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 best-fit line for the final exam grade as a function of the grade on the third-exam. We can now use the least-squares 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 (x-values) range from 65 to 75. Since 73 is between the x-values 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.
$\hat{y}=\mathrm{\u2013173.51}+4.83\left(90\right)=261.19$
The final-exam score is predicted to be 261.19. The largest the final-exam score can be is 200.
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 |
Pre-1981 | 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 pre-1981 data.
Perform linear regression. What is the linear equation? Round to the nearest whole number.
Check student’s solution.
Write the equations:
- Linear equation: __________
- a = ________
- b = ________
- r = ________
- n = ________
Solve.
- When x = 1985, ŷ = _____
- When x = 1990, ŷ =_____
- When x = 1970, ŷ =______ Why doesn’t this answer make sense?
- When x = 1985, ŷ = 25,52
- When x = 1990, ŷ = 34,275
- When x = 1970, ŷ = –725 Why doesn’t this answer make sense? The range of x values was 1981 to 2002; the year 1970 is not in this range. The regression equation does not apply, because predicting for the year 1970 is extrapolation, which requires a different process. Also, a negative number does not make sense in this context, where we are predicting AIDS cases diagnosed.
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.
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.
Write the equation: ŷ= ____________
$\widehat{y}$ = 3,448,225 + 1750x
Hand draw a smooth curve on the graph that shows the flow of the data.
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.
Graph “year” vs. “# AIDS cases diagnosed.” Do not include pre-1981. Label both axes with words. Scale both axes.
Enter your data into your calculator or computer. The pre-1981 data should not be included. Why is that so?
Write the linear equation, rounding to four decimal places:
We don’t know if the pre-1981 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 (best-fit) 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 p-value = 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 least-squares (best-fit) 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 least-squares 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 least-squares (best-fit) 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 p-value = 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 100-meter 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 least-squares 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 least-squares 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 least-squares 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.
- Introductory Statistics
- Preface
- Sampling and Data
- Descriptive Statistics
- Introduction
- Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
- Histograms, Frequency Polygons, and Time Series Graphs
- Measures of the Location of the Data
- Box Plots
- Measures of the Center of the Data
- Skewness and the Mean, Median, and Mode
- Measures of the Spread of the Data
- Descriptive Statistics
- Probability Topics
- Discrete Random Variables
- Introduction
- Probability Distribution Function (PDF) for a Discrete Random Variable
- Mean or Expected Value and Standard Deviation
- Binomial Distribution
- Geometric Distribution
- Hypergeometric Distribution
- Poisson Distribution
- Discrete Distribution (Playing Card Experiment)
- Discrete Distribution (Lucky Dice Experiment)
- Continuous Random Variables
- The Normal Distribution
- The Central Limit Theorem
- Confidence Intervals
- Hypothesis Testing with One Sample
- Hypothesis Testing with Two Samples
- The Chi-Square Distribution
- Linear Regression and Correlation
- F Distribution and One-Way ANOVA
- Appendix A: Review Exercises (Ch 3-13)
- Appendix B: Practice Tests (1-4) and Final Exams
- Appendix C: Data Sets
- Appendix D: Group and Partner Projects
- Appendix E: Solution Sheets
- Appendix F: Mathematical Phrases, Symbols, and Formulas
- Appendix G: Notes for the TI-83, 83+, 84, 84+ Calculators
- Appendix H: Tables