91Ó°ÊÓ

The following data show the retail price for 12 randomly selected laptop computers along with their corresponding processor speeds in gigahertz. $$ \begin{array}{|ccr|ccc|} \hline \text { Computer } & \text { Speed } & \text { Price } & \text { Computer } & \text { Speed } & \text { Price } \\ \hline 1 & 2.0 & \$ 1008.50 & 7 & 2.0 & \$ 1098.50 \\ 2 & 1.6 & 461.00 & 8 & 1.6 & 693.50 \\ 3 & 1.6 & 532.00 & 9 & 2.0 & 1057.00 \\ 4 & 1.8 & 971.00 & 10 & 1.6 & 1001.00 \\ 5 & 2.0 & 1068.50 & 11 & 1.0 & 468.50 \\ 6 & 1.2 & 506.00 & 12 & 1.4 & 434.50 \\ \hline \end{array} $$ a. Develop a linear equation that can be used to describe how the price depends on the processor speed b. Based on your regression equation, is there one machine that seems particularly over- or underpriced? c. Compute the correlation coefficient between the two variables. At the .05 significance level, conduct a test of hypothesis to determine if the population correlation is greater than zero.

Step by step solution

01

Organize the Data into Two Sets

First, separate the given data into two sets. One set for the processor speeds and another for corresponding prices. Speeds \( (X) \): 2.0, 1.6, 1.6, 1.8, 2.0, 1.2, 2.0, 1.6, 2.0, 1.6, 1.0, 1.4Prices \( (Y) \): 1008.50, 461.00, 532.00, 971.00, 1068.50, 506.00, 1098.50, 693.50, 1057.00, 1001.00, 468.50, 434.50

02

Calculate the Means

Calculate the mean (average) of the processor speeds and the mean of the prices. This will help in calculating the slope (\(b\)) and intercept (\(a\)) for the regression line later.

03

Compute Slope of Regression Line \( b \)

Use the formula for the slope \( b \) of the linear regression equation: \[ b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \] Substitute the values from the data to calculate \( b \).

04

Compute Intercept of Regression Line \( a \)

Calculate the intercept \( a \) using the formula: \[ a = \bar{Y} - b \bar{X} \] where \( \bar{Y} \) and \( \bar{X} \) are the means of Prices and Speeds, respectively.

05

Formulate the Regression Equation

The linear regression equation can be formed as:\[ Y = a + bX \] Plug the calculated values of \( a \) and \( b \) into this equation.This equation describes how the price depends on the processor speed.

06

Analyze for Over- or Underpriced Machines

Use the regression equation to calculate the expected price of each computer based on its speed. Compare these expected prices with actual prices to identify any machines that are significantly over- or underpriced.

07

Calculate Correlation Coefficient \( r \)

Use the formula for the correlation coefficient \( r \):\[ r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \cdot \sum (Y_i - \bar{Y})^2}} \] Calculate \( r \) using the data.

08

Hypothesis Testing on Correlation

1. **State the hypotheses:** - Null hypothesis \( H_0: \rho \leq 0 \) - Alternative hypothesis \( H_a: \rho > 0 \)2. **Calculate the test statistic using:** \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \]3. **Find the critical t-value** from the t-distribution table at \( \alpha = 0.05 \) for \( n - 2 \) degrees of freedom.4. **Conclusion:** Compare the test statistic with the critical value to determine if the population correlation is greater than zero.

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Key Concepts

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

Linear Regression Equation

Linear regression aims to model the relationship between two variables by fitting a linear equation to observed data. In this exercise, we explore the relationship between the processor speed of laptops (in gigahertz) and their prices. The regression line is characterized by the equation \[Y = a + bX\]where:

• \( Y \) is the dependent variable (Price)
• \( X \) is the independent variable (Speed)
• \( a \) is the y-intercept of the line
• \( b \) is the slope of the line

To find the slope \( b \) and intercept \( a \), we first calculate the means of the speeds and prices, denoted \( \bar{X} \) and \( \bar{Y} \), respectively. The slope \( b \) is calculated using:\[b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}\]The intercept \( a \) is computed by:\[a = \bar{Y} - b \bar{X}\]Plugging the values of \( a \) and \( b \) into the regression equation allows predicting the price for any given processor speed. This model helps us identify if certain laptops are over- or underpriced by comparing expected prices with actual prices.

Correlation Coefficient

The correlation coefficient \( r \) is a statistical measure that indicates the strength and direction of a linear relationship between two variables. It ranges from -1 to 1:

• Values near 1 imply a strong positive correlation.
• Values near -1 imply a strong negative correlation.
• Values around 0 imply little to no linear relationship.

In our context, \( r \) informs us how well processor speed explains the variability in laptop prices. The formula to compute \( r \) is:\[r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \cdot \sum (Y_i - \bar{Y})^2}}\]This formula quantifies the extent to which the two variables move in tandem. Once calculated, the correlation coefficient helps in assessing how meaningful the linear regression model is, and if a higher processor speed generally corresponds to higher prices.

Hypothesis Testing

Hypothesis testing allows us to determine whether there is enough statistical evidence to support a specific belief about a population parameter, such as the correlation coefficient \( \rho \). Here, we test whether the population correlation between processor speed and price is greater than zero, indicating a positive relationship:- **Null Hypothesis \( H_0 \):** \( \rho \leq 0 \) - **Alternative Hypothesis \( H_a \):** \( \rho > 0 \)The test statistic \( t \) for correlation is calculated using:\[t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}}\]Where \( n \) is the number of data pairs. We compare this test statistic against a critical value from the t-distribution table at a significance level \( \alpha = 0.05 \) with \( n - 2 \) degrees of freedom. If the test statistic is greater than the critical value, we reject the null hypothesis, indicating statistically significant evidence that the population correlation is indeed greater than zero.