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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