12.1 Linear Equations
Use the following information to answer the next three exercises. A vacation resort rents scuba equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.
Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.
Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late and $5 a day each day the payment remains unpaid.
Find the equation that expresses the total fee in terms of the number of days the payment is late.
Is the equation y = 10 + 5x – 3x^{2} linear? Why or why not?
Which of the following equations are linear?
a. y = 6x + 8
b. y + 7 = 3x
c. y – x = 8x^{2}
d. 4y = 8
Use the following information to answer the next exercise. Table 12.13 contains real data for the first two decades of flu reporting.
Year | Number of Flu Cases Diagnosed | Number of Flu 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 |
Use the columns Year and Number of Flu Cases Diagnosed. Why is year the independent variable and number of flu cases diagnosed the dependent variable (instead of the reverse)?
Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100x.
What are the independent and dependent variables?
Use the following information to answer the next three questions. As a result of erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000x.
What are the independent and dependent variables?
What is the y-intercept? Interpret its meaning.
Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 – 1.5x, where x is the number of hours passed in an eight-hour day of trading.
If you owned this stock, would you want a positive or negative slope? Why?
12.2 The Regression Equation
Table 12.16 below represents the relationship between the number of hours spent studying and final exam grades.
x (number of hours spent studying) | y (final exam grades) |
---|---|
3 | 50 |
5 | 72 |
1 | 45 |
2 | 51 |
6 | 80 |
8 | 96 |
4 | 65 |
7 | 90 |
Fill in the following chart as a first step in finding the line of best fit, using the median–median approach.
Group | x (no. of hours spent studying) | y (final exam grades) | Median x Value | Median y Value |
---|---|---|---|---|
1 | ||||
2 | ||||
3 |
Use the following information to answer the next five exercises. A random sample of 10 professional athletes produced the following data, where x is the number of endorsements the player has and y is the amount of money made, in millions of dollars.
x | y | x | y |
---|---|---|---|
0 | 2 | 5 | 12 |
3 | 8 | 4 | 9 |
2 | 7 | 3 | 9 |
1 | 3 | 0 | 3 |
5 | 13 | 4 | 10 |
Draw a scatter plot of the data.
Draw the line of best fit on the scatter plot.
What is the y-intercept of the line of best fit? What does it represent?
When n = 2 and r = 1, are the data significant? Explain.
12.3 Testing the Significance of the Correlation Coefficient (Optional)
When testing the significance of the correlation coefficient, what is the null hypothesis?
When testing the significance of the correlation coefficient, what is the alternative hypothesis?
If the level of significance is 0.05 and the p-value is 0.04, what conclusion can you draw?
12.4 Prediction (Optional)
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 ŷ = 101.32 + 2.48x, where ŷ is in thousands of dollars.
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 is 1,345 acres. The rate at which one person can mow is ŷ = 1350 – 1.2x, where x is the number of hours and ŷ represents the number of acres left to mow.
How many acres are left to mow after 100 hours of work?
Use the following information to answer the next 14 exercises. Table 12.17 contains real data for the first two decades of flu reporting.
Year | Number of Flu Cases Diagnosed | Number of Flu 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 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 = ________
Solve.
- When x = 1985, ŷ = _____.
- When x = 1990, ŷ = _____.
- When x = 1970, ŷ = _____. Why doesn’t this answer make sense?
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?
Hand-draw a smooth curve on the graph that shows the flow of 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?
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.
12.5 Outliers
Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.
A point is removed and the line of best fit is recalculated. The new correlation coefficient is 0.98. Does the point appear to have been an outlier? Why?
Are you more or less confident in the predictive ability of the new line of best fit?
The sum of squared errors (SSE) for a data set of 18 numbers is 49. What is the standard deviation?
The standard deviation for the SSE for a data set is 9.8. What is the cutoff for the vertical distance that a point can be from the line of best fit to be considered an outlier?