Practice Problems - Exam Four

1. A company has the following data on productivity (Y) and aptitude test score (X) (from 1 to 100) for 12 data entry personnel.
SSxx = 426; SSyy = 615.67; SSxy = 331; Sy.x = 5.98; X-bar = 36.5; Y-bar = 48.83

A) Find the slope of the regression line and interpret it.
B) Find the intercept of the regression line and interpret it.
C) Find the 95% CI for the slope, and interpret it.
D) Test the hypothesis that test scores and productivity are positively related. Assume alpha = .05.
E) Test the hypothesis that test scores can predict productivity. Assume alpha = .05.
F) Compute r², and make an interpretation of the stat.

2. A fire insurance company wants to relate the amount of fire damage in major residential fires to the distance between the residence and the nearest fire station. 15 fires
were sampled. The amount of damage in thousands of dollars (Y) and the distance in miles between the fire and the nearest fire station (X) are recorded for each fire. The
following statistics were computed:
SSxx = 34.78; SSyy = 911.52; SSxy = 171.11; Sy.x = 2.31; X-bar = 3.28; Y-bar = 26.41

A) Find the slope of the regression line and interpret it.
B) Find the intercept of the regression line and interpret it.
C) Find the 90% CI for the slope, and interpret it.
D) Test the hypothesis that amount of damage and the distance from the fire to station are positively related. Assume alpha = .05.
E) Test the hypothesis that distance from the station can positively predict amount of fire damage. Assume alpha = .01.
F) Compute r², and make an interpretation of the stat.

3. The following data resulted from a study looking at the relationship between the number of absences from class by each of 20 students (X), and their final grade at the
end of the course (Y) (converted to a 100-point scale).
SSxx = 266.4; SSyy = 1897.6; SSxy = -686.8; Sy.x = 3.96; X-bar = 7.6; Y-bar = 81.8

A) Find the slope of the regression line and interpret it.
B) Find the intercept of the regression line and interpret it.
C) Find the 99% CI for the slope, and interpret it.
D) Test the hypothesis that the number of absences is negatively related to the final grade in the course. Assume alpha = .05.
E) Test the hypothesis that the number of absences can negatively predict the final grade in the course. Assume alpha = .005.
F) Compute r², and make an interpretation of the stat.

4. A researcher wanted to investigate the potential relationship between hair length in men (X) and the amount of drugs taken by them (Y). Data were collected from 6 men,
and the following statistics were computed.SSxx = 58.0; SSyy = 66.0; SSxy = -2.0; Sy.x = 4.69; X-bar = 4.0; Y-bar = 5.0

A) Find the slope of the regression line and interpret it.
B) Find the intercept of the regression line and interpret it.
C) Find the 95% CI for the slope, and interpret it.
D) Test the hypothesis that hair length in men and amount of drugs taken are related. Assume alpha = .01.
E) Test the hypothesis that hair length can predict the amount of drugs taken by men. Assume alpha = .01.
F) Compute r², and make an interpretation of the stat.

5) A social scientist wants to determine whether marital status (divorced or not divorced) of US men is independent of their religious affiliation (or lack thereof), using a
Chi-Square test of association. A sample of 500 US men is surveyed and the results are tabulated below:

Religious Affiliation A: 39 (divorced) 172 (not divorced)
Religious Affiliation B: 19 (divorced) 61 (not divorced)
Religious Affiliation C: 12 (divorced) 44 (not divorced)
Religious Affiliation D: 28 (divorced) 70 (not divorced)
Religious Affiliation E (none): 18 (divorced) 37 (not divorced)

A) State your null and alternative hypotheses.
B) Find the critical value and state your decision rules. Assume alpha = .05.
C) Find Chi-Square_calc.
D) Draw your conclusion: reject/accept null, indicate whether p < .05 or p > .05,and state your conclusion in words.

6) Suppose 10 new paintings are shown to two art critics and each critic rates the paintings from 1 (best) to 10 (worst). We want to determine whether the critics' ratings
are positively related, using the Spearman rank-order correlation, as the data are not distributed normally, and are on an ordinal scale.

