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Homework answers / question archive / 1)If we are interested in testing whether the proportion of items in population 1 is larger than the proportion of items in population 2, the             a

1)If we are interested in testing whether the proportion of items in population 1 is larger than the proportion of items in population 2, the             a

Statistics

1)If we are interested in testing whether the proportion of items in population 1 is larger than the proportion of items in population 2, the

            a.   null hypothesis should state P1 - P2 < 0

            b.   null hypothesis should state P1 - P2 ³ 0

            c.   alternative hypothesis should state P1 - P2 > 0

            d.   alternative hypothesis should state P1 - P2 < 0

 

 

2.         The sampling distribution of  is approximated by a

            a.   normal distribution

            b.   t-distribution with n1 + n2 degrees of freedom

            c.   t-distribution with n1 + n2 – 1 degrees of freedom

            d.   t-distribution with n1 + n2 + 2 degrees of freedom

 

 

3.         When developing an interval estimate for the difference between two sample means, with sample sizes of n1 and n2,

            a.   n1 must be equal to n2

            b.   n1 must be smaller than n2

            c.   n1 must be larger than n2

            d.   n1 and n2 can be of different sizes

 

 

4.         To construct an interval estimate for the difference between the means of two populations when the standard deviations of the two populations are unknown, we must use a t distribution with (let n1 be the size of sample 1 and n2 the size of sample 2)

            a.   (n1 + n2) degrees of freedom

            b.   (n1 + n2 - 1) degrees of freedom

            c.   (n1 + n2 - 2) degrees of freedom

            d.   n1 - n2 + 2

 

 

5.         When each data value in one sample is matched with a corresponding data value in another sample, the samples are known as

            a.   corresponding samples

            b.   matched samples

            c.   independent samples

            d.   None of these alternatives is correct.

 

 

6.         Independent simple random samples are taken to test the difference between the means of two populations whose variances are not known.  The sample sizes are n1 = 32 and n2 = 40.  The correct distribution to use is the

            a.   binomial distribution

            b.   t distribution with 72 degrees of freedom

            c.   t distribution with 71 degrees of freedom

            d.   t distribution with 70 degrees of freedom

 

 

7.         Independent simple random samples are taken to test the difference between the means of two populations whose standard deviations are not known.  The sample sizes are n1 = 25 and n2 = 35.  The correct distribution to use is the

            a.   Poisson distribution

            b.   t distribution with 60 degrees of freedom

            c.   t distribution with 59 degrees of freedom

            d.   t distribution with 58 degrees of freedom

 

 

8.         If two independent large samples are taken from two populations, the sampling distribution of the difference between the two sample means

            a.   can be approximated by a Poisson distribution

            b.   will have a variance of one

            c.   can be approximated by a normal distribution

            d.   will have a mean of one

 

 

9.         The standard error of  is the

            a.   variance of

            b.   variance of the sampling distribution of

            c.   standard deviation of the sampling distribution of

            d.   difference between the two means

 

 

Exhibit 10-1

Salary information regarding male and female employees of a large company is shown below.

 

                                                                        Male                Female

            Sample Size                                          64                      36

            Sample Mean Salary (in $1,000)          44                      41

            Population Variance                            128                     72

 

10.       Refer to Exhibit 10-1.  The point estimate of the difference between the means of the two populations is

            a.   -28

            b.   3

            c.   4

            d.   -4

 

 

11.       Refer to Exhibit 10-1.  The standard error for the difference between the two means is

            a.   4

            b.   7.46

            c.   4.24

            d.   2.0

 

 

12.       Refer to Exhibit 10-1.  At 95% confidence, the margin of error is

            a.   1.96

            b.   1.645

            c.   3.920

            d.   2.000

 

 

13.       Refer to Exhibit 10-1.  The 95% confidence interval for the difference between the means of the two populations is

            a.   0 to 6.92

            b.   -2 to 2

            c.   -1.96 to 1.96

            d.   -0.92 to 6.92

 

 

