# MATHEMATICS

1. The waiting time for patients at a local hospital emergency department follows a normal distribution with a mean of 55 minutes and a population standard deviation of 15 minutes. The quality-assurance department found in a sample of 50 patients that the mean waiting time was 54 minutes. At the 0.025 significance level, decide if the sample data support the claim that the mean waiting time is less than 55 minutes.

State the null and alternative hypotheses.

State the decision rule.

Based on the sample data, state your decision in terms of the null hypothesis (reject or not reject). You may use either method, comparing the test statistic to the critical value, or the p-value approach.

2. A machine makes ball bearings for use in other industrial machinery. The mean diameter of a particular type of ball bearing is 50 millimeters. Based on a new vendor for their raw materials, the Quality Assurance manager is worried that the current production runs will be outside of specification. To test this, 100 ball bearings (n = 100) were sampled. The mean of the sample is 51.2 millimeters and the SAMPLE standard deviation is 1.223 millimeters. Decide if the sample data supports the claim that the mean diameter is 50 millimeters. Use a 0.02 level of significance.

State the null and alternative hypotheses.

State the decision rule.

Based on the sample data, state your decision in terms of the null hypothesis (reject or not reject). You may use either method, comparing the test statistic to the critical value, or the p-value approach.

3. Based on the Nielsen ratings, the local CBS affiliate claims its 11:00 PM newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers, 36% indicated that they watch the late evening news on this local CBS station. If = 0.01, what is our decision:

State the null and alternative hypotheses.

State the decision rule.

Based on the sample data, state your decision in terms of the null hypothesis (reject or not reject).

4. A study by a bank compared the average savings of customers who were depositors for three years or less, with those who had been depositors for more than three years. The results of a sample are:

< 3 Years > 3 Years

Mean Savings Balance $1,200 $1,250

Population Standard Deviation $100 $250

Sample Size 100 150

Calculate the test statistic, z.

5. Given the following hypothesis test:

H0: 1 = 2

H1: 1 ≠ 2

100 soil samples were taken from the bottom of Lake Erie, where 70 samples (X1) contained high levels of bacteria. 150 soil samples were taken from the bottom of Lake Superior, where 90 samples (X2) contained high levels of bacteria.

Using a significance level of 0.05, test the hypothesis:

State the decision rule.

What is the pooled proportion?

Compute the test statistic value.

What is the decision regarding the null hypothesis?

6. An investigation of the effectiveness of a training program to improve customer relationships included a pre-training and post-training customer survey. Seven customers were randomly selected and completed both surveys. The results follow:

Customer Pre-training survey Post-training survey

A 6 8

B 5 5

C 10 10

D 7 10

E 6 8

F 5 6

G 2 8

What is the value of the test statistic, t?

7. Suppose that an automobile manufacturer designed a radically new lightweight engine and wants to recommend the grade of gasoline that will have the best fuel economy. The four grades are: below regular, regular, premium, and super premium. The test car made three trial runs on the test track using each of the four grades, and the miles per gallon were recorded:

Below Regular Regular Premium Super Premium

36.69 39.31 38.99 40.04

40.00 39.87 40.02 39.89

41.01 39.87 39.99 39.93

a. Create and show a single-factor ANOVA table with a 0.05 level of significance ().

b. Given that the null and alternative hypotheses are:

H0: below_regular = regular = premium = super_premium

H1: below_regular ≠ regular ≠ premium ≠ super_premium

Based on the ANOVA table F statistics, what is the decision regarding the null hypothesis?

8. The college of business was interested in comparing the attendance for three different class times for a business statistics class. The data follow:

a. Create and show a two-factor ANOVA table with a 0.05 level of significance ().

b. What is the critical F statistic for testing the hypothesis of equal treatment means at the 0.05 level of significance ()?

c. Given that the null and alternative hypotheses are:

H0: 8AM = 9:30AM = 11AM

H1: 8AM ≠ 9:30AM ≠ 11AM

Based on the ANOVA table F statistics, what is the decision regarding the null hypothesis?

9. Two research labs were asked to test the battery life in hours of 3 different experimental batteries. Each lab performed 3 tests per battery type with the following results.

Battery Type 1 Battery Type 2 Battery Type 3

Lab 1 10.50 7.75 8.75

9.50 7.00 8.00

10.75 8.25 9.00

Lab 2 6.75 4.75 6.75

7.75 5.50 5.75

6.50 5.75 6.25

a. Create and show a two-factor with interaction (replication) ANOVA table with a 0.05 level of significance (). Hint: There are 3 rows per sample.

b. Based on the ANOVA table for Interaction, is there a statistically significant interaction between the labs and battery types (justify using either the F statistics or the p-value approach).

10. Given the following table:

Current Length of Employment (Years) Number of Workdays Absent

5 2

6 3

9 3

4 5

2 7

2 7

0 8

a. Develop and state an estimate (predictor) equation for Ŷ given X. You can calculate the a and b constants using any method, manual, Excel, Regression analysis, etc.

b. Based on the estimate equation in part a, what is the estimated number of workdays absent given a worker who has been employed for 7 years?

c. Determine the correlation coefficient.

d. Perform a t-Test for the significance of the correlation coefficient for a two-tailed test with 0.05 level of significance ().

e. Determine the coefficient of determination.

f. Determine the standard error of the estimate.

11. Given the following table:

Labor Hours applied Labor Cost

1000 $50755

945 $40622

850 $46111

1250 $77000

1300 $68190

1590 $75000

1425 $68444

1350 $71235

1650 $90673

1700 $101182

a. Create a scatter plot of the data.

b. Which is the independent variable?

c. Which is the dependent variable?

d. Does the data show a positive or negative correlation between X and Y?