CHAPTER 15

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CHAPTER 15

SOLUTIONS TO PROBLEMS

1. a.

The least squares regression equation for this problem is:

b. The partial slope coefficient for the variable DIVORCE is .871. This suggests that as the divorce rate per 100,000 population increases by one, the rate of violent crime per 100,000 increases by .871, controlling for the mean age of the state's population. The partial slope coefficient for the variable AGE is -.146. This suggests that as the mean age of the state's population increases by one year, the rate of violent crime per 100,000 decreases by .146, controlling for the divorce rate. The intercept is equal to 19.64. This tells us that when both the divorce rate and mean age are equal to 0, the rate of violent crime is 19.64 per 100,000.

c. There are two ways to examine the relative importance of DIVORCE and AGE in explaining the rate of violent crime. The first is to look at the standardized regression coefficients (Beta Weights). These are labeled Beta in the output. These coefficients are standardized, so that their relative size is an indication of the strength of each independent variable. The standardized coefficient for DIVORCE is .594, while that for AGE is -.133. Based on this, then, we would conclude that the divorce rate is more influential in explaining state violent crime rates than is the mean age of the population. A second way to look at the relative strength of the independent variables is to compare their respective t ratios. The larger the absolute value of the t ratio, the more influential the variable. The t ratio for DIVORCE is 4.268, while that for AGE is -3.110. Based on this, we would conclude that the divorce rate is more influential in explaining rates of violence than the mean age of a state's population.

d. The divorce rate and mean age together explain approximately 63% of the variance in rates of violent crime. The adjusted R² value is .61, indicating that 61% of the variance is explained.

e. There are two hypothesis tests here, one hypothesis concerns the overall fit of the regression model, and the second concerns the significance of the individual partial slope coefficients.

The test of the overall model is a test to determine if the overall amount of variance explained is significantly different from zero. Specifically, the null hypothesis of this hypothesis test is that all of the partial slope coefficients are equal to zero. The alternative hypothesis is that at least one of them is not equal to zero. The null and alternative hypotheses are:

H₀: b₁, b₂ = 0; or R² = 0

H₁: b₁ or b₂ ¹ 0; or R² ¹ 0.

The test statistic for this hypothesis is the F test, and the sampling distribution is the F distribution. The output provides the obtained value of the F for your data F_obt = 27.53, and the probability of that F if the null hypothesis was true. The probability of an F of 27.53, if the null hypothesis was true, is .00052. Since this probability is less than our alpha of .01, our decision is to reject the null hypothesis that all the slope coefficients are equal to zero.

H₀: b_Divorce = 0

H₁: b_Divorce > 0

The test statistic for this hypothesis is the t test, and the sampling distribution is the t distribution. The output gives you the t_obt which is 4.268, and the probability of obtaining a t this size if the null hypothesis was true is equal to .0001/2 = .00005. Since this probability is less than our alpha level of .01, we decide to reject the null hypothesis. We conclude that the population partial slope coefficient is greater than 0. Specifically, there is a significant positive relationship between a state's divorce rate and its rate of violence, controlling for mean age.

Now we can conduct a hypothesis test of the partial slope coefficient for mean AGE. Since we suspect that the younger the mean age of a state's population, the higher its rate of violence, we expect the population partial slope to be less than zero, i.e., negative. Our null and alternative hypotheses are:

H₀: b_Age = 0

H₁: b_Age < 0

As with DIVORCE, we will reject the null hypothesis if the probability of t_obt given the null hypothesis is less than .01. The t ratio for the variable AGE is t_obt = -3.11. The probability of obtaining a t statistic this low if the null hypothesis is true is equal to .0011/2 = .00055. Since this is less than .01, we will reject the null hypothesis. The partial slope coefficient between mean age and rates of violence is significantly less than zero in the population. We conclude that states with younger populations have higher rates of violence.

a. From our computational formulas, we can determine the values of the two slope coefficients as follows:

The partial slope coefficient for employee morale is -.800. This tells us that as the score on our measure of employee morale increases by 1, the number of jail escapes decreases by .800, controlling for the staff/inmate ratio. The partial slope coefficient for the staff to inmate ratio is -6.585. This indicates that as the staff/inmate ratio increases by 1 unit, the number of jail escapes decreases by 6.585.

b. The intercept can be determined as follows:

The value of the intercept is 16.657, and the full regression equation is:

c. Predict the number of escapes when the staff morale score is equal to 8, and the staff/inmate ratio is .3:

With a staff morale score of 8 and a staff to inmate ratio of .3, we would predict that there would be approximately 8 escapes a year.

d. With the partial slope coefficients, and the standard deviations for the three variables, we can calculate the standardized partial slope coefficients or beta weights. The general formula for beta weights is . From this general formula we can calculate each of the two beta weights:

The beta weight for staff morale is -.616, while the beta weight for staff/inmate ratio is -.197. Because the beta weight for morale is greater than that for the staff to inmate ratio, it has a stronger effect on the number of jail escapes.

e. We can use the following formula to determine the value of R²:

In this equation, we first let x₁ (morale) do all the explaining it can, and then let x₂ explain what it can of the remaining variance. It does not matter which independent variable is considered first, the value of R² will be the same:

