# Statistics - Hypothesis Testing

1. The General Social Survey is an annual survey given to about 1,500 U.S. adults selected at random. Each year, the survey contains several questions meant to probe respondents' views of employment. A recent survey contained the question "How important to your life is having a fulfilling job?" Of the 261 college graduates surveyed, 107 chose the response "Very important." Of the 121 people surveyed whose highest level of education was high school or less, 34 chose the response "Very important." Based on these data, can we conclude, at the 0.05 level of significance, that there is a difference between the proportion p1 of all U.S. college graduates who would answer "Very important" and the proportion p2 of all U.S. adults whose highest level of education was high school or less who would answer "Very important"?

Perform a two-tailed test. Then answer the 6 questions below.

Carry your intermediate computations to at least three decimal places and round your answers the same way.

1) The null hypothesis

2) The alternative hypothesis

3) The type of test statistic (Z, t, chi-square, etc)

4) The value of the test statistic

5) The p-value

6) (YES OR NO) Can we conclude that there is a difference between the two populations in the proportions who would answer "very important"?

2. The university data center has two main computers. The center wants to examine whether computer 1 is receiving tasks which require comparable processing time to those of computer 2. A random sample of 8 processing times from computer 1 showed a mean of 34 seconds with a standard deviation of 20 seconds, while a random sample of 14 processing times from computer 2 (chosen independently of those for computer 1) showed a mean of 52 seconds with a standard deviation of 17 seconds. Assume that the populations of processing times are normally distributed for each of the two computers, and that the variances are equal. Can we conclude, at the 0.05 level of significance, that the mean processing time of computer 1,p1 , is less than the mean processing time of computer 2, p2 ?

Perform a one-tailed test. Then answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers the same way.

1) The null hypothesis

2) The alternative hypothesis

3) The type of test statistic (Z, t, chi-square, etc)

4) The value of the test statistic

5) The p-value

6) (YES OR NO) Can we conclude that the mean processing time of computer 1 is less than the mean processing time of computer 2?

3. An industrial plant wants to determine which of two types of fuel, electric or gas, is more cost efficient (measured in cost per unit of energy). Independent random samples were taken of plants using electricity and plants using gas. These samples consisted of 11 plants using electricity, which had a mean cost per unit of $56.5 and standard deviation of $8.71, and 10 plants using gas, which had a mean of $58.4 and standard deviation of $8.75 . Assume that the populations of costs per unit are normally distributed for each type of fuel, and assume that the variances of these populations are equal. Can we conclude, at the 0.1 level of significance, that the mean cost per unit for plants using electricity, f1, differs from the mean cost per unit for plants using gas, f2 ?

Perform a two-tailed test. Then answer the 6 questions below.

Carry your intermediate computations to at least three decimal places and round your answers the same way.

1) The null hypothesis

2) The alternative hypothesis

3) The type of test statistic (Z, t, chi-square, etc)

4) The value of the test statistic

5) The two critical values at the 0.1 level of significance

6) (YES OR NO) Can we conclude that the mean cost per unit for plants using electricity differs from the mean cost per unit for plants using gas?

4. A presidential candidate's aide estimates that, among all college students, the proportion p who intend to vote in the upcoming election is at least 60%. If 119 out of a random sample of 205 college students expressed an intent to vote, can we reject the aide's estimate at the 0.05 level of significance?

Perform a one-tailed test and answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers to three as well.

1) The null hypothesis

2) The alternative hypothesis

3) The type of test statistic (ex. Z, t, Chi-Square, etc.)

4) The value of the test statistic

5) The critical value at the 0.05 level of significance

6) YES or NO. Can we reject the aide's estimate that the proportion of college students who intend to vote is at least 60%?

5. A decade-old study found that the proportion, p, of high school seniors who believed that "getting rich" was an important personal goal was 70%. A researcher decides to test whether or not that percentage still stands. He finds that, among the 205 high school seniors in his random sample, 147 believe that "getting rich" is an important goal. Can he conclude, at the 0.01 level of significance, that the proportion has indeed changed?

Perform a two-tailed test and answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers to three as well

1) The null hypothesis

2) The alternative hypothesis

3) The type of test statistic (ex. Z, t, Chi-Square, etc.)

4) The value of the test statistic

5) The p-value

6) YES or NO. Can we conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed?

6. The historical reports from two major networks showed that the mean number of commercials aired during prime time was equal for both networks last year. In order to find out whether they still air the same number of commercials on average or not, random and independent samples of 50% recent prime time airings from both networks have been considered. The first network aired an average of 111.3 commercials during prime with a standard deviation of 4.4. The second network aired 108.7 commercials with a standard deviation of 4.8. Since the sample size is quite large, assume that the population standard deviations 4.4 and 4.8 can be estimated using the sample standard deviations. At the 0.01 level of significance, is there sufficient evidence to support the claim that the average number of commercials aired during prime time by the first station, S1 is not equal to the average number of commercials aired during prime time by the second station S2

Perform a two-tailed test and answer the questions below.

Carry your intermediate computations to at least three decimal places and round your answers to three as well.

1) The null hypothesis

2) The alternative hypothesis

3) The type of test statistic (ex. Z, t, Chi-Square, etc.)

4) The value of the test statistic

5) The two critical values at the 0.01 level of significance

6) YES or NO. Can we support the claim that the mean numer of commercials aired during prime time by the first network is not equal to the mean number of commercials aired during prime time by the second network?

[See the attached questions file.]

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#### Solution Summary

Neat, step-by-step solutions are provided.

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)

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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?

_____________________

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(See attached file for full problem description)

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