Goodness of Fit Test - storage.googleapis.com · Test Statistic: _____ Rejection Region Spring Summer Fall Winter Observed Counts 92 80 120 108 Expected Counts 100 100 100 100 Goodness

Goodness of Fit Test

Lecture 23

March 19, 2018

Four Stages of Statistics

• Data Collection �

• Displaying and Summarizing Data �

• Probability �

• Inference• One Quantitative �

• One Categorical• One-Sample Proportion Test �

• Goodness of Fit Test

• One Categorical and One Quantitative

• Two Categorical

• Two Quantitative

Chi-Squared Distribution

• Chi-Squared Distribution: continuous probability distribution with the following properties:

• Unimodal and right-skewed

• Always non-negative

• Mean equal to number of degrees of freedom

• Changes shape depending on degrees of freedom• Becomes less right-skewed as df increase

Examples of Chi-Squared Distribution

Example of Chi-Squared Table Example #1: Using Chi-Squared Table

• Question: What is the chi-squared statistic with 4 degrees of freedom corresponding to 10% of the area in the upper tail?

• Answer: ___________________

______

Motivation: Goodness of Fit Test

• Scenario: Birthday effect of 400 HS football players across four seasons

• Question: Are the birthdays evenly distributed across all seasons using � = .05?

• Problem:• Variable has _________________________________

• One-sample proportion test can only __________________ ____________________________

Spring Summer Fall Winter

Sample Prop.�� = .23 �

� = .20 �� = .30 �

� = .27Hypothesized Prop. .25 .25 .25 .25

Goodness of Fit Test: Hypotheses

• Used For: • Determining if collected data is consistent with a

specified probability distribution

• Hypotheses: • �: The data is consistent with the specified

distribution.

• � : At least one �� differs from its hypothesized value

Example #2: Doing Hypothesis Test


• Hypothesis Test:1. Hypotheses:

• �: ______________________________________________________________• � : ______________________________________________________________

_____________________________


Observed Counts 92 80 120 108

Hypothesized Prop. .25 .25 .25 .25

Goodness of Fit Test: Conditions

• Conditions: • Expected counts for each category at least 5

• Expected Counts: number of observations we would expect in each category if � is true

• Hypothesized proportion for each category: ��• Total sample size: �• Expected count for category �: ��



• Hypothesis Test:2. Conditions:

• All expected counts ______________________

• Goodness of fit test ______________________




Expected Counts

Goodness of Fit Test: Test Statistic

• Test Statistic:

�� = � Actual − Expected �

Expected$

�% • Follows chi-squared distribution with & − 1 df

• Idea: Test statistic compared observed and expected counts relative to sample size

• Each group made a contribution to the test statistic

• If expected and actual are close, contribution is small

• If expected and actual are far apart, contribution is large

Number of categories


• Hypothesis Test:3. Test Statistic:

�� = ___________________________________________________________= ___________________________

= __________

Degrees of Freedom: () = __________________



Expected Counts 100 100 100 100

Goodness of Fit Test: Conclusion

• Decision: Reject � for large values of the test statistic

• Imply that actual and expected counts are far away

• Always an upper one-sided test

• Conclusion: At least one category differs from its hypothesized proportion

• Do not know which categories or how many


• Hypothesis Test:4. Critical Value: _______________

5. Decision: ____________________ (________________________)

6. Conclusion: ____________________________________________ __________________________________________________________

Test Statistic:

_______________

Rejection

Region



Expected Counts 100 100 100 100

Goodness of Fit vs. Proportion Test

• In one-sample proportion test…• Categorical variable had 2 categories

• If you hypothesize � = .60, hypothesized proportion for the other category is .40

• In goodness of fit test…• Categorical variable has 3 or more categories

• If you hypothesize � = .60 for one category, all other categories add to .40, but we don’t know how it is allocated to other categories

• Need to check accuracy of all categories simultaneously

Determining Which Categories Differ

• To determine which categories differ from their hypothesized proportions:

• Calculate a confidence interval for each category

• Determine if the interval contains the hypothesized proportion for each

Note: If � is rejected, at least one interval will not contain

its hypothesized proportion and will be the reason the null

hypothesis was rejected.

Example #3: Which Categories Differ


• Question: Which categories’ proportions differed significantly from .25?

• Strategy:• Calculate _________________________________________________

for each category

• Look for __________________________________ in the interval


Sample Prop. .23 .20 .30 .27




Season Sample Prop. Confidence Interval

Spring .23 .23 ± 1.96 .�-(.//)� = ______________

Summer .20 .20 ± 1.96 .�(.�)� = ______________

Fall .30 .30 ± 1.96 .-(./)� = ______________

Winter .27 .27 ± 1.96 .�/(./-)� = ______________



• Question: Which categories’ proportions differed significantly from .25?

• Answer: _____________________________________________

Season Sample Prop. Interval Sig. Difference?

Spring .23 (.189, .271)Summer .20 (.161, .239)Fall .30 (.255, .345)Winter .27 (.226, .314)

Example #4: Type I Error

• Results: We found a test statistic of �� = 9.28which corresponds to a p-value of .0258. This led us to reject � at the 5% level of significance.

• Question: What is the probability of making a Type I Error?

• Answer: ___________________________

• Question: What would have happened if we had made a Type I error?

• Answer: Conclude that _____________________________ when in reality ______________________________________ ________________________________________________________

Example #5: Interpreting Output

• Scenario: Mars Company claims that plain M&M’s have the following distribution of colors:

You open a bag and count each color.

• Question: Is Mars’ claim accurate at � = .05?

• Hypothesis Test:1. Hypotheses:

• �: __________________________________________________________________

_________________________________________________________________

• � : _______________________________________________________________

Color Blue Orange Green Yellow Brown Red

Prop. .24 .20 .16 .14 .13 .13


If the bags are actually being filled according to ______________

_______________, then the probability of getting __________________

______________________________ than what was observed is _______.


• Hypothesis Test:2. Conditions: _____________________________________________

3. Test Statistic: ______________________

4. P-Value: _____________

5. Decision: _________________________ (___________________)

6. Conclusion: ____________________________________________ __________________________________________________________

_____

_______

Example #6: Confidence Intervals

• Question: What do you notice about all six confidence intervals?

• Answer: All of them ________________________________ ________________________

• Reason: ___________________________________________________

Summary

• Goodness of Fit Test: used to test proportions of a categorical variable with at least three categories

• Expected Counts: how many observations we would expect in each category if the null hypothesis is true

• Test Statistic:

�� = � Actual − Expected �

Expected$

�% • Large values are evidence against �

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Goodness of Fit Test - storage.googleapis.com · Test Statistic: _____ Rejection Region Spring Summer Fall Winter Observed Counts 92 80 120 108 Expected Counts 100 100 100 100 Goodness