Together, staff morale and the staff to inmate ratio explain approximately 61% of the variance in jail escapes. If you had entered x₂ (staff/inmate ratio) first, and let it do all the explaining it could, and let x₁ (morale) explain what it could of the remaining variance, you would have obtained:

As you can see, the amount of variance explained remains the same, 61%. Notice, however, that if we let morale explain what it can of the variance first, it explains 58% of the variance, and staff/inmate ratio explains only 3% of the remaining variance. If we let the staff/inmate ratio variable explain all the variance it can first, it explains 40% of the variance in escapes, and morale explains about 21% of the remaining unexplained variance. This corroborates our conclusion that staff morale is more important in explaining the number of escapes than the staff to inmate ratio.

a. The least squares regression equation from the supplied output would be:

b. The partial slope coefficient for the environmental factors variable is -1.467. This tells us that as a person's score on the environmental causes of crime scale increases by one, their score on the punitiveness scale decreases by 1.467, controlling for religious conservatism. The partial slope coefficient for the religious conservatism scale is 1.076. As a person's score on religious conservatism increases by one unit, their score on the punitiveness scale increases by 1.076, controlling for their score on environmental factors as a cause of crime. The value of the intercept is 16.245. When both independent variables are zero, a person's score on the punitiveness scale is 16.245.

c. There are two ways to look at the relative effects of our two independent variables. One is to compare their standardized partial slope coefficients or beta weights (Beta in the output). The beta weight for ENV is -.609, indicating that when the score on the environmental factors scale increases by one standard deviation unit, the score on the punitiveness scale decreases by -.609. The beta weight for REL is .346, indicating that when the score on the religious conservatism scale increases by one standard deviation unit, the score on the punitiveness scale increases by .346 units. As you can see, a one unit change in the ENV variable produces almost twice the change in the dependent variable than the REL variable. Comparing these beta weights would lead us to conclude that the environmental factors scale is more important in explaining punitiveness scores than the religious conservatism variable.

A second way to assess the relative strengths of our independent variables is to compare their respective t ratios. The t ratio for ENV is -3.312, while that for REL is only 1.884. We would again conclude that ENV has the greater influence on the dependent variable.

d. The total variance explained is determined by the R² value. In this example, since we only have 2 independent variables and ten observations, we should use the Adjusted R² coefficient. This coefficient indicates that together, the environmental factors and religious conservatism scales explain approximately 60% of the variance in the punitiveness measure.

e. There are two hypothesis tests requested here, one hypothesis concerns the overall fit of the regression model, and the second concerns the significance of the individual partial slope coefficients.

The test of the estimated regression model is a test to determine if the overall amount of variance explained is significantly different from zero. Specifically, the null hypothesis of this hypothesis test is that all of the partial slope coefficients are equal to zero. The alternative hypothesis is that at least one of them is not equal to zero. The null and alternative hypotheses are:

H₀: b_ENV = b_REL = 0; or R² = 0

H₁: b_ENV , b_REL ¹ 0; or R² ¹ 0.

The test statistic for this hypothesis is the F test, and the sampling distribution is the F distribution. The output provides the obtained value of the F for your data F_obt = 11.59, and the probability of that F if the null hypothesis was true. The probability of an F of 11.59, if the null hypothesis was true, is .0016. Since this probability is less than our chosen alpha of .01, our decision is to reject the null hypothesis that all the slope coefficients are equal to zero.

Now we can proceed to conduct a hypothesis test for each of the two partial slope coefficients. In this test, our null hypothesis is that each of the population slope coefficients is equal to zero. We can conduct one or two-tailed hypothesis tests for the slope coefficients. We will conduct a one-tailed test in this problem. Based on the previous research on Grasmick and McGill, we predict that the partial slope coefficient for the environmental factors scale will be less than zero. This is because we think a belief in environmental causes of crime are negatively related to punitiveness. We also predict that the partial slope coefficient for religious conservatism will be greater than zero. This is because we think that the more religiously conservative one is, the more likely they are to take a punitive stance regarding the punishment of crime. With respect to the slope coefficient for ENV, our null and alternative hypotheses are:

H₀: b_ENV = 0

H₁: b_ENV < 0

The test statistic for this hypothesis is the t test, and the sampling distribution is the t distribution. The output gives you the t_obt which is -3.312, and the probability of obtaining a t this size if the null hypothesis was true is equal to .0062. Since this probability is less than our chosen alpha level of .01, we decide to reject the null hypothesis. We conclude that the population partial slope coefficient is less than 0. Specifically, there is a significant negative relationship between a persons' belief in environmental causes of crime and their expressed belief that criminal offenders should be treated punitively.

Now we can conduct a hypothesis test of the partial slope coefficient for REL. Our null and alternative hypotheses are:

H₀: b_REL = 0

H₁: b_REL > 0

As with ENV, we will reject the null hypothesis if the probability of t_obt given the null hypothesis is less than .01. The t ratio for the variable ENV is t_obt = 1.884. The probability of obtaining a t statistic of this magnitude if the null hypothesis is true is equal to .0840. Since this probability is greater than .01, our decision is to fail to reject the null hypothesis. The partial slope coefficient between religious conservatism and punitiveness toward criminal offenders is not significantly different from zero in the population, once a belief in environmental causes of crime are controlled.

f. Using our regression equation, we can predict the punitiveness score for a person with an environmental factors score of 2 and a religious conservatism score of 8:

The predicted value of y is 21.92.

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