Goodness of FitA company listed five categories and asked each employee to mark the one most important to him or her. The company wants to determine if the current distribution of responses “fits” last year’s distribution or is it different. When considering questions of this type, we are asking whether a population follows a specified distribution.

Goodness of FitA company listed five categories and asked each employee to mark the one most important to him or her. The company wants to determine if the current distribution of responses “fits” last year’s distribution or is it different. When considering questions of this type, we are asking whether a population follows a specified distribution.

The hypotheses we are testing are :

The population fits the given distribution The population has a different distribution

Goodness of FitA company listed five categories and asked each employee to mark the one most important to him or her. The company wants to determine if the current distribution of responses “fits” last year’s distribution or is it different. When considering questions of this type, we are asking whether a population follows a specified distribution.

The hypotheses we are testing are :

The population fits the given distribution The population has a different distribution

We will use the chi – square distribution to test the “goodness – of – fit” hypotheses.

Goodness of FitA company listed five categories and asked each employee to mark the one most important to him or her. The company wants to determine if the current distribution of responses “fits” last year’s distribution or is it different. When considering questions of this type, we are asking whether a population follows a specified distribution.

The hypotheses we are testing are :

The population fits the given distribution The population has a different distribution

We will use the chi – square distribution to test the “goodness – of – fit” hypotheses.

with degrees of freedom =

where # of categories in the distribution

Goodness of FitHere is the table of last year’s responses…

Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30

Salary 290Safety Regulations 70

Health & Retirement Benefits

70

Overtime Policy & Pay 40

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30

Salary 290Safety Regulations 70

Health & Retirement Benefits

70

Overtime Policy & Pay 40

To find the expected frequencies, we will multiply each % favorable from the previous population by 500…

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20

Salary 290Safety Regulations 70

Health & Retirement Benefits

70

Overtime Policy & Pay 40

4% X 500 = 0.04 X 500 = 20

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20

Salary 290 325Safety Regulations 70

Health & Retirement Benefits

70

Overtime Policy & Pay 40

4% X 500 = 0.04 X 500 = 2065% X 500 = 0.65 X 500 = 325

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20

Salary 290 325Safety Regulations 70 65

Health & Retirement Benefits

70

Overtime Policy & Pay 40

4% X 500 = 0.04 X 500 = 2065% X 500 = 0.65 X 500 = 32513% X 500 = 0.13 X 500 = 65

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20

Salary 290 325Safety Regulations 70 65

Health & Retirement Benefits

70 60

Overtime Policy & Pay 40

4% X 500 = 0.04 X 500 = 2065% X 500 = 0.65 X 500 = 32513% X 500 = 0.13 X 500 = 6512% X 500 = 0.12 X 500 = 60

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20

Salary 290 325Safety Regulations 70 65

Health & Retirement Benefits

70 60

Overtime Policy & Pay 40 30

4% X 500 = 0.04 X 500 = 2065% X 500 = 0.65 X 500 = 32513% X 500 = 0.13 X 500 = 6512% X 500 = 0.12 X 500 = 606% X 500 = 0.06 X 500 = 30

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20

Salary 290 325Safety Regulations 70 65

Health & Retirement Benefits

70 60

Overtime Policy & Pay 40 30

Now fill in the column

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100

Salary 290 325Safety Regulations 70 65

Health & Retirement Benefits

70 60

Overtime Policy & Pay 40 30

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100

Salary 290 325 1225Safety Regulations 70 65

Health & Retirement Benefits

70 60

Overtime Policy & Pay 40 30

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100

Salary 290 325 1225Safety Regulations 70 65 25

Health & Retirement Benefits

70 60

Overtime Policy & Pay 40 30

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100

Salary 290 325 1225Safety Regulations 70 65 25

Health & Retirement Benefits

70 60 100

Overtime Policy & Pay 40 30

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100

Salary 290 325 1225Safety Regulations 70 65 25

Health & Retirement Benefits

70 60 100

Overtime Policy & Pay 40 30 100

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100

Salary 290 325 1225Safety Regulations 70 65 25

Health & Retirement Benefits

70 60 100

Overtime Policy & Pay 40 30 100

Now fill in

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100 5.00

Salary 290 325 1225Safety Regulations 70 65 25

Health & Retirement Benefits

70 60 100

Overtime Policy & Pay 40 30 100

100/20 = 5

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100 5.00

Salary 290 325 1225 3.77Safety Regulations 70 65 25

Health & Retirement Benefits

70 60 100

Overtime Policy & Pay 40 30 100

1225/325 = 3.77

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100 5.00

Salary 290 325 1225 3.77Safety Regulations 70 65 25 0.38

Health & Retirement Benefits

70 60 100

Overtime Policy & Pay 40 30 100

25/65 = 0.38

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100 5.00

Salary 290 325 1225 3.77Safety Regulations 70 65 25 0.38

Health & Retirement Benefits

70 60 100 1.67

Overtime Policy & Pay 40 30 100

100/60 = 1.67

Goodness of FitHere is the table of last year’s responses…Category % of Favorable Responses

