SESSION:11 The Chi-Square DistributionSolutions

Session:11 The Chi-Square Distribution

Solutions

Introductory Business Statistics | Leadership Development – Micro-Learning Session

Rice University 2020 | Michael Laverty, Colorado State University Global Chris Littel, North Carolina State University| https://openstax.org/details/books/introductory-business-statistics

mean = 25 and standard deviation = 7.0711

when the number of degrees of freedom is greater than 90

df = 2

a test of a single variance

a left-tailed test

10.

H₀: σ² = 0.81²;

H_a: σ² > 0.81²

12.

a test of a single variance

16.

a goodness-of-fit test

18.

20.

2.04

21.

We decline to reject the null hypothesis. There is not enough evidence to suggest that the observed test scores are significantly different from the expected test scores.

23.

H₀: the distribution of AIDS cases follows the ethnicities of the general population of Santa Clara County.

25.

right-tailed

27.

2016.136

28.

Graph: Check student’s solution.

Decision: Cannot accept the null hypothesis.

Reason for the Decision: Calculated value of test statistics is either in or out of the tail of the distribution.

Conclusion (write out in complete sentences): The make-up of AIDS cases does not fit the ethnicities of the general population of Santa Clara County.

30.

a test of independence

32.

a test of independence

34.

36.

6.6

39.

Smoking level per day	African American	Native Hawaiian	Latino	Japanese Americans	White	Totals
1-10	9,886	2,745	12,831	8,378	7,650	41,490
11-20	6,514	3,062	4,932	10,680	9,877	35,065
21-30	1,671	1,419	1,406	4,715	6,062	15,273
31+	759	788	800	2,305	3,970	8,622
Totals	18,830	8,014	19,969	26,078	27,559	10,0450

Table 11.54

41.

Smoking level per day	African American	Native Hawaiian	Latino	Japanese Americans	White
1-10	7777.57	3310.11	8248.02	10771.29	11383.01
11-20	6573.16	2797.52	6970.76	9103.29	9620.27
21-30	2863.02	1218.49	3036.20	3965.05	4190.23
31+	1616.25	687.87	1714.01	2238.37	2365.49

Table 11.55

43.

10,301.8

44.

right

46.

Cannot accept the null hypothesis.
Calculated value of test statistics is either in or out of the tail of the distribution.
There is sufficient evidence to conclude that smoking level is dependent on ethnic group.

48.

test for homogeneity

50.

test for homogeneity

52.

All values in the table must be greater than or equal to five.

54.

57.

a goodness-of-fit test

59.

a test for independence

61.

Answers will vary. Sample answer: Tests of independence and tests for homogeneity both calculate the test statistic the same way $\sum_{(i j)} \frac{{(O - E)}^{2}}{E}$

$\sum_{(� �)} \frac{{(� - �)}^{2}}{�}$ . In addition, all values must be greater than or equal to five.

63.

true

65.

false

67.

225

69.

H₀: σ² ≤ 150

71.

72.

Check student’s solution.

74.

The claim is that the variance is no more than 150 minutes.

76.

a Student’s t– or normal distribution

78.

H₀: σ = 15
H_a: σ > 15
df = 42
chi-square with df = 42
test statistic = 26.88
Check student’s solution.
1. Alpha = 0.05
2. Decision: Cannot reject null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: There is insufficient evidence to conclude that the standard deviation is greater than 15.

80.

H₀: σ ≤ 3
H_a: σ > 3
df = 17
chi-square distribution with df = 17
test statistic = 28.73
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot accept the null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: There is sufficient evidence to conclude that the standard deviation is greater than three.

82.

H₀: σ = 2
H_a: σ ≠ 2
df = 14
chi-square distribution with df = 14
chi-square test statistic = 5.2094
Check student’s solution.
1. Alpha = 0.05
2. Decision: Cannot accept the null hypothesis
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: There is sufficient evidence to conclude that the standard deviation is different than 2.

84.

The sample standard deviation is $34.29.

H₀ : σ² = 25²
H_a : σ² > 25²

df = n – 1 = 7.

test statistic: $x^{2} = x_{7}^{2} = \frac{(n - 1) s^{2}}{25^{2}} = \frac{(8 - 1) {(34.29)}^{2}}{25^{2}} = 13.169$

$�^{2} = �_{7}^{2} = \frac{(� - 1) �^{2}}{25^{2}} = \frac{(8 - 1) {(34.29)}^{2}}{25^{2}} = 13.169$ ;

Alpha: 0.05

Decision: Cannot reject the null hypothesis.

Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.

Conclusion: At the 5% level, there is insufficient evidence to conclude that the variance is more than 625.

87.

Marital status	Percent	Expected frequency
Never married	31.3	125.2
Married	56.1	224.4
Widowed	2.5	10
Divorced/Separated	10.1	40.4

Table 11.56

The data fits the distribution.
The data does not fit the distribution.
3
chi-square distribution with df = 3
19.27
0.0002
Check student’s solution.
1. Alpha = 0.05
2. Decision: Cannot accept null hypothesis at the 5% level of significance
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: Data does not fit the distribution.

