week two statistics

profilegrupy1
week_two_statistics.docx
Home>Business & Finance homework help>Accounting homework help>week two statistics

Ashford 3: - Week 2 - Instructor Guidance

Week Overview:

The following video series: Against All Odds Inside Statistics is helpful if you would like to watch it.

http://www.learner.org/resources/series65.html?pop=yes&pid=3138

Back ground

You’re a plant operations manager for Springfield Cereals. You are responsible for monitoring the amount in each cereal box filled. The company specification requires a mean weight of 300 grams per box. It is your responsibility to adjust the process when the mean fill weight in the population of boxes differs from 300 grams. How can you rationally make the decision whether or not to adjust the process when it is impossible to weigh every single box as it is being filled? You begin by selecting and weighing a random sample of 30 cereal boxes. After calculating a sample mean, how do you proceed?

What is a hypothesis?

A hypothesis is a claim (assertion) about a population parameter including population mean, population proportion, standard deviation, etc.

Example: The mean monthly cell phone bill in this city is μ = $42

Example: The proportion of adults in this city with cell phones is π = 0.68

Hypothesis testing typically begins with some theory, claim or assertion about a particular parameter of a population. For example, your initial hypothesis about the cereal example is that the process is working properly, so the mean fill is 300 grams, and no corrective action is needed.

The Null Hypothesis, H0

The null hypothesis states the claim or assertion to be tested

Example: The average number of TV sets in U.S. Homes is equal to three. It is always about a population parameter, not about a sample statistic

The null hypothesis begins with the assumption that the null hypothesis is true. It is similar to the notion of innocent until proven guilty

The Alternative Hypothesis, H1

The alternative hypothesis ss the opposite of the null hypothesis, e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 ). It challenges the status quo. It may or may not be proven and is generally the hypothesis that the researcher is trying to prove.

Hypothesis Testing Process

Example

Claim: The population mean age is 50.

H0: μ = 50, H1: μ ≠ 50

Sample the population and find the sample mean.

Suppose the sample mean age was X = 20.

This is significantly lower than the claimed mean population age of 50.

If the null hypothesis were true, the probability of getting such a different sample mean would be very small, so you reject the null hypothesis .

In other words, getting a sample mean of 20 is so unlikely if the population mean was 50, you conclude that the population mean must not be 50.

The Test Statistics and Critical Value

If the sample mean is close to the assumed population mean, the null hypothesis is not rejected.

If the sample mean is far from the assumed population mean, the null hypothesis is rejected.

How far is “far enough” to reject H0?

The critical value of a test statistic creates a “line in the sand” for decision making -- it answers the question of how far is far enough.

Discussion and Assignment Background

Two Sample Test

Difference between Two Means

Discussion

You need to understand hypothesis test to answer the discussion questions.

Assignment

There are several t tests for assignment.

Problem 1,

You need to interpret the results (test results are shown). You reject Ho if p-value is less than 0.05.

Problem 2

You can use Excel to run the test, the results should look like the following.

Ho: Male mean salary = Female mean salary

Ha: Male mean salary =/= Female mean salary

Male

Female

Mean

52

38

Variance

316

334.667

Observations

25

25

Pooled Variance

325.333333

Hypothesized Mean Difference

0

df

48

t Stat

2.74421896

P(T<=t) one-tail

0.00425301

t Critical one-tail

1.6772242

P(T<=t) two-tail

0.00850602

t Critical two-tail

2.01063476

Problem 3

This is similar to #2

Ho: Male mean compa = Female mean compa

Ha: Male mean compa =/= Female mean compa

Problem 4

This is paired two sample test.

Ho: Average salary => (equal or greater than) average midpoint

Ha: Average salary < average midpoint

Ashford 3: - Week 2 - Discussion 1

Your initial discussion thread is due on Day 3 (Thursday) and you have until Day 7 (Monday) to respond to your classmates. Your grade will reflect both the quality of your initial post and the depth of your responses. Reference the Discussion Forum Grading Rubric for guidance on how your discussion will be evaluated.

Hypotheses

What is a hypothesis test? Why do we need to use them to make decisions about relating sample results to the population; why can’t we just make our decisions by the sample value?

Guided Response: Review several of your classmates’ posts. Respond to at least two classmates by commenting on the potential differences in the results and how that might affect decision making.

Ashford 3: - Week 2 - Discussion 2

Your initial discussion thread is due on Day 3 (Thursday) and you have until Day 7 (Monday) to respond to your classmates. Your grade will reflect both the quality of your initial post and the depth of your responses. Reference the Discussion Forum Grading Rubric for guidance on how your discussion will be evaluated.