Vacation Time 4%

Salary 65%

Safety Regulations 13%

Health & Retirement Benefits 12%

Overtime Policy & Pay 6%

The new year’s responses were drawn from a sample of 500 employees. We need to set up the chi – square computation table…Category

Vacation Time 30 20 100 5.00

Salary 290 325 1225 3.77Safety Regulations 70 65 25 0.38

Health & Retirement Benefits

70 60 100 1.67

Overtime Policy & Pay 40 30 100 3.33 100/30 = 3.33

Goodness of FitCategory

Vacation Time 30 20 100 5.00

Salary 290 325 1225 3.77Safety Regulations 70 65 25 0.38Health & Retirement Benefits

70 60 100 1.67

Overtime Policy & Pay 40 30 100 3.33

∑ = 14.15

Now sum that column…

Goodness of FitCategory

Vacation Time 30 20 100 5.00

Salary 290 325 1225 3.77Safety Regulations 70 65 25 0.38Health & Retirement Benefits

70 60 100 1.67

Overtime Policy & Pay 40 30 100 3.33

∑ = 14.15

Now sum that column…

with

Goodness of FitCategory

Vacation Time 30 20 100 5.00

Salary 290 325 1225 3.77Safety Regulations 70 65 25 0.38Health & Retirement Benefits

70 60 100 1.67

Overtime Policy & Pay 40 30 100 3.33

∑ = 14.15

Now sum that column…

with

is between 0.005 and 0.010 and we will use

Goodness of FitCategory

Vacation Time 30 20 100 5.00

Salary 290 325 1225 3.77Safety Regulations 70 65 25 0.38Health & Retirement Benefits

70 60 100 1.67

Overtime Policy & Pay 40 30 100 3.33

∑ = 14.15

Now sum that column…

with

is between 0.005 and 0.010 and we will use

Since we reject the null hypothesis

Goodness of FitSo we are basically doing the chi – square steps. The only difference is our expected outcomes are based on a previous population’s random sample. Let’s try another example…

Goodness of FitEXAMPLE # 2 : The age of distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of Red Lake Village are shown below. Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

Age ( years ) % Canadian Pop. Observed Number in Red Lake Village

Under 5 7.2% 47

5 to 14 13.6% 75

15 to 64 67.1% 288

65 and older 12.1% 45

Goodness of FitEXAMPLE # 2 : The age of distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of Red Lake Village are shown below. Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

Age ( years ) % Canadian Pop. Observed Number in Red Lake Village

Under 5 7.2% 47

5 to 14 13.6% 75

15 to 64 67.1% 288

65 and older 12.1% 45

Age

Under 5 47

5 – 14 75

15 – 65 288

65 or older 45

∑ =

Set up your chi – square computation table and fill in the observed frequencies and the rest of the cells…

Goodness of FitEXAMPLE # 2 : The age of distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of Red Lake Village are shown below. Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

Age ( years ) % Canadian Pop. Observed Number in Red Lake Village

Under 5 7.2% 47

5 to 14 13.6% 75

15 to 64 67.1% 288

65 and older 12.1% 45

Age

Under 5 47 33

5 – 14 75 62

15 – 65 288 305

65 or older 45 55

∑ =

455 X 0.072 = 33455 X 0.136 = 62455 X 0.671 = 305455 X 0.121 = 55

Goodness of FitEXAMPLE # 2 : The age of distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of Red Lake Village are shown below. Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

Age ( years ) % Canadian Pop. Observed Number in Red Lake Village

Under 5 7.2% 47

5 to 14 13.6% 75

15 to 64 67.1% 288

65 and older 12.1% 45

Age

Under 5 47 33 196

5 – 14 75 62 169

15 – 65 288 305 289

65 or older 45 55 100

∑ =

Goodness of FitEXAMPLE # 2 : The age of distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of Red Lake Village are shown below. Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

Age ( years ) % Canadian Pop. Observed Number in Red Lake Village

Under 5 7.2% 47

5 to 14 13.6% 75

15 to 64 67.1% 288

65 and older 12.1% 45

Age

Under 5 47 33 196 5.94

5 – 14 75 62 169 2.73

15 – 65 288 305 289 0.95

65 or older 45 55 100 1.82

∑ = 11.44

Goodness of Fit The population fits the specified distribution of categories The population has a different distribution

Age

Under 5 47 33 196 5.94

5 – 14 75 62 169 2.73

15 – 65 288 305 289 0.95

65 or older 45 55 100 1.82

∑ = 11.44

𝑥2=11.44 h𝑤𝑖𝑡 𝑑 . 𝑓 .=𝑘−1=4−1=3

Goodness of Fit The population fits the specified distribution of categories The population has a different distribution

Age

Under 5 47 33 196 5.94

5 – 14 75 62 169 2.73

15 – 65 288 305 289 0.95

65 or older 45 55 100 1.82

∑ = 11.44

Goodness of Fit The population fits the specified distribution of categories The population has a different distribution

Age

Under 5 47 33 196 5.94

5 – 14 75 62 169 2.73

15 – 65 288 305 289 0.95

65 or older 45 55 100 1.82

∑ = 11.44

Since 0.05, we reject null hypothesis