89.

H₀: The local results follow the distribution of the U.S. AP examinee population
H_a: The local results do not follow the distribution of the U.S. AP examinee population
df = 5
chi-square distribution with df = 5
chi-square test statistic = 13.4
Check student’s solution.
1. Alpha = 0.05
2. Decision: Cannot accept null when a = 0.05
3. Reason for Decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: Local data do not fit the AP Examinee Distribution.
5. Decision: Do not reject null when a = 0.01
6. Conclusion: There is insufficient evidence to conclude that local data do not follow the distribution of the U.S. AP examinee distribution.

91.

H₀: The actual college majors of graduating females fit the distribution of their expected majors
H_a: The actual college majors of graduating females do not fit the distribution of their expected majors
df = 10
chi-square distribution with df = 10
test statistic = 11.48
Check student’s solution.
1. Alpha = 0.05
2. Decision: Cannot reject null when a = 0.05 and a = 0.01
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: There is insufficient evidence to conclude that the distribution of actual college majors of graduating females fits the distribution of their expected majors.

94.

true

96.

false

98.

The hypotheses for the goodness-of-fit test are:

H₀: Surveyed obese fit the distribution of expected obese
H_a: Surveyed obese do not fit the distribution of expected obese

Use a chi-square distribution with df = 4 to evaluate the data.

The test statistic is X² = 9.85

The P-value = 0.0431

At 5% significance level, α = 0.05. For this data, P < α. Reject the null hypothesis.

At the 5% level of significance, from the data, there is sufficient evidence to conclude that the surveyed obese do not fit the distribution of expected obese.

100.

H₀: Car size is independent of family size.
H_a: Car size is dependent on family size.
df = 9
chi-square distribution with df = 9
test statistic = 15.8284
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot reject the null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: At the 5% significance level, there is insufficient evidence to conclude that car size and family size are dependent.

102.

H₀: Honeymoon locations are independent of bride’s age.
H_a: Honeymoon locations are dependent on bride’s age.
df = 9
chi-square distribution with df = 9
test statistic = 15.7027
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot reject the null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: At the 5% significance level, there is insufficient evidence to conclude that honeymoon location and bride age are dependent.

104.

H₀: The types of fries sold are independent of the location.
H_a: The types of fries sold are dependent on the location.
df = 6
chi-square distribution with df = 6
test statistic =18.8369
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot accept the null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: At the 5% significance level, There is sufficient evidence that types of fries and location are dependent.

106.

H₀: Salary is independent of level of education.
H_a: Salary is dependent on level of education.
df = 12
chi-square distribution with df = 12
test statistic = 255.7704
Check student’s solution.
Alpha: 0.05

Decision: Cannot accept the null hypothesis.

Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.

Conclusion: At the 5% significance level, there is sufficient evidence to conclude that salary and level of education are dependent.

108.

true

110.

true

112.

H₀: Age is independent of the youngest online entrepreneurs’ net worth.
H_a: Age is dependent on the net worth of the youngest online entrepreneurs.
df = 2
chi-square distribution with df = 2
test statistic = 1.76
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot reject the null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: At the 5% significance level, there is insufficient evidence to conclude that age and net worth for the youngest online entrepreneurs are dependent.

114.

H₀: The distribution for personality types is the same for both majors
H_a: The distribution for personality types is not the same for both majors
df = 4
chi-square with df = 4
test statistic = 3.01
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot reject the null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: There is insufficient evidence to conclude that the distribution of personality types is different for business and social science majors.

116.

H₀: The distribution for fish caught is the same in Green Valley Lake and in Echo Lake.
H_a: The distribution for fish caught is not the same in Green Valley Lake and in Echo Lake.
3
chi-square with df = 3
11.75
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot accept the null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: There is evidence to conclude that the distribution of fish caught is different in Green Valley Lake and in Echo Lake

118.

H₀: The distribution of average energy use in the USA is the same as in Europe between 2005 and 2010.
H_a: The distribution of average energy use in the USA is not the same as in Europe between 2005 and 2010.
df = 4
chi-square with df = 4
test statistic = 2.7434
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot reject the null hypothesis.
3. Reason for decision: Calculated value of test statistics is either in or out of the tail of the distribution.
4. Conclusion: At the 5% significance level, there is insufficient evidence to conclude that the average energy use values in the US and EU are not derived from different distributions for the period from 2005 to 2010.

120.

H₀: The distribution for technology use is the same for community college students and university students.
H_a: The distribution for technology use is not the same for community college students and university students.
2
chi-square with df = 2
7.05
p-value = 0.0294
Check student’s solution.
1. Alpha: 0.05
2. Decision: Cannot accept the null hypothesis.
3. Reason for decision: p-value < alpha
4. Conclusion: There is sufficient evidence to conclude that the distribution of technology use for statistics homework is not the same for statistics students at community colleges and at universities.

122.

The test statistic is always positive and if the expected and observed values are not close together, the test statistic is large and the null hypothesis will be rejected.
Testing to see if the data fits the distribution “too well” or is too perfect.

Session:11 The Chi-Square Distribution

Solutions

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