10.6 Exercises
1. If the number of degrees of freedom for a [latex]\chi^2[/latex]-distribution is 25, what is the population mean and standard deviation?
2. Where is [latex]\mu[/latex] located on a chi-square curve?
3.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 |
At the 5% significance level, do the actual grades match the teacher’s assumed distribution?
4. The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in the table below.
| 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 the table below.
| Ethnicity | Percentage of total county population |
| White | 42.9% |
| Hispanic | 26.7% |
| Black/African-American | 2.6% |
| Asian, Pacific Islander | 27.8% |
| Total = 100% |
At the 5% significance level, does it appear that the pattern of AIDS cases in Santa Clara County corresponds to the distribution of ethnic groups in this county?
5. A six-sided die is rolled 120 times and the results are recorded in the table below. At the 5% significance level, determine if the die is fair. (Hint: in a fair die, each of the faces is equally likely to occur.)
| Face Value | Frequency |
| 1 | 15 |
| 2 | 29 |
| 3 | 16 |
| 4 | 15 |
| 5 | 30 |
| 6 | 15 |
6. The marital status distribution of the U.S. male population, ages 15 and older, is as shown in the table below.
| Marital Status | Percent |
| never married | 31.3 |
| married | 56.1 |
| widowed | 2.5 |
| divorced/separated | 10.1 |
Suppose that a random sample of 400 U.S. young adult males, 18 to 24 years old, yielded the following frequency distribution. At the 5% significance level, test if this age group of males fits the distribution of the U.S. adult population.
| Marital Status | Frequency |
| never married | 140 |
| married | 238 |
| widowed | 2 |
| divorced/separated | 20 |
7. The columns in the table below contain the Race/Ethnicity of U.S. Public Schools for a recent year, the percentages for the Advanced Placement Examinee Population for that class, and the Overall Student Population. Suppose the right column contains the result of a survey of 1,000 local students from that year who took an AP Exam.
| Race/Ethnicity | AP Examinee Population | Overall Student Population | Survey Frequency |
|---|---|---|---|
| Asian, Asian American, or Pacific Islander | 10.2% | 5.4% | 113 |
| Black or African-American | 8.2% | 14.5% | 94 |
| Hispanic or Latino | 15.5% | 15.9% | 136 |
| American Indian or Alaska Native | 0.6% | 1.2% | 10 |
| White | 59.4% | 61.6% | 604 |
| Not reported/other | 6.1% | 1.4% | 43 |
- At the 5% significance level, determine if the local results follow the distribution of the U.S. overall student population based on ethnicity.
- At the 5% significance level, determine if the local results follow the distribution of U.S. AP examinee population, based on ethnicity.
8. UCLA conducted a survey of more than 263,000 college freshmen from 385 colleges in fall 2005. The results of students’ expected majors by gender were reported in The Chronicle of Higher Education (2/2/2006). Suppose a survey of 5,000 graduating females and 5,000 graduating males was done as a follow-up last year to determine what their actual majors were. The results are shown in the tables below. The second column in each table does not add to 100% because of rounding.
- At the 5% significance level, determine if the actual college majors of graduating females fit the distribution of their expected majors.
| Major | Women – Expected Major | Women – Actual Major |
|---|---|---|
| Arts & Humanities | 14.0% | 670 |
| Biological Sciences | 8.4% | 410 |
| Business | 13.1% | 685 |
| Education | 13.0% | 650 |
| Engineering | 2.6% | 145 |
| Physical Sciences | 2.6% | 125 |
| Professional | 18.9% | 975 |
| Social Sciences | 13.0% | 605 |
| Technical | 0.4% | 15 |
| Other | 5.8% | 300 |
| Undecided | 8.0% | 420 |
- At the 5% significance level determine if the actual college majors of graduating males fit the distribution of their expected majors.
| Major | Men – Expected Major | Men – Actual Major |
|---|---|---|
| Arts & Humanities | 11.0% | 600 |
| Biological Sciences | 6.7% | 330 |
| Business | 22.7% | 1130 |
| Education | 5.8% | 305 |
| Engineering | 15.6% | 800 |
| Physical Sciences | 3.6% | 175 |
| Professional | 9.3% | 460 |
| Social Sciences | 7.6% | 370 |
| Technical | 1.8% | 90 |
| Other | 8.2% | 400 |
| Undecided | 6.6% | 340 |
9. The table below contains information from a survey among 499 participants classified according to their age groups. The second column shows the percentage of obese people per age class among the study participants. The last column comes from a different study at the national level that shows the corresponding percentages of obese people in the same age classes in the USA. At the 5% significance level to determine whether the survey participants are a representative sample of the USA obese population.
