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### AP Statistics

Chi – Square Tests

Chi-Square GOF Example

- Rolling a die…

MC Practice Use for 1 – 3.

- A random sample of 100 traffic tickets given to motorists in a large city is examined. The tickets were classified according to the race of the driver. There results are summarized below:
WhiteBlackHispanicOther

# 46 37 11 6

The proportion of the population of the city is…

WhiteBlackHispanicOther

Prop. 0.65 .30 0.03 0.02

MC cont.

1) We wish to test whether the racial distribution of traffic tickets in the city is the same as the racial distribution of the population of the city. To do so we use the X2 statistic. The component of the X2 Statistic corresponding to the Hispanic category is:

A) C)

B) D)

MC cont.

2) We wish to test whether the racial distribution of traffic tickets in the city is the same as the racial distribution of the population of the city. To do so we use the X2 statistic. Assume that this X2 statistic has a approximately a χ2 distribution, the P-value of our test is:

A) greater than 0.10 C) between 0.05 and 0.01

B) between 0.10 and 0.05 D) less than 0.01

MC cont.

3) We wish to test whether the racial distribution of traffic tickets in the city is the same as the racial distribution of the population of the city. To do so we use the X2 statistic. Why may we assume the X2 statistic has a approximately a χ2 distribution?

A) The number of tickets given in each race category is greater than 5.

B) The sample size is 100, which is large enough.

C) The number of categories is small relative to the number of observations.

D) We should not make this assumption in this context.

Notes: Inference for 2-way Tables

- When we want to compare more than 2 groups we use a Chi-Square test. Often the Chi-Square test will be used for categorical variables described in a two-way table.
- There are two types of tests:
- Chi-Square test for homogeneity of proportions
- Chi-Square test for association/independence

Test for Homogeneity

- Homogeneity means “the same”. It is used when we have more than two groups and we want to know if the category proportions are the same for each group.
- Our data will be organized in a two way table, which we will put into a matrix in our calculator to do the grunt work.
- Degrees of Freedom: df = (r – 1)(c – 1).

Test for homogeneity cont.

- The expected count in any cell of a 2-way table when H0 is true is
expected count = (row total) x (column total)

(table total)

Conditions:

- Expected counts are all larger than 5.
- Data must be an SRS to generalize to population.

Example 1

- The table below shows the number of Del Riomante students who passed the AP Calculus AB exam. Has the distribution of passing scores changed over those three years?
Year

Score200520062007

3 11 15 18

4 11 12 13

5 13 14 12

Ex 1 cont.

- Chi – Square Test for homogeneity
- Check conditions:
- We do not have a SRS, but no generalization is intended.
- Expected cell counts are all greater than 5, see table below.

- Hypotheses:
- H0: The passing scores on the Calculus AB exam have the same distribution for 2005, 2006, 2007.
- HA: The passing scores on the Calculus AB exam do not have the same distribution for 2005, 2006, 2007.

Ex 1. cont.

- Mechanics:
- df = (3 – 1)(3 – 1) = 4
- Expected Counts from Calculator.
- p-value = P(X2 > 1.138) = 0.888 (calculator)

Ex 1 cont.

- Conclusion: The p-value is very large. The is not sufficient evidence to reject the null hypothesis that the passing grades for the AP Calculus AB exam for the years 2005, 2006, 2007 have the same distribution.

HW p748, p756

- 13.14: No test in this problem, just a review of 2-way table concepts.
- 13.16: refers back to 13.14 and does the full Chi-Square Test.
- 13.18: examine the situation, make comparative bar graphs, comment on the graphs, run the test.
- You may want to skim read through Ch 13 just to pick up any loose ends and fill gaps.

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