11.11: Practice
- Page ID
- 6136
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)11.2 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?
5. Is it more likely the df is 90, 20, or two in the graph?
11.3 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 (data is measured in distance from the center of the target). An observer claims the standard deviation is less.
6. What type of test should be used?
7. State the null and alternative hypotheses.
8. Is this a right-tailed, left-tailed, or two-tailed test?
Use the following information to answer the next three exercises: The standard deviation of heights for students in a school is 0.81. A random sample of 50 students is taken, and the standard deviation of heights of the sample is 0.96. A researcher in charge of the study believes the standard deviation of heights for the school is greater than 0.81.
Use the following information to answer the next four exercises: The average waiting time in a doctor’s office varies. The standard deviation of waiting times in a doctor’s office is 3.4 minutes. A random sample of 30 patients in the doctor’s office has a standard deviation of waiting times of 4.1 minutes. One doctor believes the variance of waiting times is greater than originally thought.
11.4 Goodness-of-Fit Test
Determine the appropriate test to be used in the next three exercises.
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 Table \(\PageIndex{1}\).
| Grade | Proportion |
|---|---|
| A | 0.25 |
| B | 0.30 |
| C | 0.35 |
| D | 0.10 |
Table \(\PageIndex{1}\)
The actual distribution for a class of 20 is in Table \(\PageIndex{2}\)
| Grade | Frequency |
|---|---|
| A | 7 |
| B | 7 |
| C | 5 |
| D | 1 |
Table \(\PageIndex{2}\)
Use the following information to answer the next nine exercises: The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table \(\PageIndex{3}\)
| Ethnicity | Number of cases |
|---|---|
| White | 2,229 |
| Hispanic | 1,157 |
| Black/African-American | 457 |
| Asian, Pacific Islander | 232 |
| Total = 4,075 |
Table \(\PageIndex{3}\)
The percentage of each ethnic group in Santa Clara County is as in Table \(\PageIndex{4}\)
| 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% |
Table \(\PageIndex{4}\)
Perform a goodness-of-fit test to determine whether the occurrence of AIDS cases follows the ethnicities of the general population of Santa Clara County.
28. Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the confidence level.
Let α = 0.05
Decision: ________________
Reason for the Decision: ________________
Conclusion (write out in complete sentences): ________________
11.4 Test of Independence
Determine the appropriate test to be used in the next three exercises.
30. 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.
31. 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.
32. 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 as well as the brand of shoes they were wearing.
Use the following information to answer the next seven exercises: Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table \(\PageIndex{1}\) shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.
| 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 |
33. State the hypotheses.
\(H_0\): _______
\(H_a\): _______
34. \(df\) = _______
35. How many passengers are expected to travel between 201 and 300 miles and purchase second-class tickets?
36. How many passengers are expected to travel between 401 and 500 miles and purchase first-class tickets?
37. What is the test statistic?
38. What can you conclude at the 5% level of significance?
Use the following information to answer the next eight exercises: An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.
39. Complete the table.
| Smoking level per day | African American | Native Hawaiian | Latino | Japanese Americans | White | Totals |
|---|---|---|---|---|---|---|
| 1-10 | ||||||
| 11-20 | ||||||
| 21-30 | ||||||
| 31+ | ||||||
| Totals |
40. State the hypotheses.
\(H_0\): _______
\(H_a\): _______
41. Enter expected values in Table \(\PageIndex{26}\). Round to two decimal places.
Calculate the following values:
43. \(\chi^2\) test statistic = ______
44. Is this a right-tailed, left-tailed, or two-tailed test? Explain why.
45. Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the confidence level.
State the decision and conclusion (in a complete sentence) for the following preconceived levels of \alpha.
46.
\(\alpha = 0.05\)
- Decision: ___________________
- Reason for the decision: ___________________
- Conclusion (write out in a complete sentence): ___________________
47.
\(\alpha = 0.01\)
11.6 Test for Homogeneity
48. A math teacher wants to see if two of her classes have the same distribution of test scores. What test should she use?
49. What are the null and alternative hypotheses for Table \(\PageIndex{7}\).
| 20–30 | 30–40 | 40–50 | 50–60 | |
|---|---|---|---|---|
| Private practice | 16 | 40 | 38 | 6 |
| Hospital | 8 | 44 | 59 | 39 |
53. State the null and alternative hypotheses.
54. \(df\) = _______
55. What is the test statistic?
11.7 Comparison of the Chi-Square Tests
57. Which test do you use to decide whether an observed distribution is the same as an expected distribution?
58. What is the null hypothesis for the type of test from Exercise \(\PageIndex{57}\)?
59. Which test would you use to decide whether two factors have a relationship?
60. Which test would you use to decide if two populations have the same distribution?
61. How are tests of independence similar to tests for homogeneity?
62. How are tests of independence different from tests for homogeneity?