| Age Class (Years) | Obese (Percentage) | Expected USA average (Percentage) |
| 20–30 | 75.0 | 32.6 |
| 31–40 | 26.5 | 32.6 |
| 41–50 | 13.6 | 36.6 |
| 51–60 | 21.9 | 36.6 |
| 61–70 | 21.0 | 39.7 |
10. Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. The table below shows the results. At the 5% significance level determine 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 |
11. A recent debate about where in the United States skiers believe the skiing is best prompted the following survey. At the 5% significance level test to see if the best ski area is independent of the level of the skier.
| U.S. Ski Area | Beginner | Intermediate | Advanced |
| Tahoe | 20 | 30 | 40 |
| Utah | 10 | 30 | 60 |
| Colorado | 10 | 40 | 50 |
12. Car manufacturers are interested in whether there is a relationship between the size of car an individual drives and the number of people in the driver’s family (that is, whether car size and family size are independent). To test this, suppose that 800 car owners were randomly surveyed with the results in the table. Conduct a test of independence. Use a 5% significance level.
| Family Size | Sub & Compact | Mid-size | Full-size | Van & Truck |
| 1 | 20 | 35 | 40 | 35 |
| 2 | 20 | 50 | 70 | 80 |
| 3–4 | 20 | 50 | 100 | 90 |
| 5+ | 20 | 30 | 70 | 70 |
13. College students may be interested in whether or not their majors have any effect on starting salaries after graduation. Suppose that 300 recent graduates were surveyed as to their majors in college and their starting salaries after graduation. The table below shows the data. Conduct a test of independence. Use a 5% significance level.
| Major | < $50,000 | $50,000 – $68,999 | $69,000 + |
| English | 5 | 20 | 5 |
| Engineering | 10 | 30 | 60 |
| Nursing | 10 | 15 | 15 |
| Business | 10 | 20 | 30 |
| Psychology | 20 | 30 | 20 |
14. Some travel agents claim that honeymoon hot spots vary according to age of the bride. Suppose that 280 recent brides were interviewed as to where they spent their honeymoons. The information is recorded in the table below. Conduct a test of independence. Use a 5% significance level.
| Location | 20–29 | 30–39 | 40–49 | 50 and over |
| Niagara Falls | 15 | 25 | 25 | 20 |
| Poconos | 15 | 25 | 25 | 10 |
| Europe | 10 | 25 | 15 | 5 |
| Virgin Islands | 20 | 25 | 15 | 5 |
| Sport | 18 – 25 | 26 – 30 | 31 – 40 | 41 and over |
| racquetball | 42 | 58 | 30 | 46 |
| tennis | 58 | 76 | 38 | 65 |
| swimming | 72 | 60 | 65 | 33 |
16. A major food manufacturer is concerned that the sales for its skinny french fries have been decreasing. As a part of a feasibility study, the company conducts research into the types of fries sold across the country to determine if the type of fries sold is independent of the area of the country. The results of the study are shown in the table. Conduct a test of independence. Use a 5% significance level.
| Type of Fries | Northeast | South | Central | West |
| skinny fries | 70 | 50 | 20 | 25 |
| curly fries | 100 | 60 | 15 | 30 |
| steak fries | 20 | 40 | 10 | 10 |
17. According to Dan Lenard, an independent insurance agent in the Buffalo, N.Y. area, the following is a breakdown of the amount of life insurance purchased by males in the following age groups. He is interested in whether the age of the male and the amount of life insurance purchased are independent events. Conduct a test for independence. Use a 5% significance level.
| Age of Males | None | < $200,000 | $200,000–$400,000 | $401,001–$1,000,000 | $1,000,001+ |
| 20–29 | 40 | 15 | 40 | 0 | 5 |
| 30–39 | 35 | 5 | 20 | 20 | 10 |
| 40–49 | 20 | 0 | 30 | 0 | 30 |
| 50+ | 40 | 30 | 15 | 15 | 10 |
18. Suppose that 600 thirty-year-olds were surveyed to determine whether or not there is a relationship between the level of education an individual has and salary. Conduct a test of independence. Use a 5% significance level.
| Annual Salary | Not a high school graduate | High school graduate | College graduate | Masters or doctorate |
| < $30,000 | 15 | 25 | 10 | 5 |
| $30,000–$40,000 | 20 | 40 | 70 | 30 |
| $40,000–$50,000 | 10 | 20 | 40 | 55 |
| $50,000–$60,000 | 5 | 10 | 20 | 60 |
| $60,000+ | 0 | 5 | 10 | 150 |
| U.S. region/Flavor | Strawberry | Chocolate | Vanilla | Rocky Road | Mint Chocolate Chip | Pistachio | Row total |
|---|---|---|---|---|---|---|---|
| West | 12 | 21 | 22 | 19 | 15 | 8 | 97 |
| Midwest | 10 | 32 | 22 | 11 | 15 | 6 | 96 |
| East | 8 | 31 | 27 | 8 | 15 | 7 | 96 |
| South | 15 | 28 | 30 | 8 | 15 | 6 | 102 |
| Column Total | 45 | 112 | 101 | 46 | 60 | 27 | 391 |
20. The table provides a recent survey of the youngest online entrepreneurs whose net worth is estimated at one million dollars or more. Their ages range from 17 to 30. Each cell in the table illustrates the number of entrepreneurs who correspond to the specific age group and their net worth. Are the ages and net worth independent? Perform a test of independence at the 5% significance level.