14.       Refer to Exhibit 10-1.  If you are interested in testing whether or not the average salary of males is significantly greater than that of females, the test statistic is

            a.   2.0

            b.   1.5

            c.   1.96

            d.   1.645

 

 

15.       Refer to Exhibit 10-1.  The p-value is

            a.   0.0668

            b.   0.0334

            c.   1.336

            d.   1.96

 

 

16.       Refer to Exhibit 10-1.  At 95% confidence, the conclusion is the

            a.   average salary of males is significantly greater than females

            b.   average salary of males is significantly lower than females

            c.   salaries of males and females are equal

            d.   None of these alternatives is correct.

 

 

Exhibit 10-2

The following information was obtained from matched samples.

The daily production rates for a sample of workers before and after a training program are shown below.

 

            Worker            Before             After

                 1                      20                   22

                 2                      25                   23

                 3                      27                   27

                 4                      23                   20

                 5                      22                   25

                 6                      20                   19

                 7                      17                   18

 

17.       Refer to Exhibit 10-2.  The point estimate for the difference between the means of the two populations is

            a.   -1

            b.   -2

            c.   0

            d.   1

 

 

18.       Refer to Exhibit 10-2.  The null hypothesis to be tested is H0md = 0.  The test statistic is

            a.   -1.96

            b.   1.96

            c.   0

            d.   1.645

 

 

19.       Refer to Exhibit 10-2.  Based on the results of question 18, the

            a.   null hypothesis should be rejected

            b.   null hypothesis should not be rejected

            c.   alternative hypothesis should be accepted

            d.   None of these alternatives is correct.

 

 

Exhibit 10-3

A statistics teacher wants to see if there is any difference in the abilities of students enrolled in statistics today and those enrolled five years ago.  A sample of final examination scores from students enrolled today and from students enrolled five years ago was taken.  You are given the following information.

 

                                                Today             Five Years Ago

                                                82                          88

                                             112.5                       54

            n                                     45                          36

 

20.       Refer to Exhibit 10-3.  The point estimate for the difference between the means of the two populations is

            a.   58.5

            b.   9

            c.   -9

            d.   -6

 

 

21.       Refer to Exhibit 10-3.  The standard error of  is

            a.   12.9

            b.   9.3

            c.   4

            d.   2

 

 

22.       Refer to Exhibit 10-3.  The 95% confidence interval for the difference between the two population means is

            a.   -9.92 to -2.08

            b.   -3.92 to 3.92

            c.   -13.84 to 1.84

            d.   -24.228 to 12.23

 

 

23.       Refer to Exhibit 10-3.  The test statistic for the difference between the two population means is

            a.   -.47

            b.   -.65

            c.   -1.5

            d.   -3

 

 

24.       Refer to Exhibit 10-3.  The p-value for the difference between the two population means is

            a.   .0014

            b.   .0028

            c.   .4986

            d.   .9972

 

 

25.       Refer to Exhibit 10-3.  What is the conclusion that can be reached about the difference in the average final examination scores between the two classes?  (Use a .05 level of significance.)

            a.   There is a statistically significant difference in the average final examination scores between the two classes.

            b.   There is no statistically significant difference in the average final examination scores between the two classes.

            c.   It is impossible to make a decision on the basis of the information given.

            d.   There is a difference, but it is not significant.

 

 

Exhibit 10-4

The following information was obtained from independent random samples.

Assume normally distributed populations with equal variances.

 

                                                            Sample 1         Sample 2

  Sample Mean 45                                      42

  Sample Variance                                      85                    90

  Sample Size   10                                      12

 

26.       Refer to Exhibit 10-4.  The point estimate for the difference between the means of the two populations is

            a.   0

            b.   2

            c.   3

            d.   15

 

 

27.       Refer to Exhibit 10-4.  The standard error of  is

            a.   3.0

            b.   4.0

            c.   8.372

            d.   19.48

 

 

28.       Refer to Exhibit 10-4.  The degrees of freedom for the t-distribution are

            a.   22

            b.   21

            c.   20

            d.   19

 

