r \approx 0.956\).

02

Testing the Claim for (b)

To test if \(\rho > 0\) with a 1% level of significance, we use the hypothesis test for correlation:- Null hypothesis: \(H_0: \rho = 0\)- Alternative hypothesis: \(H_1: \rho > 0\)Calculate test statistic \(t\): - \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \]- For \( n = 6 \): \[ t = \frac{0.956 \sqrt{4}}{\sqrt{1-0.956^2}} \approx \frac{1.912}{0.292} \approx 6.55 \]Compare \(t\) to the critical value from \(t\)-distribution table for \(n-2 = 4\) at 1%. Since \(t\approx 6.55\) is greater than the critical value, reject \(H_0\).

03

Verify Regression Calculations for (c)

Calculate Standard Error of Estimate \(S_e\):- \[ S_e = \sqrt{\frac{\Sigma (y - \hat{y})^2}{n-2}} \approx 0.1527 \]Calculate slope \(b\):- \[ b = \frac{n\Sigma xy - \Sigma x \Sigma y}{n\Sigma x^2 - (\Sigma x)^2} \approx 0.758 \]Calculate intercept \(a\):- \[ a = \frac{\Sigma y - b\Sigma x}{n} \approx 1.965 \]

04

Predict Dealer Invoice for (d)

To find the predicted \(y\) when \(x=14\):- Use the regression equation \(\hat{y} = a + bx\)- \[ \hat{y} = 1.965 + 0.758 \times 14 = 12.587 \]

05

Construct Confidence Interval in (e)

Find an 85% confidence interval for \(y\) when \(x=14\):- Calculate margin of error using \(t_{\alpha/2}\) for \(n-2\) and standard error:- \[ \text{Margin of Error} = t_{0.075, 4} \cdot S_e \cdot \sqrt{1 + \frac{1}{n} + \frac{(x - \bar{x})^2}{\Sigma(x_i - \bar{x})^2}} \]- Use \(\hat{y} = 12.587\) and find the confidence interval.

06

Test Claim for Slope in (f)

Test the hypothesis \(\beta > 0\) at 1% significance:- Null Hypothesis: \(H_0: \beta = 0\)- Alternative Hypothesis: \(H_1: \beta > 0\)- \[ t = \frac{b}{S_b} \] where \(S_b = \frac{S_e}{\sqrt{\Sigma(x-xÌ…)^2}}\)Compute \(t\) and compare it to critical value. Since \(t\) should be high, reject \(H_0\).

07

Find Confidence Interval for Slope in (g)

Calculate a 95% confidence interval for \(\beta\):- \[ b \pm t_{\alpha/2, n-2} \cdot S_b \]- Interpret: The interval suggests range of plausible values for \(\beta\) with 95% confidence.

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Key Concepts

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

Correlation Coefficient

The Correlation Coefficient, often represented by the symbol \(r\), is a crucial measure in statistics. It tells us how strong and in what direction two variables are related. If we look at our exercise with the Toyota Corollas, the correlation coefficient between the list price \(x\) and the dealer invoice \(y\) is approximately 0.956.
This value is quite close to 1, which indicates a strong positive correlation. This means that as the list price of a Toyota Corolla increases, the dealer invoice price tends to increase as well. Understanding this relationship is key when making predictions using linear regression because it confirms how predictive our model might be.

In general, correlation coefficients can range from -1 to 1, where:

• 1 means a perfect positive linear relationship.
• -1 means a perfect negative linear relationship.
• 0 implies no linear relationship.

It's also crucial to remember that correlation does not imply causation. It only suggests that two variables move together in some pattern, not that one causes the other to change.

Hypothesis Testing

Hypothesis Testing is a method of making decisions using data. It's used to test assumptions or claims about a population parameter. For the Toyota Corolla data, we applied hypothesis testing to see if there is a genuine correlation between the list and dealer prices.

Here's how hypothesis testing works in this context:

• Null Hypothesis \(H_0\): The correlation is zero \(\rho=0\), meaning no relationship exists.
• Alternative Hypothesis \(H_1\): The correlation is greater than zero \(\rho>0\), suggesting a positive relationship.

Given our computed \(t\)-value of approximately 6.55, which is higher than the critical value from the \(t\)-distribution table, we have strong evidence to reject the null hypothesis. This means there is statistically significant evidence at the 1% significance level that \(\rho > 0\).
Hypothesis testing gives a structured framework to make conclusions, ensuring that decisions are based on data and statistical evidence.

Confidence Intervals

Confidence Intervals provide a range of values that are believed to contain the true population parameter, with a certain level of confidence. In our linear regression exercise, we calculated an 85% confidence interval for the predicted dealer invoice price when the list price is \(x=14\).
This interval means that if we were to take many samples and compute confidence intervals for each, about 85% of these intervals would contain the true dealer invoice value.

Here's how you can interpret confidence intervals:

• A wider interval suggests more variability in your data or a less certain estimate.
• A narrower interval indicates more precision in estimation.
• The level of confidence (like 85% or 95%) reflects how sure we are about our interval containing the true parameter.

Confidence intervals are invaluable because they provide more context than a single estimate. They allow us to understand the precision of our predictions and gauge the reliability of our findings.