Exam 11: Simple Linear Regression

Consider the following pairs of measurements: x 1 3 4 6 7 y 3 6 8 12 13 a. Construct a scattergram for the data. b. What does the scattergram suggest about the relationship between $x$ and $y$ ? c. Find the least squares estimates of $\beta _ { 0 }$ and $\beta _ { 1 }$ . d. Plot the least squares line on your scattergram. Does the line appear to fit the data well?

What is the relationship between diamond price and carat size? 307 diamonds were sampled (ranging in size from 0.18 to 1.1 carats)and a straight-line relationship was hypothesized between Y = diamond price (in dollars)and x = size of the diamond (in carats). The simple linear regression For the analysis is shown below: Least Squares Linear Regression of PRICE Predictor Interpret the estimated $y$ -intercept of the regression line.

Is there a relationship between the raises administrators at State University receive and their performance on the job? A faculty group wants to determine whether job rating $( x )$ is a useful linear predictor of raise $( y )$ . Consequently, the group considered the straight-line regression model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x .$ Using the method of least squares, the faculty group obtained the following prediction equation: $\hat { y } = 14,000 - 2,000 x$ Interpret the estimated slope of the line.

A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses in a section of the county called East Meadow. One of the many variables thought to be An important predictor of appraised value is the number of rooms in the house. Consequently, the Appraiser decided to fit the simple linear regression model: $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ where $\mathrm { y } =$ appraised value of the house (in thousands of dollars) and $x =$ number of rooms. Using data collected for a sample of $n = 74$ houses in East Meadow, the following results were obtained: $\hat { y } = 74.80 + 19.72 x \quad R = 0.539 \quad R ^ { 2 } = 0.290 \quad s = 58.031$ Give a practical interpretation of the estimate of $\sigma$ , the standard deviation of the random error term in the model.

For the situation above, give a practical interpretation of $\hat { \beta } _ { 1 } = 228$ .

If a least squares line were determined for the data set in each scattergram, which would have the smallest variance?

(0, 3)and (3, 0)

The coefficient of correlation is a useful measure of the linear relationship between two variables.

Consider the data set shown below. Find the 95% confidence interval for the slope of the regression line. 0 3 2 3 8 10 11 -2 0 2 4 6 8 10

In team-teaching, two or more teachers lead a class. An researcher tested the use of team-teaching in mathematics education. Two of the variables measured on each sample of 165 mathematics teachers were years of teaching experience x and reported success rate y (measured as a percentage)of team-teaching mathematics classes. a. The researcher hypothesized that mathematics teachers with more years of experience will report higher perceived success rates in team-taught classes. State this hypothesis in terms of the parameter of a linear model relating x to y. b. The correlation coefficient for the sample data was reported as r = -0.31. Interpret this result. c. Does the value of r support the hypothesis? Test using α = .05.

A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses in a section of the county called East Meadow. One of the many variables thought to be An important predictor of appraised value is the number of rooms in the house. Consequently, the Appraiser decided to fit the simple linear regression model: $E ( y ) = \beta _ { 0 } + \beta 1 ^ { x }$ where $y =$ appraised value of the house (in thousands of dollars) and $x =$ number of rooms. Using data collected for a sample of $n = 74$ houses in East Meadow, the following results were obtained: $\hat { y } = 74.80 + 19.72 x$ Give a practical interpretation of the estimate of the $y$ -intercept of the least squares line.

(4, 3)and (6, 9)

A low value of the correlation coefficient r implies that x and y are unrelated.

Consider the data set shown below. Find the estimate of the slope of the least squares regression line. 0 3 2 3 8 10 11 -2 0 2 4 6 8 10

For the situation above, give a practical interpretation of $t = 6.67$ .

For the situation above, write the equation of the least squares line.

Locate the values of $S S E , s ^ { 2 }$ , and $s$ on the printout below. $\text { Model Summary }$ Model R R Square Adjusted R Square Std. Error of the Estimate 1 .859 .737 .689 11.826 ANOVA Model Sum of Squares df Mean Square F Sig. 1 Regression 4512.024 1 4512.024 32.265 .001 Residual 1678.115 12 139.843 Total 6190.139 13

For the situation above, give a practical interpretation of $r ^ { 2 } = .66$ .

Suppose you fit a least squares line to 22 data points and the calculated value of SSE is .678. a. Find $s ^ { 2 }$ , the estimator of $\sigma ^ { 2 }$ . b. Find $s$ , the estimator of $\sigma$ . c. What is the largest deviation you might expect between any one of the 22 points and the least squares line?

Is the number of games won by a major league baseball team in a season related to the teamʹs batting average? Data from 14 teams were collected and the summary statistics yield: $\sum y = 1,134 , \sum ^ { x = 3.642 , } \sum y ^ { 2 } = 93,110 , \sum ^ { 2 } = .948622 , \text { and } \sum x y = 295.54$ Find the least squares prediction equation for predicting the number of games won, y, using a straight-line relationship with the teamʹs batting average, x.