 

29.       Refer to Exhibit 10-4.  The 95% confidence interval for the difference between the two population means is

            a.   -5.372 to 11.372

            b.   -5 to 3

            c.   -4.86 to 10.86

            d.   -2.65 to 8.65

 

 

Exhibit 10-5

Two major automobile manufacturers have produced compact cars with the same size engines.  We are interested in determining whether or not there is a significant difference in the MPG (miles per gallon) of the two brands of automobiles.  A random sample of eight cars from each manufacturer is selected, and eight drivers are selected to drive each automobile for a specified distance.  The following data show the results of the test.

 

Driver

Manufacturer A

Manufacturer B

1

32

28

2

27

22

3

26

27

4

26

24

5

25

24

6

29

25

7

31

28

8

25

27

 

30.       Refer to Exhibit 10-5.  The mean for the differences is

            a.   0.50

            b.   1.5

            c.   2.0

            d.   2.5

 

 

31.       Refer to Exhibit 10-5.  The test statistic is

            a.   1.645

            b.   1.96

            c.   2.096

            d.   2.256

 

 

32.       Refer to Exhibit 10-5.  At 90% confidence the null hypothesis

            a.   should not be rejected

            b.   should be rejected

            c.   should be revised

            d.   None of these alternatives is correct.

 

 

Exhibit 10-6

The results of a recent poll on the preference of shoppers regarding two products are shown below.

 

                                                                          Shoppers Favoring

            Product          Shoppers Surveyed         This Product

                 A                             800                                560

                 B                             900                                612

 

33.       Refer to Exhibit 10-6.  The point estimate for the difference between the two population proportions in favor of this product is

            a.   52

            b.   100

            c.   0.44

            d.   0.02

 

 

34.       Refer to Exhibit 10-6.  The standard error of  is

            a.   52

            b.   0.044

            c.   0.0225

            d.   100

 

 

35.       Refer to Exhibit 10-6.  At 95% confidence, the margin of error is

            a.   0.064

            b.   0.044

            c.   0.0225

            d.   52

 

 

36.       Refer to Exhibit 10-6.  The 95% confidence interval estimate for the difference between the populations favoring the products is

            a.   -0.024 to 0.064

            b.   0.6 to 0.7

            c.   0.024 to 0.7

            d.   0.02 to 0.3

 

 

Exhibit 10-7

An insurance company selected samples of clients under 18 years of age and over 18 and recorded the number of accidents they had in the previous year.  The results are shown below.

 

            Under Age of 18                    Over Age of 18

            n1 = 500                                   n2 = 600

            Number of accidents = 180    Number of accidents = 150

 

            We are interested in determining if the accident proportions differ between the two age groups.

 

37.       Refer to Exhibit 10-7 and let pu represent the proportion under and po the proportion over the age of 18.  The null hypothesis is

            a.   pu - po £ 0

            b.   pu - po ³ 0

            c.   pu - po ¹ 0

            d.   pu - po = 0

 

 

38.       Refer to Exhibit 10-7.  The pooled proportion is

            a.   0.305

            b.   0.300

            c.   0.027

            d.   0.450

 

 

39.       Refer to Exhibit 10-7.  The test statistic is

            a.   0.96

            b.   1.96

            c.   2.96

            d.   3.96

 

 

40.       Refer to Exhibit 10-7.  The p-value is

            a.   less than 0.001

            b.   more than 0.10

            c.   0.0228

            d.   0.3

 

 

 

PROBLEMS

 

1.         In order to estimate the difference between the average Miles per Gallon of two different models of automobiles, samples are taken and the following information is collected.

 

                                                      Model A                     Model B

            Sample Size                             60                                55

            Sample Mean                          28                                25

            Sample Variance                     16                                  9

 

            a.   At 95% confidence develop an interval estimate for the difference between the average Miles per Gallon for the two models.

            b.   Is there conclusive evidence to indicate that one model gets a higher MPG than the other?  Why or why not?  Explain.

