Linear Regression and Correlation
Prediction
OpenStaxCollege
[latexpage]
Recall the third exam/final exam example.
We examined the scatterplot and showed that the correlation coefficient is significant. We found the equation of the bestfit line for the final exam grade as a function of the grade on the thirdexam. We can now use the leastsquares regression line for prediction.
Suppose you want to estimate, or predict, the mean final exam score of statistics students who received 73 on the third exam. The exam scores (xvalues) range from 65 to 75. Since 73 is between the xvalues 65 and 75, substitute x = 73 into the equation. Then:
We predict that statistics students who earn a grade of 73 on the third exam will earn a grade of 179.08 on the final exam, on average.
Recall the third exam/final exam example.
a. What would you predict the final exam score to be for a student who scored a 66 on the third exam?
a. 145.27
b. What would you predict the final exam score to be for a student who scored a 90 on the third exam?
b. The x values in the data are between 65 and 75. Ninety is outside of the domain of the observed x values in the data (independent variable), so you cannot reliably predict the final exam score for this student. (Even though it is possible to enter 90 into the equation for x and calculate a corresponding y value, the y value that you get will not be reliable.)
To understand really how unreliable the prediction can be outside of the observed x values observed in the data, make the substitution x = 90 into the equation.
\(\stackrel{^}{y}=–173.51+4.83\left(90\right)=261.19\)
The finalexam score is predicted to be 261.19. The largest the finalexam score can be is 200.
The process of predicting inside of the observed x values observed in the data is called interpolation. The process of predicting outside of the observed x values observed in the data is called extrapolation.
Data are collected on the relationship between the number of hours per week practicing a musical instrument and scores on a math test. The line of best fit is as follows:
ŷ = 72.5 + 2.8x
What would you predict the score on a math test would be for a student who practices a musical instrument for five hours a week?
86.5
References
Data from the Centers for Disease Control and Prevention.
Data from the National Center for HIV, STD, and TB Prevention.
Data from the United States Census Bureau. Available online at http://www.census.gov/compendia/statab/cats/transportation/motor_vehicle_accidents_and_fatalities.html
Data from the National Center for Health Statistics.
Chapter Review
After determining the presence of a strong correlation coefficient and calculating the line of best fit, you can use the least squares regression line to make predictions about your data.
Use the following information to answer the next two exercises. An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where x is the day. The model can be written as follows:
ŷ = 101.32 + 2.48x where ŷ is in thousands of dollars.
What would you predict the sales to be on day 60?
💲250,120
What would you predict the sales to be on day 90?
Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,345 acres. The rate at which one person can mow is as follows:
ŷ = 1350 – 1.2x where x is the number of hours and ŷ represents the number of acres left to mow.
How many acres will be left to mow after 20 hours of work?
1,326 acres
How many acres will be left to mow after 100 hours of work?
How many hours will it take to mow all of the lawns? (When is ŷ = 0?)
1,125 hours, or when x = 1,125
[link] contains real data for the first two decades of AIDS reporting.
Year  # AIDS cases diagnosed  # AIDS deaths 
Pre1981  91  29 
1981  319  121 
1982  1,170  453 
1983  3,076  1,482 
1984  6,240  3,466 
1985  11,776  6,878 
1986  19,032  11,987 
1987  28,564  16,162 
1988  35,447  20,868 
1989  42,674  27,591 
1990  48,634  31,335 
1991  59,660  36,560 
1992  78,530  41,055 
1993  78,834  44,730 
1994  71,874  49,095 
1995  68,505  49,456 
1996  59,347  38,510 
1997  47,149  20,736 
1998  38,393  19,005 
1999  25,174  18,454 
2000  25,522  17,347 
2001  25,643  17,402 
2002  26,464  16,371 
Total  802,118  489,093 
Graph “year” versus “# AIDS cases diagnosed” (plot the scatter plot). Do not include pre1981 data.
Perform linear regression. What is the linear equation? Round to the nearest whole number.
Check student’s solution.
Write the equations:
Solve.
Does the line seem to fit the data? Why or why not?
What does the correlation imply about the relationship between time (years) and the number of diagnosed AIDS cases reported in the U.S.?
Also, the correlation r = 0.4526. If r is compared to the value in the 95% Critical Values of the Sample Correlation Coefficient Table, because r > 0.423, r is significant, and you would think that the line could be used for prediction. But the scatter plot indicates otherwise.
<!– From 12.4 Move to 12.8 –>
Plot the two given points on the following graph. Then, connect the two points to form the regression line.
Obtain the graph on your calculator or computer.
<!– From 12.5 MOVE to 12.8 –>
Write the equation: ŷ= ____________
\(\stackrel{^}{y}\) = 3,448,225 + 1750x
Hand draw a smooth curve on the graph that shows the flow of the data.
<!– From 12.6 Move to 12.8 –>
Does the line seem to fit the data? Why or why not?
There was an increase in AIDS cases diagnosed until 1993. From 1993 through 2002, the number of AIDS cases diagnosed declined each year. It is not appropriate to use a linear regression line to fit to the data.
Do you think a linear fit is best? Why or why not?
What does the correlation imply about the relationship between time (years) and the number of diagnosed AIDS cases reported in the U.S.?
Since there is no linear association between year and # of AIDS cases diagnosed, it is not appropriate to calculate a linear correlation coefficient. When there is a linear association and it is appropriate to calculate a correlation, we cannot say that one variable “causes” the other variable.
<!– From 12.7 MOVE to 12.8 –>
Graph “year” vs. “# AIDS cases diagnosed.” Do not include pre1981. Label both axes with words. Scale both axes.
Enter your data into your calculator or computer. The pre1981 data should not be included. Why is that so?
Write the linear equation, rounding to four decimal places:
We don’t know if the pre1981 data was collected from a single year. So we don’t have an accurate x value for this figure.
Regression equation: ŷ (#AIDS Cases) = –3,448,225 + 1749.777 (year)
Coefficients  

