Solutions

mean = 25 and standard deviation = 7.0711

when the number of degrees of freedom is greater than 90

df = 2

a goodness-of-fit test

3

2.04

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.

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

right-tailed

2016.136

Graph: Check student’s solution.

Decision: Reject the null hypothesis.

Reason for decision: p-value < alpha

Conclusion: The make-up of cases does not fit the ethnicities of the general population of Santa Clara County.

a test of independence

a test of independence

8

6.6

0.0435

10,301.8

right-tailed

1. Reject the null hypothesis.
2. p-value < alpha
3. There is sufficient evidence to conclude that product use is dependent on ethnic group.

test for homogeneity

test for homogeneity

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

3

0.00005

a goodness-of-fit test

a test for independence

Answers will vary. Sample answer: Tests of independence and tests for homogeneity both calculate the test statistic the same way ${\displaystyle \sum }_{(ij)}\frac{{(O-E)}^2}{E}$. In addition, all values must be greater than or equal to five.

a test of a single variance

a left-tailed test

H₀: σ² = 0.81²;

H_a: σ² > 0.81²

a test of a single variance

0.0542

true

false

1. H₀: The local results follow the distribution of the U.S. AP examinee population.
2. H_a: The local results do not follow the distribution of the U.S. AP examinee population.
3. df = 5
4. chi-square distribution with df = 5
5. chi-square test statistic = 13.4
6. p-value = 0.0199
7. Check student’s solution.
8. 1. Alpha = 0.05
   2. Decision: Reject null when a = 0.05.
   3. Reason for decision: p-value < alpha
   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.

1. H₀: The actual college majors of graduating females fit the distribution of their expected majors.
2. H_a: The actual college majors of graduating females do not fit the distribution of their expected majors.
3. df = 10
4. chi-square distribution with df = 10
5. test statistic = 11.48
6. p-value = 0.3211
7. Check student’s solution.
8. 1. Alpha = 0.05
   2. Decision: Do not reject null hypothesis when a = 0.05 and a = 0.01.
   3. Reason for decision: p-value > alpha
   4. Conclusion: There is insufficient evidence to conclude that the distribution of actual college majors of graduating females do not fit the distribution of their expected majors.

true

true

false

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 χ² = 9.85
• The p-value = 0.0431
• At the 5% significance level, α = 0.05. For this data, p < α.
• 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.

1. H₀: Car size is independent of family size.
2. H_a: Car size is dependent on family size.
3. df = 9
4. chi-square distribution with df = 9
5. test statistic = 15.8284
6. p-value = 0.0706
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Do not reject the null hypothesis.
   3. Reason for decision: p-value > alpha
   4. Conclusion: At the 5 percent significance level, there is insufficient evidence to conclude that car size and family size are dependent.

1. H₀: Honeymoon locations are independent of bride’s age.
2. H_a: Honeymoon locations are dependent on bride’s age.
3. df = 9
4. chi-square distribution with df = 9
5. test statistic = 15.7027
6. p-value = 0.0734
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Do not reject the null hypothesis.
   3. Reason for decision: p-value > alpha
   4. Conclusion: At the 5 percent significance level, there is insufficient evidence to conclude that honeymoon location and bride age are dependent.

1. H₀: The types of fries sold are independent of the location.
2. H_a: The types of fries sold are dependent on the location.
3. df = 6
4. chi-square distribution with df = 6
5. test statistic =18.8369
6. p-value = 0.0044
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Reject the null hypothesis.
   3. Reason for decision: p-value < alpha
   4. Conclusion: At the 5 percent significance level, there is sufficient evidence that types of fries and location are dependent.

1. H₀: Salary is independent of level of education.
2. H_a: Salary is dependent on level of education.
3. df = 12
4. chi-square distribution with df = 12
5. test statistic = 255.7704
6. p-value = 0
7. Check student’s solution.

8. Alpha: 0.05

   Decision: Reject the null hypothesis.

