# Sample Statistics and Coefficients Problems

Suppose that you are interested in examining the determinants of the use of plea bargaining by state prosecutors. During a one-month period, you record for each state in the US (1) the number plea bargains per 1,000 cases (y), (2) the average number of cases per prosecutor (x1), and (3) the average number of empty beds in the state's correctional institution (x2). Your hypothesis is that the average number of cases per prosecutor and the average number of empty correctional beds may explain some of the variation in the use of plea bargaining across states. More precisely, you hypothesize that as the average number cases per prosecutor increases, plea bargaining may also increase. On the other hand, you hypothesize that as the average number of empty beds in the state's correctional institution increases, the use of plea bargaining may decrease. After collecting your data, you obtain the following statistics:

(see attachment)

a. What would the values of b1 and b2 be from the above sample statistics? Interpret these coefficients.

b. From your calculated partial slope coefficients and sample means, solve for the value of the intercept (a). What is the complete multiple regression equation?

c. Using the above multiple regression equation, predict the value of Y (number of plea bargains per 1,000) if the average number of cases per prosecutor is 150 and the average number of empty beds is two.

d. Calculate the Beta weights for each of the partial slope coefficients above. What do they tell you about the relative importance of each independent variable?

e. Calculate the multiple coefficient of determination from these sample statistics. What does this coefficient indicate?

https://brainmass.com/statistics/correlation-and-regression-analysis/sample-statistics-and-coefficients-problems-64509

#### Solution Summary

The solution answers a multipart question about hypothesis testing for the use of the plea bargain. Attached as Word document.

Business Statistics

Please answer all questions. Please show all your work (detailed solution) so i can understand how you came up with the anwer. Thanks

(See attached file for full problem description)

---

EXHIBIT 1: A manager at a local bank analyzed the relationship between monthly salary and three independent variables: length of service (measured in months), gender (0=female, 1=male) and job type (0=clerical, 1=technical). The following tables summarizes the regression results:

df SS MS F

Regression SS 3 1004346.771 334782.257 5.96

Unexplained SS 26 1461134.596 56197.48445

Coefficients Standard Error t Stat P-Value

Constant 784.92 322.25 2.44 0.02

Service 9.19 3.20 2.87 0.01

Gender 222.78 89.00 2.50 0.02

Job -28.21 89.61 -0.31 0.76

(1) Referencing Exhibit 1, based on the SS table and 0.05 significance level, the global null hypothesis test of the multiple regression model (2 points):

(a) Will be rejected and conclude that monthly salary is related to all the independent variables

(b) Will be rejected and conclude that monthly salary is related to at least one of the independent variables

(c) Will not be rejected

(d) Will show a high multiple coefficient of determination

(e) Cannot be ascertained given the above information ______________

(2) Referencing Exhibit 1, the adjusted multiple coefficient of determination is (3 points):

(a) 5.93%

(b) 59.3 %

(c) 40.7 %

(d) 33.9%

(e) None of the above ________________

(3) Referencing Exhibit 1, based on hypothesis tests for the individual regression coefficients at a 5% significance level (2 points),

(a) All the regression coefficients are not equal to zero,

(b) "Job" is the only significant independent variable in the model

(c) Only "Service" and "Gender" are significantly related to monthly salary

(d) "Service" is the only significant variable in the model

(e) None of the variables are significant ________________

(4) Referencing Exhibit 1, which of the variables are dummy variables in the model (2 points)?

(a) Salary

(b) Service

(c) Service and gender

(d) Gender and job

(e) Service, gender and job ________________

(5) Referencing Exhibit 1, the results for the variable "Gender" show (2 points)

(a) Males average $222.78 more than females in monthly salary

(b) Females average $222.78 more than males in monthly salary

(c) Gender is not related to monthly salary

(d) Gender and months of service are correlated ________________

(6) A survey of 144 retail stores revealed that a particular brand and model of a VCR retails for $375 with a standard deviation of $20. What is a 99% confidence interval to estimate the true cost of the VCR?

_____________________

(7) A survey of an urban university (population 25,450) showed that 870 of 1100 students sampled supported a fee increase to fund improvements to the student recreation center. Using the 95% level of confidence, what is the confidence interval for the true proportion who support the fee increase?

_____________________

(8)The average cost of tuition, room and board at small private liberal arts colleges is reported to be $8,500 per term but a financial administrator believes that the average cost is higher. A study conducted using 150 small liberal arts colleges showed that the average cost per term is $9,000 with a standard deviation of $1,200. What is the decision at 0.05 level of significance?

_____________________

(9) The mean weight of newborn infants at a community hospital is 6.6 pounds. A sample of seven infants is randomly selected and their weights are recorded as 9.0; 7.3; 6.0; 8.8; 6.8; 8.4; and 6.6 pounds. What is the decision for a statistical significant change in average weights at birth at the 5% level of significance?

_____________________

(10) A manufacturer of automobile transmissions uses three different processes. The management ordered a study of the production costs to see if there is a difference among the three processes. A summary of the findings is shown below.

Process 1 Process 2 Process 3

Process Totals ($ 100's) 137 108 107

Sample Size 10 10 10

Sum of Squares 1893 1188 1175

What is the sum of squares for treatments?

_____________________

For problem 10, what is the sum of squares for error?

_____________________

For problem 10, what is your conclusion at alpha = .05?

_____________________

(11) A sales manager for an advertising agency believes there is a relationship between the number of contacts and the amount of sales. To verify this belief, the following data was collected:

Salesperson # of Contacts Sales ($ '000)

1 14 24

2 12 14

3 20 28

4 16 30

5 46 80

6 23 30

7 48 90

8 50 85

9 55 120

10 50 110

What is the value of coefficient of determination ?

_____________________

For problem 11, what is the regression equation?

_____________________

---

(See attached file for full problem description)

View Full Posting Details