Intercept  –3,448,225 
X Variable 1  1,749.777 
Calculate the following:
 a = _____
 b = _____
 correlation = _____
 n = _____
Homework
Recently, the annual number of driver deaths per 100,000 for the selected age groups was as follows:
Age  Number of Driver Deaths per 100,000 

17.5  38 
22  36 
29.5  24 
44.5  20 
64.5  18 
80  28 
 For each age group, pick the midpoint of the interval for the x value. (For the 75+ group, use 80.)
 Using “ages” as the independent variable and “Number of driver deaths per 100,000” as the dependent variable, make a scatter plot of the data.
 Calculate the least squares (best–fit) line. Put the equation in the form of: ŷ = a + bx
 Find the correlation coefficient. Is it significant?
 Predict the number of deaths for ages 40 and 60.
 Based on the given data, is there a linear relationship between age of a driver and driver fatality rate?
 What is the slope of the least squares (bestfit) line? Interpret the slope.

Age Number of Driver Deaths per 100,000 16–19 38 20–24 36 25–34 24 35–54 20 55–74 18 75+ 28  Check student’s solution.
 ŷ = 35.5818045 – 0.19182491x
 r = –0.57874
For four df and alpha = 0.05, the LinRegTTest gives pvalue = 0.2288 so we do not reject the null hypothesis; there is not a significant linear relationship between deaths and age.
Using the table of critical values for the correlation coefficient, with four df, the critical value is 0.811. The correlation coefficient r = –0.57874 is not less than –0.811, so we do not reject the null hypothesis.
 if age = 40, ŷ (deaths) = 35.5818045 – 0.19182491(40) = 27.9
if age = 60, ŷ (deaths) = 35.5818045 – 0.19182491(60) = 24.1
 For entire dataset, there is a linear relationship for the ages up to age 74. The oldest age group shows an increase in deaths from the prior group, which is not consistent with the younger ages.
 slope = –0.19182491
[link] shows the life expectancy for an individual born in the United States in certain years.
Year of Birth  Life Expectancy 

1930  59.7 
1940  62.9 
1950  70.2 
1965  69.7 
1973  71.4 
1982  74.5 
1987  75 
1992  75.7 
2010  78.7 
 Decide which variable should be the independent variable and which should be the dependent variable.
 Draw a scatter plot of the ordered pairs.
 Calculate the least squares line. Put the equation in the form of: ŷ = a + bx
 Find the correlation coefficient. Is it significant?
 Find the estimated life expectancy for an individual born in 1950 and for one born in 1982.
 Why aren’t the answers to part e the same as the values in [link] that correspond to those years?
 Use the two points in part e to plot the least squares line on your graph from part b.
 Based on the data, is there a linear relationship between the year of birth and life expectancy?
 Are there any outliers in the data?
 Using the least squares line, find the estimated life expectancy for an individual born in 1850. Does the least squares line give an accurate estimate for that year? Explain why or why not.
 What is the slope of the leastsquares (bestfit) line? Interpret the slope.
The maximum discount value of the Entertainment® card for the “Fine Dining” section, Edition ten, for various pages is given in [link]
Page number  Maximum value (💲) 