Painting / Critic 1 Rating / Critic 2 Rating
1 / 2 / 5
2 / 1 / 1
3 / 9 / 10
4 / 5 / 5
5 / 2 / 1
6 / 8 / 9
7 / 7 / 7
8 / 2 / 3
9 / 6 / 4
10 / 8 / 7

A) State your null and alternative hypotheses.
B) Find your r_s_crit (alpha is .05), and state your decision rules?
C) Find r_s_calc.
D) Draw your conclusion: reject/accept null, indicate whether p < .05 or p > .05,and state your conclusion in words.

ANSWERS:

1A) b1=.78; for each additional point on the aptitude test, productivity increases .78.
B) bo=20.36; can't logically interpret the intercept value, since aptitude test scores cannot be 0.
C) 95% CI = (.13,1.43); 95% confident that the true population slope falls between .13 and 1.43; 95% confidence that each additional point on the aptitude test results in
increases productivity between .13 and 1.43
D) Ho: rho < 0; Ha: rho > 0; r_calc=.65; r_crit = .4973; reject null; p < .05; there is a significant moderate, positive correlation between aptitude scores and productivity
E) Ho: beta_1 = 0; Ha: beta_1 not = 0; t_calc = 2.69; t_crit = 2.228; reject null; p < .05; aptitude test scores predict productivity.
F)r²=.42; 42% of the variability in productivity can be accounted for by aptitude test scores.

2A) b1=4.92; for each additional mile the station is away from the fire, the amount of fire damage goes up $4,920.
B) bo=10.27; when the fire station is 0 miles away from the fire, the amount of damage is $10,270.
C) 90% CI = (4.23, 5.60); 90% confident that the true population slope falls between 4.23 and 5.6; 90% confident that fire damage increases between $4230 and $5600
with each additional mile the station is from the fire
D) Ho: rho < 0; Ha: rho > 0; r_calc=.96; r_crit = .4409; reject the null; p < .05; There is a significant strong, positive correlation between distance from firestation and
amount of damage.
E) Ho: beta_1 < 0; Ha: beta_1 > 0; t_calc =12.62; t_crit =2 .650; reject the null; p < .01; distance from the station positively predicts amount of fire damage
F) r²=.92; 92% of the variability in amount of fire damage is accounted for by distance of the fire station from the fire.

3A) b1=-2.58; with each additional absence, the final grade in the course decreases by 2.58%.
B) bo=101.41; can't logically interpret since the final grade cannot be over 100.
C) 99% CI = (-3.28, -1.88); 99% confident that the true population slope falls between -3.28 and -1.88; 99% confident that final grade decreases between 3.28 and 1.88%
with each additional absence
D) Ho: rho > 0; Ha: rho < 0; r_calc= -.97; r_crit = -.3783; reject the null; p < .05; There is a significant strong, negative correlation between times absent and the final
grade in the class.
E) Ho: beta_1 > 0; Ha: beta_1 < 0; t_calc = -10.75; t_crit = -2.878; reject the null; p < .005; Number of days absent from class significantly negatively predicts the final
grade.
F) r²=.94; 94% of the variability in final grades is accounted for by number of absences.

4A) b1=- -.03; for each additional inch of hair, the amount of drugs done decreases by .03
B) bo=5.12; When hair length is 0, the amount of drugs taken is 5.12.
C) 95% CI = (-1.740, 1.676); 95% confident that the true population slope falls between 1.740 and
1.676; 95% confident that each additional inch of hair results in change in drugs used between -1.74 and 1.676D) Ho: rho = 0; Ha: rho not = 0; r_calc= -.032; r_crit = -.9172 and .9172; do not reject the null; p > .01; There is not a significant correlation between hair length and
amount of drugs used
E) Ho: beta_1 = 0; Ha: beta_1 not = 0; t_calc = -.05; t_crit = -4.604 and 4.604; do not reject the null; p > .01; Hair length does not significantly predict amount of drug
use.
F) r²=.001; .1% of drug use is accounted for by hair length.

5A) Ho: religious affiliation and marital status are independent; Ha: religious affiliation and marital status are dependent
5B) Chi-square_crit = 9.49; reject null if Chi-square_calc > 9.49; accept null if Chi square_calc < 9.49
5C) Chi-Square_calc = 7.135
5D) accept null; p > .05; religious affiliation and marital status independent

6A) Ho: rho_s_calc < 0; Ha: rho_s_calc > 0
6B) r_s_crit = .564; reject null if r_s_calc > .564; accept null if r_s_calc < .564
6C) r_s_calc = .912
6D) reject null; p < .05; judges' ratings are very strongly positively correlated