Questions: How does elevation affect average January temperature? The following data reports elevation in feet (x) and average January temperature in degrees Fahrenheit (y) for 10 US cities. (a) Calculate the correlation coefficient, r. Round answer to THREE decimal places. r= type your answer... (b) Does this correlation suggest there is a linear relation between elevation and average January temperature? (For the first answer blank, write Yes or No. For the second answer blank, write greater or less. For the third blank, write the appropriate critical value from Table II.) type your answer..., since the absolute value of the correlation coefficient is type your answer... than the critical value of type your answer.. (c) Find the equation of the regression line. Round slope and y-intercept to THREE decimal places. y= type your answer... x+ type your answer... (d) Use the regression equation to predict the value of y at x=7000 ft. Round answer to one decimal place, if appropriate. If you should not predict, write 'outside scope' in the answer blank. The predicted average January temperature for 7000 ft is type your answer... degrees Fahrenheit. (e) Use the regression equation to predict the value of y at x=800 ft. Round answer to one decimal place, if appropriate. If you should not predict, write 'outside scope' in the answer blank. The predicted average January temperature for 800 ft is type your answer... degrees Fahrenheit.

How does elevation affect average January temperature? The following data reports elevation in feet (x) and average January temperature in degrees Fahrenheit (y) for 10 US cities. (a) Calculate the correlation coefficient, r. Round answer to THREE decimal places. r= type your answer... (b) Does this correlation suggest there is a linear relation between elevation and average January temperature? (For the first answer blank, write Yes or No. For the second answer blank, write greater or less. For the third blank, write the appropriate critical value from Table II.) type your answer..., since the absolute value of the correlation coefficient is type your answer... than the critical value of type your answer.. (c) Find the equation of the regression line. Round slope and y-intercept to THREE decimal places. y= type your answer... x+ type your answer... (d) Use the regression equation to predict the value of y at x=7000 ft. Round answer to one decimal place, if appropriate. If you should not predict, write 'outside scope' in the answer blank. The predicted average January temperature for 7000 ft is type your answer... degrees Fahrenheit. (e) Use the regression equation to predict the value of y at x=800 ft. Round answer to one decimal place, if appropriate. If you should not predict, write 'outside scope' in the answer blank. The predicted average January temperature for 800 ft is type your answer... degrees Fahrenheit.
Transcript text: How does elevation affect average January temperature? The following data reports elevation in feet ( $x$ ) and average January temperature in degrees Fahrenheit ( $y$ ) for 10 US cities. (a) Calculate the correlation coefficient, $r$. Round answer to THREE decimal places. $r=$ $\square$ type your answer... (b) Does this correlation suggest there is a linear relation between elevation and average January temperature? (For the first answer blank, write Yes or No. For the second answer blank, write greater or less. For the third blank, write the appropriate critical value from Table II.) type your answer... $\square$ , since the absolute value of the correlation coefficient is $\square$ type your answer... than the critical value of $\square$ type your answer.. (c) Find the equation of the regression line. Round slope and $y$-intercept to THREE decimal places. $\hat{y}=$ $\square$ type your answer... $x+$ $\square$ type your answer... (d) Use the regression equation to predict the value of $y$ at $x=7000 \mathrm{ft}$. Round answer to one decimal place, if appropriate. If you should not predict, write 'outside scope' (without the quotes) in the answer blank. The predicted average January temperature for 7000 ft is $\square$ type your answer... degrees Fahrenheit. (e) Use the regression equation to predict the value of $y$ at $x=800 \mathrm{ft.}$ Round answer to one decimal place, if appropriate. If you should not predict, write 'outside scope' (without the quotes) in the answer blank. The predicted average January temperature for 800 ft is $\square$ type your answer... degrees Fahrenheit.
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Solution

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Solution Steps

To solve the given problem, we need to follow these steps:

(a) Calculate the correlation coefficient, \( r \), which measures the strength and direction of the linear relationship between elevation and average January temperature.

(b) Determine if the correlation coefficient suggests a linear relationship by comparing its absolute value to a critical value from a statistical table.

(c) Find the equation of the regression line, which involves calculating the slope and y-intercept of the best-fit line through the data points.

Solution Approach

  1. Calculate the correlation coefficient \( r \):

    • Use the formula for the Pearson correlation coefficient.

  2. Determine if there is a linear relationship:

    • Compare the absolute value of \( r \) to a critical value from a statistical table.

  3. Find the equation of the regression line:

    • Use the least squares method to calculate the slope and y-intercept.
Step 1: Calculate the Correlation Coefficient

The correlation coefficient \( r \) measures the strength and direction of the linear relationship between elevation (\( x \)) and average January temperature (\( y \)). Using the given data, we find: \[ r = -0.811 \]

Step 2: Determine if There is a Linear Relationship

To determine if there is a linear relationship, we compare the absolute value of \( r \) to the critical value from a statistical table. For 10 data points, the degrees of freedom is \( 10 - 2 = 8 \). The critical value for 8 degrees of freedom at a 0.05 significance level is approximately \( 0.632 \).

Since \( |r| = 0.811 \) is greater than \( 0.632 \), we conclude that there is a linear relationship.

Step 3: Find the Equation of the Regression Line

The equation of the regression line is given by: \[ \hat{y} = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. Using the least squares method, we find: \[ m = -0.003 \] \[ b = 43.605 \]

Thus, the equation of the regression line is: \[ \hat{y} = -0.003x + 43.605 \]

Final Answer

(a) The correlation coefficient \( r \) is: \[ \boxed{r = -0.811} \]

(b) There is a linear relationship: \[ \boxed{\text{Yes, since the absolute value of the correlation coefficient is greater than the critical value of } 0.632} \]

(c) The equation of the regression line is: \[ \boxed{\hat{y} = -0.003x + 43.605} \]

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