1. Chi-squared tests
Goodness of fit: Does the actual frequency distribution of some data agree with an assumption?
Test of Independence: Are two characteristics of a set of data dependent or not?
Test of homogeneity: Do different populations have the same characteristic in the same proportion?
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2. Example: The distribution of colors in a bag of M&M’s is the same
Goodness-of-Fit Test
Test the hypothesis that an observed frequency distribution fits some claimed distribution
Example: The distribution of colors in a bag of M&M’s is the same
Another example: The distribution of colors in a bag of M&M’s is the following:
30% Brown 10% Green
20% Red 10% Blue
20% Yellow 10% Orange
The name for this test comes from the idea that we are testing how well an observed frequency distribution fits some specified theoretical distribution.

3. Categories have equal frequencies
Hypothesis Testing
Categories have equal frequencies
H0: p1 = p2 = p3 = = pk
H1: at least one of the probabilities is different
Categories have unequal frequencies
H0 : p1 , p2, p3, , pk are as claimed
H1 : at least one of the above proportions is not the claimed value
Examples on page of text.

4. Claim: The distribution of colors is the same:
Expected Frequencies
Brown
Yellow
Red
Orange
Green
Blue
Observed frequency (O) 33 26 21 8 7 5
Expected frequency (E)
Claim: The distribution of colors is the same:
First, we collect our sample (Observed frequency)
Next, we calculate our Expected frequency:
If all expected frequencies are equal:
If all expected frequencies are not all equal:
Each category’s expected frequency is the product the total and the category’s probability
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5. Test Statistics and Critical Values
Brown
Yellow
Red
Orange
Green
Blue
Observed frequency (O) 33 26 21 8 7 5
Expected frequency (E)
(O-E)2/E
Test Statistic:
This formula is a measure of the discrepancies between the O and E values.
The chi-distribution was first presented in Chapter 6, Section 6-6, page 343.

6. Critical Values in Chi Square table
Fail to Reject
Reject
Discuss with students that the test does not identify where the significant discrepancies are. However, the figure on the following slide helps to visualize where they might be.
Heading: Alpha
Goodness-of-fit hypothesis tests are always right-tailed.
Degrees of freedom: number of categories minus 1

7. Uneven Distribution Example
Mars claims its M&M candies are distributed with the color percentages of 30% brown, 20% yellow, 20% red, 10% orange, 10% green, and 10% blue.
At the 0.05 significance level, test the claim that the color distribution is as claimed by Mars, Inc.
H0 (and claim): p1 = 0.30, p2 = 0.20, p3 = 0.20, p4 = 0.10, p5 = 0.10, p6 = 0.10
H1: At least one of the proportions is different from the claimed value.

8. Uneven Distribution Test statistic Critical Value Brown Yellow Red
Orange
Green
Blue
Observed frequency(O) 33 26 21 8 7 5
Expected frequency
(O-E)2/E
Test statistic
Critical Value

9. Locating the test statistic
 = 0.05
Fail to Reject
Reject
Discuss with students that the test does not identify where the significant discrepancies are. However, the figure on the following slide helps to visualize where they might be.

10. For example Even distribution: Uneven distribution:
The population of South students is evenly distributed among the four classes
Uneven distribution:
The population of South students is distributed as follows:
15% Lincroft 15% River Plaza
20% Nut Swamp 20% Navesink
20% Leonard 5% Village
5% Private

11. Live example Leonardo 20 Frosh 29 Lincroft 18 Sophomore 24 Navesink 19
Junior
Nut Swamp 21
Senior 23
River Plaza 10
Private 7
Village 5

12. Your Turn
Month
Frequency 1 2 3 5 4 6 7 9 8 10 11 12
Claim: students’ favorite month is equally distributed throughout the year

13. Your Turn, again
Claim: students’ favorite colors distribution are are 25% green, 25% blue, and 10% the remaining colors
red 8
green 10
blue 13
orange 2
purple 7
black
yellow 4