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Statistics LibreTexts

14.E: Regression (Exercises)

  • Page ID
    2650
  • General Questions

    Q1

    What is the equation for a regression line? What does each term in the line refer to? (relevant section)

    Q2

    The formula for a regression equation based on a sample size of \(25\) observations is \(Y' = 2X + 9\).

    1. What would be the predicted score for a person scoring \(6\) on \(X\)?
    2. If someone's predicted score was \(14\), what was this person's score on \(X\)? (relevant section)

    Q3

    What criterion is used for deciding which regression line fits best? (relevant section)

    Q4

    What does the standard error of the estimate measure? What is the formula for the standard error of the estimate? (relevant section)

    Q5

    1. In a regression analysis, the sum of squares for the predicted scores is \(100\) and the sum of squares error is \(200\), what is \(R^2\)?
    2. In a different regression analysis, \(40\%\) of the variance was explained. The sum of squares total is \(1000\). What is the sum of squares of the predicted values? (relevant section)

    Q6

    For the \(X,Y\) data below, compute:

    1. \(r\) and determine if it is significantly different from zero.
    2. the slope of the regression line and test if it differs significantly from zero.
    3. the \(95\%\) confidence interval for the slope.

    (relevant section)

    X
    Y
    2
    5
    4
    6
    4
    7
    5
    11
    6
    12

    Q7

    What assumptions are needed to calculate the various inferential statistics of linear regression? (relevant section)

    Q8

    The correlation between years of education and salary in a sample of \(20\) people from a certain company is \(0.4\). Is this correlation statistically significant at the \(0.05\) level? (relevant section)

    Q9

    A sample of \(X\) and \(Y\) scores is taken, and a regression line is used to predict \(Y\) from \(X\). If \(SSY' = 300\), \(SSE = 500\), and \(N = 50\), what is: (relevant section relevant section)

    1. \(SSY\)?
    2. the standard error of the estimate?
    3. \(R^2\)?

    Q10

    Using linear regression, find the predicted post-test score for someone with a score of \(43\) on the pre-test. (relevant section)

    Pre Post
    59 56
    52 63
    44 55
    51 50
    42 66
    42 48
    41 58
    45 36
    27 13
    63 50
    54 81
    44 56
    50 64
    47 50
    55 63
    49 57
    45 73
    57 63
    46 46
    60 60
    65 47
    64 73
    50 58
    74 85
    59 44

    Q11

    The equation for a regression line predicting the number of hours of TV watched by children (\(Y\)) from the number of hours of TV watched by their parents (\(X\)) is \(Y' = 4 + 1.2X\). The sample size is \(12\).

    1. If the standard error of \(b\) is \(0.4\), is the slope statistically significant at the \(0.05\) level? (relevant section)
    2. If the mean of \(X\) is \(8\), what is the mean of \(Y\)? (relevant section)

    Q12

    Based on the table below, compute the regression line that predicts \(Y\) from \(X\). (relevant section)

    MX
    MY
    sX sY r
    10
    12
    2.5 3.0 -0.6

    Q13

    Does \(A\) or \(B\) have a larger standard error of the estimate? (relevant section)

    se_est_graph.gif

    Q14

    True/false: If the slope of a simple linear regression line is statistically significant, then the correlation will also always be significant. (relevant section)

    Q15

    True/false: If the slope of the relationship between \(X\) and \(Y\) is larger for \(\text{Population 1}\) than for \(\text{Population 2}\), the correlation will necessarily be larger in \(\text{Population 1}\) than in \(\text{Population 1}\). Why or why not? (relevant section)

    Q16

    True/false: If the correlation is \(0.8\), then \(40\%\) of the variance is explained. (relevant section)

    Q17

    True/false: If the actual \(Y\) score was \(31\), but the predicted score was \(28\), then the error of prediction is \(3\). (relevant section)

    Questions from Case Studies

    The following question is from the Angry Moods (AM) case study.

    Q18

    (AM#23) Find the regression line for predicting Anger-Out from Control-Out.

    1. What is the slope?
    2. What is the intercept?
    3. Is the relationship at least approximately linear?
    4. Test to see if the slope is significantly different from \(0\).
    5. What is the standard error of the estimate?

    (relevant section, relevant section, relevant section)

    The following question is from the SAT and GPA (SG) case study.

    Q19

    (SG#3) Find the regression line for predicting the overall university GPA from the high school GPA.

    1. What is the slope?
    2. What is the \(y\)-intercept?
    3. If someone had a \(2.2\) GPA in high school, what is the best estimate of his or her college GPA?
    4. If someone had a \(4.0\) GPA in high school, what is the best estimate of his or her college GPA?

    (relevant section)

    The following questions are from the Driving (D) case study.

    Q20

    (D#5) What is the correlation between age and how often the person chooses to drive in inclement weather? Is this correlation statistically significant at the \(0.01\) level? Are older people more or less likely to report that they drive in inclement weather? (relevant section, relevant section )

    Q21

    (D#8) What is the correlation between how often a person chooses to drive in inclement weather and the percentage of accidents the person believes occur in inclement weather? Is this correlation significantly different from \(0\)? (relevant section, relevant section )

    Q22

    (D#10) Use linear regression to predict how often someone rides public transportation in inclement weather from what percentage of accidents that person thinks occur in inclement weather. (Pubtran by Accident)

    1. Create a scatter plot of this data and add a regression line.
    2. What is the slope?
    3. What is the intercept?
    4. Is the relationship at least approximately linear?
    5. Test if the slope is significantly different from \(0\).
    6. Comment on possible assumption violations for the test of the slope.
    7. What is the standard error of the estimate?

    (relevant section, relevant section, relevant section)

    Selected Answers

    S2

    1. \(21\)

    S5

    1. \(0.33\)

    S6

    1. \(b = 1.91\) 

    S9

    1. \(800\)

    S12

    \(a = 19.2\) 

    S18

    1. \(3.45\)

    S19

    1. \(2.6\)

    S20

    \(r = 0.43\)

    S22

    1. \(0.35\)

    Contributor

    • Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.