4  16 
14  19 
25  15 
32  17 
43  19 
57  15 
72  16 
85  15 
90  17 
 Decide which variable should be the independent variable and which should be the dependent variable.
 Draw a scatter plot of the ordered pairs.
 Calculate the leastsquares line. Put the equation in the form of: ŷ = a + bx
 Find the correlation coefficient. Is it significant?
 Find the estimated maximum values for the restaurants on page ten and on page 70.
 Does it appear that the restaurants giving the maximum value are placed in the beginning of the “Fine Dining” section? How did you arrive at your answer?
 Suppose that there were 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200?
 Is the least squares line valid for page 200? Why or why not?
 What is the slope of the leastsquares (bestfit) line? Interpret the slope.
 We wonder if the better discounts appear earlier in the book so we select page as X and discount as Y.
 Check student’s solution.
 ŷ = 17.21757 – 0.01412x
 r = – 0.2752
For seven df and alpha = 0.05, using LinRegTTest pvalue = 0.4736 so we do not reject; there is a not a significant linear relationship between page and discount.
Using the table of critical values for the correlation coefficient, with seven df, the critical value is 0.666. The correlation coefficient xi = –0.2752 is not less than 0.666 so we do not reject.
 page 10: 17.08 page 70: 16.23
 There is not a significant linear correlation so it appears there is no relationship between the page and the amount of the discount.
 page 200: 14.39
 No, using the regression equation to predict for page 200 is extrapolation.
 slope = –0.01412
As the page number increases by one page, the discount decreases by 💲0.01412
[link] gives the gold medal times for every other Summer Olympics for the women’s 100meter freestyle (swimming).
Year  Time (seconds) 

1912  82.2 
1924  72.4 
1932  66.8 
1952  66.8 
1960  61.2 
1968  60.0 
1976  55.65 
1984  55.92 
1992  54.64 
2000  53.8 
2008  53.1 
 Decide which variable should be the independent variable and which should be the dependent variable.
 Draw a scatter plot of the data.
 Does it appear from inspection that there is a relationship between the variables? Why or why not?
 Calculate the least squares line. Put the equation in the form of: ŷ = a + bx.
 Find the correlation coefficient. Is the decrease in times significant?
 Find the estimated gold medal time for 1932. Find the estimated time for 1984.
 Why are the answers from part f different from the chart values?
 Does it appear that a line is the best way to fit the data? Why or why not?
 Use the leastsquares line to estimate the gold medal time for the next Summer Olympics. Do you think that your answer is reasonable? Why or why not?
State  # letters in name  Year entered the Union  Rank for entering the Union  Area (square miles) 

Alabama  7  1819  22  52,423 
Colorado  8  1876  38  104,100 
Hawaii  6  1959  50  10,932 
Iowa  4  1846  29  56,276 
Maryland  8  1788  7  12,407 
Missouri  8  1821  24  69,709 
New Jersey  9  1787  3  8,722 
Ohio  4  1803  17  44,828 
South Carolina  13  1788  8  32,008 
Utah  4  1896  45  84,904 
Wisconsin  9  1848  30  65,499 
We are interested in whether or not the number of letters in a state name depends upon the year the state entered the Union.
 Decide which variable should be the independent variable and which should be the dependent variable.
 Draw a scatter plot of the data.
 Does it appear from inspection that there is a relationship between the variables? Why or why not?
 Calculate the leastsquares line. Put the equation in the form of: ŷ = a + bx.
 Find the correlation coefficient. What does it imply about the significance of the relationship?
 Find the estimated number of letters (to the nearest integer) a state would have if it entered the Union in 1900. Find the estimated number of letters a state would have if it entered the Union in 1940.
 Does it appear that a line is the best way to fit the data? Why or why not?
 Use the leastsquares line to estimate the number of letters a new state that enters the Union this year would have. Can the least squares line be used to predict it? Why or why not?
 Year is the independent or x variable; the number of letters is the dependent or y variable.
 Check student’s solution.
 no
 ŷ = 47.03 – 0.0216x
 –0.4280
 6; 5
 No, the relationship does not appear to be linear; the correlation is not significant.
 current year: 2013: 3.55 or four letters; this is not an appropriate use of the least squares line. It is extrapolation.