Section Exercises
Facts About the Chi-Square Distribution
1. If the number of degrees of freedom for a chi-square distribution is 25, what is the population mean and standard deviation?
3. When does the chi-square curve approximate a normal distribution?
4. Where is μ located on a chi-square curve?
5. Is it more likely the df is 90, 20, or two in the graph?
Decide whether the following statements are true or false.
6. As the number of degrees of freedom increases, the graph of the chi-square distribution looks more and more symmetrical.
7. The standard deviation of the chi-square distribution is twice the mean.
8. The mean and the median of the chi-square distribution are the same if df = 24.
Goodness-of-Fit Test
Determine the appropriate test to be used in the next three exercises.
9. 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.
10. 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.
11. 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 that the distribution of grades on the final exam will be and they are recorded in the table.
| Grade | Proportion |
|---|---|
| A | 0.25 |
| B | 0.30 |
| C | 0.35 |
| D | 0.10 |
The actual distribution for a class of 20 is in the table below.
| Grade | Frequency |
|---|---|
| A | 7 |
| B | 7 |
| C | 5 |
| D | 1 |
12. df= ______
13. State the null and alternative hypotheses.
14. χ2 test statistic = ______
15. p-value = ______
16. At the 5% 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 inthe table below.
| Ethnicity | Percentage of total county population | Number expected (round to two decimal places) |
|---|---|---|
| White | 42.9% | 1748.18 |
| Hispanic | 26.7% | |
| Black/African-American | 2.6% | |
| Asian, Pacific Islander | 27.8% | |
| Total = 100% |
17. If the ethnicities of AIDS victims followed the ethnicities of the total county population, fill