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Statistics LibreTexts

Prediction (Exercises)

  • Page ID
    5909
  • 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:

    \[\hat{y} = 101.32 + 2.48x\] where \(\hat{y}\) is in thousands of dollars.

    Exercise 12.6.2

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

    Answer

    $250,120

    Exercise 12.6.3

    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:

    \[\hat{y} = 1350 - 1.2x\] where \(x\) is the number of hours and \(\hat{y}\) represents the number of acres left to mow.

    Exercise 12.6.4

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

    Answer

    1,326 acres

    Exercise 12.6.5

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

    Exercise 12.6.7

    How many hours will it take to mow all of the lawns? (When is \(\hat{y} = 0\)?)

    Answer

    1,125 hours, or when \(x = 1,125\)

    Table contains real data for the first two decades of AIDS reporting.

    Adults and Adolescents only, United States
    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

    Exercise 12.6.8

    Graph “year” versus “# AIDS cases diagnosed” (plot the scatter plot). Do not include pre-1981 data.

    Exercise 12.6.9

    Perform linear regression. What is the linear equation? Round to the nearest whole number.

    Answer

    Check student’s solution.

    Exercise 12.6.10

    Write the equations:

    1. Linear equation: __________
    2. \(a =\) ________
    3. \(b =\) ________
    4. \(r =\) ________
    5. \(n =\) ________

    Exercise 12.6.11

    Solve.

    1. When \(x = 1985\), \(\hat{y} =\) _____
    2. When \(x = 1990\), \(\hat{y} =\)_____
    3. When \(x = 1970\), \(\hat{y} =\)______ Why doesn’t this answer make sense?

    Answer

    1. When \(x = 1985\), \(\hat{y} = 25,52\)
    2. When \(x = 1990\), \(\hat{y} = 34,275\)
    3. When \(x = 1970\), \(\hat{y} = –725\) 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.

    Exercise 12.6.11

    Does the line seem to fit the data? Why or why not?

    Exercise 12.6.12

    What does the correlation imply about the relationship between time (years) and the number of diagnosed AIDS cases reported in the U.S.?

    Answer

    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.

    Exercise 12.6.13

    Plot the two given points on the following graph. Then, connect the two points to form the regression line.

    Blank graph with horizontal and vertical axes.
    Figure 12.6.1.

    Obtain the graph on your calculator or computer.

    Exercise 12.6.14

    Write the equation: \(\hat{y} =\) ____________

    Answer

    \(\hat{y} = 3,448,225 + 1750x\)

    Exercise 12.6.15

    Hand draw a smooth curve on the graph that shows the flow of the data.

    Exercise 12.6.16

    Does the line seem to fit the data? Why or why not?

    Answer

    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.

    Exercise 12.6.17

    Do you think a linear fit is best? Why or why not?

    Exercise 12.6.18

    What does the correlation imply about the relationship between time (years) and the number of diagnosed AIDS cases reported in the U.S.?

    Answer

    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.

    Exercise 12.6.19

    Graph “year” vs. “# AIDS cases diagnosed.” Do not include pre-1981. Label both axes with words. Scale both axes.

    Exercise 12.6.20

    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:

    Answer

    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: \(\hat{y} \text{(#AIDS Cases)} = -3,448,225 + 1749.777 \text{(year)}\)

    Coefficients
    Intercept –3,448,225
    \(X\) Variable 1 1,749.777

    Exercise 12.6.21

    Calculate the following:

    1. \(a =\) _____
    2. \(b =\) _____
    3. correlation = _____
    4. \(n =\) _____