Variation

Variation exists in virtually all parts of our lives. We often see variation in results in what we spend (utility costs each month, food costs, business supplies, etc.). Consider the measures and data you use (in either your personal or job activities). When are differences (between one time period and another, between different production lines, etc.) between average or actual results important? How can you or your department decide whether or not the observed differences over time are important? How could using a mean difference test help? Guided Response: Review several of your classmates’ posts. Respond to at least two classmates and comment on the use of the test.

Ashford 3: - Week 2 - Assignment

Problem Set Week Two In the Week Two Assignment sheet, complete the problems below and submit your work in an Excel document. Be sure to show all of your work and clearly label all calculations. All statistical calculations will use the Employee Salary Data Set and Week 2 assignment sheet. (Note: Questions 1- 4 have additional elements to respond to below the analysis results and included in the Week Two Assignment sheet are 2 one-sample t-tests comparing male and female average salaries to the overall sample mean.)

Carefully review the Grading Rubric for the criteria that will be used to evaluate your assignment

See comments at the right of the data set.

ID

Salary

Compa

Midpoint

Age

Performance Rating

Service

Gender

Raise

Degree

Gender1

Grade

8

23

1.000

23

32

90

9

1

5.8

0

F

A

The ongoing question that the weekly assignments will focus on is: Are males and females paid the same for equal work (under the Equal Pay Act)?

10

22

0.956

23

30

80

7

1

4.7

0

F

A

Note: to simplfy the analysis, we will assume that jobs within each grade comprise equal work.

11

23

1.000

23

41

100

19

1

4.8

0

F

A

14

24

1.043

23

32

90

12

1

6

0

F

A

The column labels in the table mean:

15

24

1.043

23

32

80

8

1

4.9

0

F

A

ID – Employee sample number

Salary – Salary in thousands

23

23

1.000

23

36

65

6

1

3.3

1

F

A

Age – Age in years

Performance Rating – Appraisal rating (Employee evaluation score)

26

24

1.043

23

22

95

2

1

6.2

1

F

A

Service – Years of service (rounded)

Gender: 0 = male, 1 = female

31

24

1.043

23

29

60

4

1

3.9

0

F

A

Midpoint – salary grade midpoint

Raise – percent of last raise

35

24

1.043

23

23

90

4

1

5.3

1

F

A

Grade – job/pay grade

Degree (0= BS\BA 1 = MS)

36

23

1.000

23

27

75

3

1

4.3

1

F

A

Gender1 (Male or Female)

Compa - salary divided by midpoint

37

22

0.956

23

22

95

2

1

6.2

1

F

A

42

24

1.043

23

32

100

8

1

5.7

0

F

A

3

34

1.096

31

30

75

5

1

3.6

0

F

B

18

36

1.161

31

31

80

11

1

5.6

1

F

B

20

34

1.096

31

44

70

16

1

4.8

1

F

B

39

35

1.129

31

27

90

6

1

5.5

1

F

B

7

41

1.025

40

32

100

8

1

5.7

0

F

C

13

42

1.050

40

30

100

2

1

4.7

1

F

C

22

57

1.187

48

48

65

6

1

3.8

0

F

D

24

50

1.041

48

30

75

9

1

3.8

1

F

D

45

55

1.145

48

36

95

8

1

5.2

0

F

D

17

69

1.210

57

27

55

3

1

3

0

F

E

48

65

1.140

57

34

90

11

1

5.3

1

F

E

28

75

1.119

67

44

95

9

1

4.4

1

F

F

43

77

1.149

67

42

95

20

1

5.5

1

F

F

19

24

1.043

23

32

85

1

0

4.6

1

M

A

25

24

1.043

23

41

70

4

0

4

0

M

A

40

25

1.086

23

24

90

2

0

6.3

0

M

A

2

27

0.870

31

52

80

7

0

3.9

0

M

B

32

28

0.903

31

25

95

4

0

5.6

0

M

B

34

28

0.903

31

26

80

2

0

4.9

1

M

B

16

47

1.175

40

44

90

4

0

5.7

0

M

C

27

40

1.000

40

35

80

7

0

3.9

1

M

C

41

43

1.075

40

25

80

5

0

4.3

0

M

C

5

47

0.979

48

36

90

16

0

5.7

1

M

D

30

49

1.020

48

45

90

18

0

4.3

0

M

D

1

58

1.017

57

34

85

8

0

5.7

0

M

E

4

66

1.157

57

42

100

16

0

5.5

1

M

E

12

60

1.052

57

52

95

22

0

4.5

0

M

E

33

64

1.122

57

35

90

9

0

5.5

1

M

E

38

56

0.982

57

45

95

11

0

4.5

0

M

E

44

60

1.052

57

45

90

16

0

5.2

1

M

E

46

65

1.140

57

39

75

20

0

3.9

1

M

E

47

62

1.087

57

37

95

5

0

5.5

1

M

E

49

60

1.052

57

41

95

21

0

6.6

0

M

E

50

66

1.157

57

38

80

12

0

4.6

0

M

E

6

76

1.134

67

36

70

12

0

4.5

1

M

F

9

77

1.149

67

49

100

10

0

4

1

M

F

21

76

1.134

67

43

95

13

0

6.3

1

M

F

29

72

1.074

67

52

95

5

0

5.4

0

M

F

Score:

Week 2

Testing means - T-tests

Q3

In questions 2 and 3, be sure to include the null and alternate hypotheses you will be testing.