 

           

 

2.         Independent random samples taken on two university campuses revealed the following information concerning the average amount of money spent on textbooks during the fall semester.

 

                                                University A               University B

            Sample Size                           50                                40

            Average Purchase                $260                            $250

            Standard Deviation (s)         $20                              $23

 

            We want to determine if, on the average, students at University A spent more on textbooks then the students at University B.

 

            a.   Compute the test statistic.

            b.   Compute the p-value.

            c.   What is your conclusion?  Let a = .05.

 

           

 

3.         The management of Recover Fast Hospital (RFH) claims that the average length of stay in their hospital after a major surgery is less than the average length of stay at General Hospital (GH).  The following data have been accumulated to test their claim.

 

                                                         RFH                   GH

            Sample size                             45                      58

            Mean (in days)                        .6                       4.9

            Standard Deviation (s)         0.5                      0.6

 

            a.   Formulate the hypotheses.

            b.   Compute the test statistic.

            c.   Using the p-value approach, test to see if the average length of stay in RFH is significantly less than the average length of stay in GH.   Let a = 0.05.

 

           

 

4.         In order to determine whether or not a driver's education course improves the scores on a driving exam, a sample of 6 students were given the exam before and after taking the course.  The results are shown below.

            Let d = Score After - Score Before.

 

                                               Score                        Score

            Student           Before the Course     After the Course

                 1                              83                               87

                 2                              89                               88

                 3                              93                               91

                 4                              77                               77

                 5                              86                               93

                 6                              79                               83

 

            a.   Compute the test statistic.

            b.   At 95% confidence using the p-value approach, test to see if taking the course actually increased scores on the driving exam.

 

           

 

5.         Of 200 UTC seniors surveyed, 60 were planning on attending Graduate School.  At UTK, 400 seniors were surveyed and 100 indicated that they were planning to attend Graduate School.

 

            a.   Determine a 95% confidence interval estimate for the difference between the proportion of seniors at the two universities that were planning to attend Graduate School.

            b.   Is there conclusive evidence to prove that the proportion of students from UTC who plan to go to Graduate School is significantly more than those from UTK?  Explain.

 

 

6.         Of 300 female registered voters surveyed, 120 indicated they were planning to vote for the incumbent president; while of 400 male registered voters, 140 indicated they were planning to vote for the incumbent president.

 

            a.   Compute the test statistic.

            b.   At alpha = .05, test to see if there is a significant difference between the proportions of females and males who plan to vote for the incumbent president.  (Use the p-value approach.)

 

 

7.         A random sample of 89 tourists in the Grand Bahamas showed that they spent an average of $2,860 (in a week) with a standard deviation of $126; and a sample of 64 tourists in New Province showed that they spent an average of $2,935 (in a week) with a standard deviation of $138.  We are interested in determining if there is any significant difference between the average expenditures of those who visited the two islands?

 

            a.   Determine the degrees of freedom for this test.

            b.   Compute the test statistic.

            c.   Compute the p-value.

            d.   What is your conclusion?  Let a = .05.

 

           

 

8.         Consider the following results for two samples randomly taken from two populations.

 

                                                            Sample A              Sample B

            Sample Size                                    31                         35

            Sample Mean                                106                       102

            Sample Standard Deviation              8                           7

 

            a.   Determine the degrees of freedom for the t-distribution.

            b.   Develop a 95% confidence interval for the difference between the two population means.

 

           

 

9.         Consider the following results for two samples randomly taken from two populations.

 

                                                            Sample A        Sample B

            Sample Size                                   25                    38

            Sample Mean                                66                    60

            Sample Standard Deviation           5                      7

 

            a.   What are the degrees of freedom for the t distribution?

            b.   At 95% confidence, compute the margin of error.

            c.   Develop a 95% confidence interval for the difference between the two population means.

 

           

 

10.       Independent random samples of managers’ yearly salaries (in $1000) taken from governmental and private organizations provided the following information.  At 95% confidence, test to determine if there is a significant difference between the average salaries of the managers in the two sectors.

