AP Statistics Unit 8 Test: Inference for Categorical Data: Chi-Square

Test 8 for AP Stats.

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Question 1

Which is true about chi-square distributions?

A
Values are positive, skewed right.
B
Values are negative, skewed right.
C
Values are positive, skewed left.
D
Values are positive and symmetric.
Question 1 Explanation: 
Chi-square distributions are positive and skewed right. Answer A is correct.
Question 2

A teacher has a theory that as students advance in grades, more students will have jobs. The teacher’s theory is that students having jobs will follow the ratio $1:2:2:3$. The teacher takes a sample of 100 students from each grade and finds the following number of students with jobs in each grade: 9th grade - 12, 10th grade - 22, 11th grade - 27, 12th grade - 35. $x^2$Calculate for the sample.

A
0.361 with df=3
B
0.361 with df=4
C
1.694 with df=3
D
1.694 with df=4
Question 2 Explanation: 
Based on the theoretical ratio, we would get the following numbers (based on a total sum of $12+22+27+35=96$):

Grade Experiment Theoretical
9 12 12
10 22 24
11 30 24
12 35 36

$x^2 = Σ \dfrac{(observed-expected)^2}{expected}$

$ = \dfrac{(12-12)^2}{12} + \dfrac{(22-24)^2}{24} + \dfrac{(30-24)^2}{24} + \dfrac{(35-36)^2}{36}$

$ = 0 + \dfrac{4}{24} + \dfrac{36}{24} + \dfrac{(1}{36} ≈ 1.694$

There are 4 categories, so $df = 4-1=3$. Answer C is correct.
Question 3

A high school principal surveyed students in grades 9-12 about their satisfaction with school lunches. The results are below:

Grade

Satisfied

Not Satisfied

Total

9

20

5

25

10

18

8

26

11

24

5

29

12

14

6

20

Total

76

24

100

What is the expected count for the cell corresponding to 9th graders not satisfied with school lunch?

A
5
B
6
C
20
D
21
Question 3 Explanation: 
The expected count for 9th graders not satisfied with school lunch would be $(24÷100)*25 = 6$. Answer B is correct.
Question 4

What is an appropriate null and alternative hypotheses for a chi-square test for independence?

A
$H_0$: There is no difference in distributions of movies watched between adults and children. $H_a$ : There is a difference in distributions of movies watched between adults and children.
B
$H_0$: There is a difference in distributions of movies watched between adults and children. $H_a$ : There is no difference in distributions of movies watched between adults and children.
C
$H_0$: There is an association between movies watched and age. $H_a$: There is no association between movies watched and age.
D
$H_0$: There is no association between movies watched and age. $H_a$: There is an association between movies watched and age.
Question 4 Explanation: 
The appropriate hypotheses for a chi-square test for independence have a null hypothesis stating no association between categorical variables and an alternative hypothesis stating that there is an association. Answer A would be a potential null and alternative hypothesis for a chi-square test for homogeneity. Answer D is correct.
Question 5

Which of these is NOT a condition for making statistical inferences when testing a chi-square distribution for independence or homogeneity?

A
Data should be collected using a simple or stratified random sample.
B
For sampling without replacement, $n≥10%N$
C
Expected counts should be greater than 5.
D
All of these are conditions.
Question 5 Explanation: 
Answer choice B is NOT a condition. For sampling without replacement, $n≤10%N$.
Question 6

What is the p-value for a chi-square test for independence or homogeneity if the number of rows is 4 and the number of columns is 3 with 14.92? Use the excerpt of the table below.

df

0.25

0.20

0.15

0.10

0.05

0.025

0.02

6

7.84

8.56

9.45

10.64

12.59

14.45

15.03

7

9.04

9.80

10.75

12.02

14.07

16.01

16.62

8

10.22

11.03

12.03

13.36

15.51

17.53

18.17

9

11.39

12.24

13.29

14.68

16.92

19.02

19.68

10

12.55

13.44

14.53

15.99

18.31

20.48

21.16

11

13.70

14.63

15.77

17.28

19.68

21.92

22.62

12

14.85

15.81

16.99

18.55

21.03

23.34

24.05
A
$0.025<p<0.02$
B
$0.20<p<0.05$
C
$0.10<p<0.05$
D
$0.25<p<0.20$
Question 6 Explanation: 
To find the appropriate p-value range, first find $ = (rows-1)(columns-1)$. In this case, $3*2=6$ so $df=6$. In the $df=6$ row, $14.92$ falls between $0.025$ and 0.02. Answer A is correct.
Question 7

What are the degrees of freedom for the following table?

Pepperoni

Cheese

Sausage

Total

Grade 8

9

5

20

34

Grade 9

20

8

10

38

Grade 10

16

12

14

42

Grade 11

12

8

5

25

Grade 12

16

11

12

39

Total

73

44

61

178
A
8
B
10
C
12
D
15
Question 7 Explanation: 
For a chi-square test for homogeneity or independence, $df = (r-1)(c-1) = 4*2=8$. Answer A is correct.
Question 8

Which of the following is a condition for making statistical inferences when testing goodness of fit for a chi-square distribution?

A
Data is collected using a stratified random sample.
B
Data is collected using a simple random sample.
C
Expected counts are less than 5.
D
None of these are conditions.
Question 8 Explanation: 
When testing goodness of fit, the following conditions must be true: data is collected using a simple random sample, expected counts are at least five, and $n≤10%N$ if sampling without replacement. Answer B is correct.
Question 9

Which of the following statements are true about the relationship between the p-value, , and the null and alternative hypotheses?

A
If $p-value > α$ then assume there was an error in calculation.
B
If $p-value < α$ then reject the null hypothesis.
C
If $p-value > α$ then reject the null hypothesis.
D
If $p-value < α$ then assume there was an error in calculation.
Question 9 Explanation: 
A decision to either reject or fail to reject the null hypothesis is based on comparing the p-value to $α$. If the p-value is less than the significance level, then there is sufficient evidence to reject the null hypothesis. Answer B is correct.
Question 10

What would not be appropriate null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data?

A
$H_0$: The distribution of students competing in a state track meet is consistent with the distribution of the state’s population by region.. $H_a$: The distribution of students competing in a state track meet is not consistent with the state’s population by region.
B
$H_0$: The proportion of students participating in each fall sport at a given school will match the ratio of participation across the state. $H_a$: The proportion of students participating in each fall sport at a given school will not match the ratio of participation across the state.
C
$H_0$: The distribution of flavors in a mixed candy bag will be exactly equal. $H_a$: The distribution of flavors in a mixed candy bag will match the distribution at the factory.
D
$H_0$: The proportion of pink to red flowers in a garden will match the proportion of seeds planted. $H_a$: The proportion of pink to red flowers in a garden will not match the proportion of seeds planted.
Question 10 Explanation: 
Answer choice C is not appropriate. The null hypothesis may or may not be exactly equal, depending on the distribution at the factor. If the factory has an equal distribution of flavors, then the alternative hypothesis should be that the distribution will not be equal. If the factory does not use an equal distribution of flavors, then the listed Ha would actually be $H_0$ and the alternative hypothesis should be that the distribution in the bag will not match the distribution at the factory.
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