   Reason for decision: p-value < alpha

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

true

true

1. H₀: Age is independent of the youngest online entrepreneurs’ net worth.
2. H_a: Age is dependent on the net worth of the youngest online entrepreneurs.
3. df = 2
4. chi-square distribution with df = 2
5. test statistic = 1.76
6. p-value = 0.4144
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Do not reject the null hypothesis.
   3. Reason for decision: p-value > alpha
   4. Conclusion: At the 5 percent significance level, there is insufficient evidence to conclude that age and net worth for the youngest online entrepreneurs are dependent.

1. H₀: The distribution for personality types is the same for both majors.
2. H_a: The distribution for personality types is not the same for both majors.
3. df = 4
4. chi-square with df = 4
5. test statistic = 3.01
6. p-value = 0.5568
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Do not reject the null hypothesis.
   3. Reason for decision: p-value > alpha
   4. Conclusion: There is insufficient evidence to conclude that the distribution of personality types is different for business and social science majors.

1. H₀: The distribution for fish caught is the same in Green Valley Lake and in Echo Lake.
2. H_a: The distribution for fish caught is not the same in Green Valley Lake and in Echo Lake.
3. 3
4. chi-square with df = 3
5. 11.75
6. p-value = 0.0083
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Reject the null hypothesis.
   3. Reason for decision: p-value < alpha
   4. Conclusion: There is evidence to conclude that the distribution of fish caught is different in Green Valley Lake and in Echo Lake.

1. H₀: The distribution of average energy use in the United States is the same as in Europe between 2005 and 2010.
2. H_a: The distribution of average energy use in the United States is not the same as in Europe between 2005 and 2010.
3. df = 4
4. chi-square with df = 4
5. test statistic = 2.7434
6. p-value = 0.7395
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Do not reject the null hypothesis.
   3. Reason for decision: p-value > alpha
   4. Conclusion: At the 5 percent significance level, there is insufficient evidence to conclude that the average energy use values in the United States and EU are not derived from different distributions for the period from 2005 to 2010.

1. H₀: The distribution for technology use is the same for community college students and university students.
2. H_a: The distribution for technology use is not the same for community college students and university students.
3. 2
4. chi-square with df = 2
5. 7.05
6. p value = 0.0294
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Reject 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.

225

H₀: σ² ≤ 150

36

Check student’s solution.

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

a student's t or normal distribution

1. H₀: σ = 15
2. H_a: σ > 15
3. df = 42
4. chi-square with df = 42
5. test statistic = 26.88
6. p-value = 0.9663
7. Check student’s solution.
8. 1. Alpha = 0.05
   2. Decision: Do not reject null hypothesis.
   3. Reason for decision: p-value > alpha
   4. Conclusion: There is insufficient evidence to conclude that the standard deviation is greater than 15.

1. H₀: σ ≤ 3
2. H_a: σ > 3
3. df = 17
4. chi-square distribution with df = 17
5. test statistic = 28.73
6. p-value = 0.0371
7. Check student’s solution.
8. 1. Alpha: 0.05
   2. Decision: Reject the null hypothesis.
   3. Reason for decision: p-value < alpha
   4. Conclusion: There is sufficient evidence to conclude that the standard deviation is greater than three.

1. H₀: σ = 2
2. H_a: σ ≠ 2
3. df = 14
4. chi-square distiribution with df = 14
5. chi-square test statistic = 5.2094
6. p-value = 0.0346
7. Check student’s solution.
8. 1. Alpha = 0.05
   2. Decision: Reject the null hypothesis
   3. Reason for decision: p-value < alpha
   4. Conclusion: There is sufficient evidence to conclude that the standard deviation is different than two.

The sample standard deviation is $34.29.

H₀ : σ² = 25²
H_a : σ² > 25²

df = n – 1 = 7

Test statistic: $x^2= x_7^2= \frac{(n–1)s^2}{{25}^2}= \frac{(8–1){(34.29)}^2}{{25}^2}=13.169$;

p-value: $P\left(x_7^2>13.169\right)=1–P\left(x_7^2 \le 13.169\right)=.0681$

Alpha: 0.05

Decision: Do not reject the null hypothesis.

Reason for decision: p-value > alpha

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

1. 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.
2. Testing to see if the data fits the distribution too well or is too perfect.