Ho

Female

Male

Female

In the first 3 questions use alpha = 0.05 in making your decisions on rejecting or not rejecting the null hypothesis.

45

34

1.017

1.096

45

41

0.870

1.025

<1 point>

1

Below are 2 one-sample t-tests comparing male and female average salaries to the overall sample mean.

45

23

1.157

1.000

(Note: a one-sample t-test in Excel can be performed by selecting the 2-sample unequal variance t-test and making the second variable = Ho value -- see column S)

45

22

0.979

0.956

Based on our sample, how do you interpret the results and what do these results suggest about the population means for male and female average salaries?

45

23

1.134

1.000

Males

Females

45

42

1.149

1.050

Ho: Mean salary = 45

Ho: Mean salary = 45

45

24

1.052

1.043

Ha: Mean salary =/= 45

Ha: Mean salary =/= 45

45

24

1.175

1.043

45

69

1.043

1.210

Note: While the results both below are actually from Excel's t-Test: Two-Sample Assuming Unequal Variances,

45

36

1.134

1.161

having no variance in the Ho variable makes the calculations default to the one-sample t-test outcome - we are tricking Excel into doing a one sample test for us.

45

34

1.043

1.096

 

Male

Ho

 

Female

Ho

45

57

1.000

1.187

Mean

52

45

Mean

38

45

45

23

1.074

1.000

Variance

316

0

Variance

334.667

0

45

50

1.020

1.041

Observations

25

25

Observations

25

25

45

24

0.903

1.043

Hypothesized Mean Difference

0

Hypothesized Mean Difference

0

45

75

1.122

1.119

df

24

df

24

45

24

0.903

1.043

t Stat

1.96890383

t Stat

-1.9132

45

24

0.982

1.043

P(T<=t) one-tail

0.03030785

P(T<=t) one-tail

0.03386

45

23

1.086

1.000

t Critical one-tail

1.71088208

t Critical one-tail

1.71088

45

22

1.075

0.956

P(T<=t) two-tail

0.0606157

P(T<=t) two-tail

0.06772

45

35

1.052

1.129

t Critical two-tail

2.06389856

 

t Critical two-tail

2.0639

 

45

24

1.140

1.043

Conclusion: Do not reject Ho; mean equals 45

Conclusion: Do not reject Ho; mean equals 45

45

77

1.087

1.149

Is this a 1 or 2 tail test?

Is this a 1 or 2 tail test?

- why?

- why?

P-value is:

P-value is:

45

55

1.052

1.145

Is P-value > 0.05?

Is P-value > 0.05?

45

65

1.157

1.140

Why do we not reject Ho?

Why do we not reject Ho?

Interpretation:

<1 point>

2

Based on our sample data set, perform a 2-sample t-test to see if the population male and female average salaries could be equal to each other.

(Since we have not yet covered testing for variance equality, assume the data sets have statistically equal variances.)

Ho:

Ha:

Test to use:

Place B43 in Outcome range box.

P-value is:

Is P-value < 0.05?

Reject or do not reject Ho:

If the null hypothesis was rejected, what is the effect size value:

Meaning of effect size measure:

Interpretation:

b.

Since the one and two sample t-test results provided different outcomes, which is the proper/correct apporach to comparing salary equality? Why?

<1 point>

3

Based on our sample data set, can the male and female compas in the population be equal to each other? (Another 2-sample t-test.)

Ho:

Ha:

Statistical test to use:

Place B75 in Outcome range box.

What is the p-value:

Is P-value < 0.05?

Reject or do not reject Ho:

If the null hypothesis was rejected, what is the effect size value:

Meaning of effect size measure:

Interpretation:

<1 point>

4

Since performance is often a factor in pay levels, is the average Performance Rating the same for both genders?

Ho:

Ha:

Test to use:

Place B106 in Outcome range box.

What is the p-value:

Is P-value < 0.05?

Do we REJ or Not reject the null?

If the null hypothesis was rejected, what is the effect size value:

Meaning of effect size measure:

Interpretation:

<2 points>

5

If the salary and compa mean tests in questions 2 and 3 provide different results about male and female salary equality,

which would be more appropriate to use in answering the question about salary equity? Why?

What are your conclusions about equal pay at this point?