 

                                Government                  Private

                                     80                               75

            s                             9                               10

            n                           28                               31

 

           

 

11.       Independent random samples taken at two local malls provided the following information regarding purchases by patrons of the two malls.

 

                                                Hamilton Place          Eastgate

            Sample Size                          85                            93

            Average Purchase             $143                        $150

            Standard Deviation            $22                          $18

 

            We want to determine whether or not there is a significant difference between the average purchases by the patrons of the two malls.

 

            a.   Give the hypotheses for the above.

            b.   Compute the test statistic.

            c.   At 95% confidence, test the hypotheses.

 

           

 

12.       Recently, a local newspaper reported that part time students are older than full time students.  In order to test the validity of its statement, two independent samples of students were selected.

 

                        Full Time              Part Time

                    26                          24

            s                  2                            3

            n                42                          31

 

            a.   Give the hypotheses for the above.

            b.   Determine the degrees of freedom.

            c.   Compute the test statistic.

            d.   At 95% confidence, test to determine whether or not the average age of part time students is significantly more than full time students.

 

           

 

13.       The daily production rates for a sample of factory workers before and after a training program are shown below.  Let d = After – Before.

 

            Worker        Before          After

                  1                  6                   9

                  2                10                 12

                  3                  9                  10

                  4                  8                  11

                  5                  7                   9

 

            We want to determine if the training program was effective.

 

            a.   Give the hypotheses for this problem.

            b.   Compute the test statistic.

            c.   At 95% confidence, test the hypotheses.  That is, did the training program actually increase the production rates?

 

 

14.       In a sample of 100 Republicans, 60 favored the President's anti-drug program.  While in a sample of 150 Democrats, 84 favored his program.  At 95% confidence, test to see if there is a significant difference in the proportions of the Democrats and the Republicans who favored the President's anti-drug program.

 

 

15.       During the recent primary elections, the democratic presidential candidate showed the following pre-election voter support in Alabama and Mississippi.

 

                                                                          Voters Favoring the

            State                Voters Surveyed        Democratic Candidate

            Alabama                   800                                 440

            Mississippi                600                                 360

 

            a.   We want to determine whether or not the proportions of voters favoring the Democratic candidate were the same in both states.  Provide the hypotheses.

            b.   Compute the test statistic.

            c.   Determine the p-value; and at 95% confidence, test the above hypotheses.

 

           

 

16.       A test on world history was given to a group of individuals before and also after a film on the history of the world was presented.  The results are given below.  We want to determine if the film significantly increased the test scores.  (For the following matched samples, let the difference "d" be d = after - before.)

 

                              Individual          After               Before

                                    1                      92                      86

                                    2                      86                      88

                                    3                      89                      84

                                    4                      90                      90

                                    5                      93                      85

                                    6                      88                      90

                                    7                      97                      91

 

            a.   Give the hypotheses for this problem.

            b.   Compute the test statistic.

            c.   At 95% confidence, test the hypotheses.

 

           

 

17.       The Dean of Students at UTC has said that the average grade of UTC students is higher than that of the students at GSU.  Random samples of grades from the two schools are selected, and the results are shown below.

 

                                                                        UTC              GSU

            Sample Size                                           14                    12

            Sample Mean                                      2.85                 2.61

            Sample Standard Deviation                0.40                 0.35

            Sample Mode                                      2.5                   3.0

 

            a.   Give the hypotheses.

            b.   Compute the test statistic.

            c.   At a 0.1 level of significance, test the Dean of Students' statement.

 

           

 

18.       The following shows the monthly sales in units of six salespersons before and after a bonus plan was introduced.  At 95% confidence, determine whether the bonus plan has increased sales significantly.  (For the following matched samples, let the difference "d" be:  d = after - before.)

 

                                                Monthly Sales

            Salesperson                After   Before

                    1                              94        90

                    2                              82        84

                    3                              90        84

                    4                              76        70

                    5                              79        80

                    6                              85        80

 

19.       The office of records at a university has stated that the proportion of incoming female students who major in business has increased.  A sample of female students taken several years ago is compared with a sample of female students this year.  Results are summarized below.  Has the proportion increased significantly?  Test at alpha = .10.

