Practice
11.1 Facts About the Chi-Square Distribution
If the number of degrees of freedom for a chi-square distribution is 25, what is the population mean and standard deviation?
If df > 90, the distribution is _____________. If df = 15, the distribution is ________________.
When does the chi-square curve approximate a normal distribution?
Where is μ located on a chi-square curve?
Is it more likely the df is 90, 20, or 2 in the graph?
11.2 Goodness-of-Fit Test
Determine the appropriate test to be used in the next three exercises.
An archeologist is calculating the distribution of the frequency of the number of artifacts she finds in a dig site. Based on previous digs, the archeologist creates an expected distribution broken down by grid sections in the dig site. Once the site has been fully excavated, she compares the actual number of artifacts found in each grid section to see if her expectation was accurate.
An economist is deriving a model to predict outcomes on the stock market. He creates a list of expected points on the stock market index for the next two weeks. At the close of each day’s trading, he records the actual points on the index. He wants to see how well his model matched what actually happened.
A personal trainer is putting together a weight-lifting program for her clients. For a 90-day program, she expects each client to lift a specific maximum weight each week. As she goes along, she records the actual maximum weights her clients lifted. She wants to know how well her expectations met with what was observed.
Use the following information to answer the next five exercises: A teacher predicts the distribution of grades on the final exam. The predictions are shown in Table 11.27.
| Grade | Proportion |
|---|---|
| A | 0.25 |
| B | 0.30 |
| C | 0.35 |
| D | 0.10 |
The actual distribution for a class of 20 is in Table 11.28.
| Grade | Frequency |
|---|---|
| A | 7 |
| B | 7 |
| C | 5 |
| D | 1 |
______
State the null and alternative hypotheses.
χ2 test statistic = ______
p-value = ______
At the 5 percent significance level, what can you conclude?
| Ethnicity | Number of Cases |
|---|---|
| White | 2,229 |
| Hispanic | 1,157 |
| Black/African American | 457 |
| Asian, Pacific Islander | 232 |
| Total = 4,075 |
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.
| Ethnicity | % of Total County Population | Number Expected (round to two decimal places) |
|---|---|---|
| White | 42.9% | 1,748.18 |
| Hispanic | 26.7% | |
| Black/African American | 2.6% | |
| Asian, Pacific Islander | 27.8% | |
| Total = 100% |
If the ethnicities of patients followed the ethnicities of the total county population, fill in the expected number of cases per ethnic group.
H0: _______
Ha: _______
Is this a right-tailed, left-tailed, or two-tailed test?
degrees of freedom = _______
χ2 test statistic = _______
p-value = _______
Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the p-value.
Let α = 0.05.
Decision: ________________
Reason for the decision: ________________
Conclusion (write out in complete sentences): ________________
Does it appear that the pattern of disease cases in Santa Clara County corresponds to the distribution of ethnic groups in this county? Why or why not?
11.3 Test of Independence
Determine the appropriate test to be used in the next three exercises.
A pharmaceutical company is interested in the relationship between age and presentation of symptoms for a common viral infection. A random sample is taken of 500 people with the infection across different age groups.
The owner of a baseball team is interested in the relationship between player salaries and team winning percentage. He takes a random sample of 100 players from different organizations.
A marathon runner is interested in the relationship between the brand of shoes runners wear and their run times. She takes a random sample of 50 runners and records their run times and the brand of shoes they were wearing.
| Traveling Distance | Third Class | Second Class | First Class | Total |
|---|---|---|---|---|
| 1–100 miles | 21 | 14 | 6 | 41 |
| 101–200 miles | 18 | 16 | 8 | 42 |
| 201–300 miles | 16 | 17 | 15 | 48 |
| 301–400 miles | 12 | 14 | 21 | 47 |
| 401–500 miles | 6 | 6 | 10 | 22 |
| Total | 73 | 67 | 60 | 200 |
State the hypotheses.
df = _______
How many passengers are expected to travel between 201 and 300 miles and purchase second-class tickets?
How many passengers are expected to travel between 401 and 500 miles and purchase first-class tickets?
What is the test statistic?
What is the p-value?
What can you conclude at the 5 percent level of significance?
Complete the table.
| Product use Per Day | African American | Native Hawaiian | Latino | Japanese American | White | TOTALS |
|---|---|---|---|---|---|---|
| 1–10 | ||||||
| 11–20 | ||||||
| 21–30 | ||||||
| 31+ | ||||||
| TOTALS |
State the hypotheses.
Enter expected values in Table 11.32. Round to two decimal places.
Calculate the following values:
df = _______
test statistic = ______
p-value = ______
Is this a right-tailed, left-tailed, or two-tailed test? Explain why.
Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the p-value.
State the decision and conclusion (in a complete sentence) for the following preconceived levels of α:
α = 0.05
- Decision: ___________________
- Reason for the decision: ___________________
- Conclusion (write out in a complete sentence): ___________________
α = 0.01
- Decision: ___________________
- Reason for the decision: ___________________
- Conclusion (write out in a complete sentence): ___________________
11.4 Test for Homogeneity
A math teacher wants to see if two of her classes have the same distribution of test scores. What test should she use?
A market researcher wants to see if two different stores have the same distribution of sales throughout the year. What type of test should he use?
A meteorologist wants to know if East and West Australia have the same distribution of storms. What type of test should she use?
What condition must be met to use the test for homogeneity?
Use the following information to answer the next five exercises: Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in Table 11.33.
| 20–30 | 30–40 | 40–50 | 50–60 | |
|---|---|---|---|---|
| Private Practice | 16 | 40 | 38 | 6 |
| Hospital | 8 | 44 | 59 | 39 |
State the null and alternative hypotheses.
df = _______
What is the test statistic?
What is the p-value?
What can you conclude at the 5 percent significance level?
11.5 Comparison of the Chi-Square Tests
Which test do you use to decide whether an observed distribution is the same as an expected distribution?
Which test would you use to decide whether two factors have a relationship?
Which test would you use to decide if two populations have the same distribution?
How are tests of independence similar to tests for homogeneity?
How are tests of independence different from tests for homogeneity?
11.6 Test of a Single Variance
Use the following information to answer the next three exercises: An archer’s standard deviation for his hits is six, where the data are measured in distance from the center of the target. An observer claims the standard deviation is less than six.
What type of test should be used?
State the null and alternative hypotheses.
Is this a right-tailed, left-tailed, or two-tailed test?
What type of test should be used?
State the null and alternative hypotheses.
df = ________
What type of test should be used?
What is the test statistic?
What is the p-value?
What can you conclude at the 5 percent significance level?