Prediction (Optional)

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. 90 is outside 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 how unreliable the prediction can be outside the x values observed in the data, make the substitution x = 90 into the equation:

$ŷ=\mathrm{–173.51}+4.83\left(90\right)=261.19\text{.}$

The final exam score is predicted to be 261.19. The most points that can be awarded for the final exam are 200.

What would you predict the sales to be on day 60?

$250,120

What would you predict the sales to be on day 90?

How many acres are left to mow after 20 hours of work?

1326 acres

How many acres are left to mow after 100 hours of work?

How many hours does it take to mow all the lawns, or when is ŷ = 0?

1125 hours, or when x = 1125

Graph year versus number of flu cases diagnosed (plot the scatter plot). Do not include pre-1981 data.

Perform a linear regression. What is the linear equation? Round to the nearest whole number. Find the following:

Write the equations:

• Linear equation: __________
• a = ________
• b = ________
• r = ________
• n = ________

Check student solution.

Solve.

1. When x = 1985, ŷ = _____.
2. When x = 1990, ŷ = _____.
3. When x = 1970, ŷ = _____. Why doesn’t this answer make sense?

1. When x = 1985, ŷ = 25,52.
2. When x = 1990, ŷ = 34,275.
3. 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, when we are predicting flu 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 flu cases reported in the United States?

Also, the correlation r = 0.4526. If r is compared with the value in the 95 Percent 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 points on the graph. Then, connect the two points to form the regression line.

Check student’ solution.

Write the equation: ŷ = ____________.

$ŷ$ = 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 flu cases diagnosed until 1993. From 1993 through 2002, the number of flu 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 flu cases reported in the United States?

Because there is no linear association between year and number of flu 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. number flu 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 were collected from a single year. So, we don’t have an accurate x value for this figure.

Regression equation: ŷ (number of flu cases) = –3,448,225 + 1749.777 (year).

Calculate the following:

• a = _____
• b = _____
• correlation = _____
• n = _____

• a = –3,488,225
• b = 1,750
• correlation = 0.4526
• n = 22

Recently, the annual numbers of driver deaths per 100,000 people for the selected age groups are as follows:

1. For each age group, pick the midpoint of the interval for the x value. For the 75+ group, use 80.
2. Using age as the independent variable and number of driver deaths per 100,000 people as the dependent variable, make a scatter plot of the data.
3. Calculate the least-squares (best–fit) line. Put the equation in the form ŷ = a + bx.
4. Find the correlation coefficient. Is it significant?
5. Predict the number of deaths for ages 40 years and 60 years.
6. Based on the given data, is there a linear relationship between age of a driver and driver fatality rate?
7. What is the slope of the least-squares (best-fit) line? Interpret the slope.

b. Check student solution.

c. ŷ = 35.5818045 – 0.19182491x

d. r = –0.57874
For four degrees of freedom and alpha = 0.05, the LinRegTTest gives a p value of 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 degrees of freedom, 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.

f. There is not a linear relationship between the two variables, as evidenced by a p value greater than 0.05.

idasdgf10411945 shows the life expectancy for an individual born in the United States in certain years.

1. Decide which variable should be the independent variable and which should be the dependent variable.
2. Draw a scatter plot of the ordered pairs.
3. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
4. Find the correlation coefficient. Is it significant?
5. Find the estimated life expectancy for an individual born in 1950 and for one born in 1982.
6. Why aren’t the answers to Part E the same as the values in idasdgf10411945 that correspond to those years?
7. Use the two points in Part E to plot the least-squares line on your graph from Part B.
8. Based on the data, is there a linear relationship between the year of birth and life expectancy?
9. Are there any outliers in the data?
10. 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.
11. 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, 10th edition, for various pages is given in id9216hh080.

1. Decide which variable should be the independent variable and which should be the dependent variable.
2. Draw a scatter plot of the ordered pairs.
3. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
4. Find the correlation coefficient. Is it significant?
5. Find the estimated maximum values for the restaurants on page 10 and on page 70.
6. 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?
7. Suppose there are 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200?
8. Is the least-squares line valid for page 200? Why or why not?
9. What is the slope of the least-squares (best-fit) line? Interpret the slope.

a. We wonder if the better discounts appear earlier in the book, so we select page as x and discount as y.

b. Check student solution.

c. ŷ = 17.21757 – 0.01412x

d. r = – 0.2752
For seven degrees of freedom and alpha = 0.05, LinRegTTest gives a 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 gives degrees of freedom, the critical value is 0.666. The correlation coefficient xi = –0.2752 is not less than 0.666, so we do not reject.

f. There is not a significant linear correlation so it appears there is no relationship between the page and the amount of the discount.

As the page number increases by one page, the discount decreases by $0.01412.

id9978lok311 gives the gold medal times for every other Summer Olympics for the women’s 100-meter freestyle in swimming.

1. Decide which variable should be the independent variable and which should be the dependent variable.
2. Draw a scatter plot of the data.
3. Does it appear from inspection that there is a relationship between the variables? Why or why not?
4. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
5. Find the correlation coefficient. Is the decrease in times significant?
6. Find the estimated gold medal time for 1932. Find the estimated time for 1984.
7. Why are the answers from Part F different from the chart values?
8. Does it appear that a line is the best way to fit the data? Why or why not?
9. Use the least-squares line to estimate the gold medal time for the next Summer Olympics. Do you think your answer is reasonable? Why or why not?

We are interested in whether the number of letters in a state name depends on the year the state entered the Union.

1. Decide which variable should be the independent variable and which should be the dependent variable.
2. Draw a scatter plot of the data.
3. Does it appear from inspection that there is a relationship between the variables? Why or why not?
4. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
5. Find the correlation coefficient. What does it imply about the significance of the relationship?
6. Find the estimated number of letters (to the nearest integer) a state name would have if it entered the Union in 1900. Find the estimated number of letters a state name would have if it entered the Union in 1940.
7. Does it appear that a line is the best way to fit the data? Why or why not?
8. Use the least-squares line to estimate the number of letters for a new state that enters the Union this year. Can the least-squares line be used to predict it? Why or why not?

a. Year is the independent or x variable; the number of letters is the dependent or y variable.

b. Check student’s solution.

c. No.

d. ŷ = 47.03 – 0.0216x

e. –0.4280. The r value indicates that there is not a significant correlation between the year the state entered the Union and the number of letters in the name.

g. No. The relationship does not appear to be linear; the correlation is not significant.