 

                                                     Sample Size             No. Majoring in Business

            Previous Sample                      250                                   50

            Present Sample                        300                                   69

 

           

 

20.       The following data present the number of computer units sold per day by a sample of 6 salespersons before and after a bonus plan was implemented.

 

            Salesperson            Before              After

                    1                           3                      6

                    2                           7                      5

                    3                           6                      6

                    4                           8                      7

                    5                           7                      8

                    6                           9                      8

 

            At 95% confidence, test to see if the bonus plan was effective.  That is, did the bonus plan actually increase sales?

 

           

 

21.       Zip, Inc. manufactures Zip drives on two different manufacturing processes.  Because the management of this company is interested in determining if process 1 takes less manufacturing time, they selected independent samples from each process.  The results of the samples are shown below.

 

                                                            Process 1                    Process 2

            Sample Size                                   27                                22

            Sample Mean (in minutes)            10                                14

            Sample Variance                           16                                25

 

            a.   State the null and alternative hypotheses.

            b.   Determine the degrees of freedom for the t test.

            c.   Compute the test statistic

            d.   At 95% confidence, test to determine if there is sufficient evidence to indicate that process 1 takes a significantly shorter time to manufacture the Zip drives.

 

           

 

22.       A recent Time magazine reported the following information about a sample of workers in Germany and the United States.

 

                                                                              United States     Germany

            Average length of workweek (hours)               42                      38

            Sample Standard Deviation                               5                        6

            Sample Size                                                     600                    700

 

            We want to determine whether or not there is a significant difference between the average workweek in the United States and the average workweek in Germany.

 

            a.   State the null and the alternative hypotheses.

            b.   Compute the test statistic.

            c.   Compute the p-value.  What is your conclusion?

 

           

 

23.       Allied Corporation is trying to determine whether to purchase Machine A or B.  It has leased the two machines for a month.  A random sample of 5 employees has been taken.  These employees have gone through a training session on both machines.  Below you are given information on their productivity rate on both machines.  (Let the difference "d" be d = A - B.)

 

                                                Productivity Rate

                    Person           Machine A     Machine B

                        1                        47                    52

                        2                        53                    58

                        3                        50                    47

                        4                        55                    60

                        5                        45                    53

 

            a.   State the null and alternative hypotheses for a two-tailed test.

            b.   Find the mean and standard deviation for the difference.

            c.   Compute the test statistic.

            d.   Test the null hypothesis stated in Part a at the 10% level.

 

           

 

24.       A company attempts to evaluate the potential for a new bonus plan by selecting a sample of 4 salespersons to use the bonus plan for a trial period.  The weekly sales volume before and after implementing the bonus plan is shown below.  (For the following matched samples, let the difference "d" be  d = after - before.)

 

                                                               Weekly Sales

                             Salesperson         Before             After

                                    1                      48                    44

                                    2                      48                    40

                                    3                      38                    36

                                    4                      44                    50

 

            a.   State the hypotheses.

            b.   Compute the test statistic.

            c.   Use Alpha = .05 and test to see if the bonus plan will result in an increase in the mean weekly sales.

 

           

 

25.       The following information was obtained from matched samples regarding the productivity of four individuals using two different methods of production.

 

Individual

Method 1

Method 2

1

6

8

2

9

5

3

7

6

4

7

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9

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3

 

            Let d = Method 1 - Method 2.  Is there a significant difference between the productivity of the two methods?  Let a = 0.05.

 

           

 

26.       The results of a recent poll on the preference of voters regarding presidential candidates are shown below.

 

                                               Voters       Voters Favoring

                  Candidate        Surveyed     This Candidate

                        A                       400                 192

                        B                       450                 225

 

            At 95% confidence, test to determine whether or not there is a significant difference between the preferences for the two candidates.

 

           

 

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