| Age Group\ Net Worth Value (in millions of US dollars) | 1–5 | 6–24 | ≥25 | Row Total |
|---|---|---|---|---|
| 17–25 | 8 | 7 | 5 | 20 |
| 26–30 | 6 | 5 | 9 | 20 |
| Column Total | 14 | 12 | 14 | 40 |
21. A 2013 poll in California surveyed people about taxing sugar-sweetened beverages. The results are presented in the table, and are classified by ethnic group and response type. Are the poll responses independent of the participants’ ethnic group? Conduct a test of independence at the 5% significance level.
| Opinion/Ethnicity | Asian-American | White/Non-Hispanic | African-American | Latino | Row Total |
|---|---|---|---|---|---|
| Against tax | 48 | 433 | 41 | 160 | 628 |
| In Favor of tax | 54 | 234 | 24 | 147 | 459 |
| No opinion | 16 | 43 | 16 | 19 | 84 |
| Column Total | 118 | 710 | 71 | 272 | 1171 |
22. 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. At the 5% significance level, test the observer’s claim.
23. The variance of heights for students in a school is 0.66. 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 variation of heights for the school is greater than 0.66. At the 5% significance level, determine if the variance in the heights for students in the school is greater than 0.66.
24. The average waiting time in a doctor’s office varies. 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.
- Construct a 96% confidence interval for the variation in the wait times at the doctor’s office.
- Interpret the confidence interval found in part (a).
- One of the doctors believes that the variance in the wait times is greater than 12. Is the doctor’s claim reasonable? Explain.
26. A plant manager is concerned her equipment may need recalibrating. It seems that the actual weight of the 15 oz. cereal boxes it fills has been fluctuating. In order to determine if the machine needs to be recalibrated, 84 randomly selected boxes of cereal from the next day’s production were weighed. The standard deviation of the 84 boxes was 0.54.
- Construct a 99% confidence interval for the variance in the weight of the cereal boxes.
- Interpret the confidence interval found in part (a).
- If the variance in the weight of the cereal boxes is supposed to be at most 25, does the machine need to be recalibrated?
27. Consumers may be interested in whether the cost of a particular calculator varies from store to store. Based on surveying 43 stores, which yielded a sample mean of $84 and a sample standard deviation of $12, test the claim that the standard deviation is greater than $15. Use a 5% significance level.
28. Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. They are also interested in the variation of the number of babies. Suppose that an airline executive believes the average number of babies on flights is six with a variance of nine at most. The airline conducts a survey. The results of the 18 flights surveyed give a sample average of 6.4 with a sample standard deviation of 3.9. Conduct a hypothesis test of the airline executive’s belief. Use a 5% significance level.
29. According to an avid aquarist, the average number of fish in a 20-gallon tank is 10, with a variance of four. His friend, also an aquarist, does not believe that the standard deviation is two. She counts the number of fish in 15 other 20-gallon tanks. Based on the results that follow, do you think that the variance is different from four? Use a 5% significance level. Data: 11; 10; 9; 10; 10; 11; 11; 10; 12; 9; 7; 9; 11; 10; 11
30. The manager of “Frenchies” is concerned that patrons are not consistently receiving the same amount of French fries with each order. The chef claims that the variation for a ten-ounce order of fries is 2.25., but the manager thinks that it may be higher. He randomly weighs 49 orders of fries, which yields a mean of 11 oz. and a standard deviation of two oz. At the 5% significance level, determine if the variation in the amount of fries per order is higher than claimed.
31. You want to buy a specific computer. A sales representative of the manufacturer claims that retail stores sell this computer at an average price of $1,249 with a variance of 625. You find a website that has a price comparison for the same computer at a series of stores as follows: $1,299; $1,229.99; $1,193.08; $1,279; $1,224.95; $1,229.99; $1,269.95; $1,249. Can you argue that pricing has a larger variation than claimed by the manufacturer? Use the 5% significance level. As a potential buyer, what would be the practical conclusion from your analysis?
32. A company packages apples by weight. One of the weight grades is Class A apples. A batch of apples is selected to be included in a Class A apple package.
- Construct a 95% confidence interval for the variation in the weight of apples in the package.
- Interpret the confidence interval found in part (a).
Weights in selected apple batch (in grams): 158; 167; 149; 169; 164; 139; 154; 150; 157; 171; 152; 161; 141; 166; 172;
Attribution
“Chapter 11 Practice” in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.
