Chapter 11: The Chi-Square DistributionContent check — Introductory Statistics, Chapter 11: The Chi-Square Distribution

Determine the appropriate test to be used in the next three exercises.

An economist is deriving a model to predict outcomes on the stock market. They create a list of expected points on the stock market index for the next two weeks. At the close of each day’s trading, The economist records the actual points on the index. They want to see how well the model matched what actually happened.

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 11.27.

The actual distribution for a class of 20 is in Table 11.28.

$df=$ ______

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 11.27.

The actual distribution for a class of 20 is in Table 11.28.

χ² test statistic = ______

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 11.27.

The actual distribution for a class of 20 is in Table 11.28.

At the 5% significance level, what can you conclude?

Use the following information to answer the next nine exercises: The cumulative number of COVID-19 related cases reported for Santa Clara County for a certain time period is broken down by ethnicity as in Table 11.29.

The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

H₀: _______

Use the following information to answer the next nine exercises: The cumulative number of COVID-19 related cases reported for Santa Clara County for a certain time period is broken down by ethnicity as in Table 11.29.

The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

Is this a right-tailed, left-tailed, or two-tailed test?

Use the following information to answer the next nine exercises: The cumulative number of COVID-19 related cases reported for Santa Clara County for a certain time period is broken down by ethnicity as in Table 11.29.

The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

χ² test statistic = _______

Use the following information to answer the next nine exercises: The cumulative number of COVID-19 related cases reported for Santa Clara County for a certain time period is broken down by ethnicity as in Table 11.29.

The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

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): ________________

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.

Determine the appropriate test to be used in the next three exercises.

A marathon runner is interested in the relationship between the brand of shoes runners wear and their run times. They take 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 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

df = _______

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 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

How many passengers are expected to travel between 401 and 500 miles and purchase first-class tickets?

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 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

What is the p-value?

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 Black people, 2,745 Native Hawaiian people, 12,831 Hispanic/Latino people, 8,378 Japanese American people and 7,650 White people. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 Black people, 3,062 Native Hawaiian people, 4,932 Hispanic/Latino people, 10,680 Japanese American people, and 9,877 White people. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 Black people, 1,419 Native Hawaiian people, 1,406 Hispanic/Latino people, 4,715 Japanese American people, and 6,062 White people. Of the people smoking at least 31 cigarettes per day, there were 759 Black people, 788 Native Hawaiian people, 800 Latino people, 2,305 Japanese American people, and 3,970 White people.

${\chi }^{\text{2}}$ test statistic = ______

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 Black people, 2,745 Native Hawaiian people, 12,831 Hispanic/Latino people, 8,378 Japanese American people and 7,650 White people. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 Black people, 3,062 Native Hawaiian people, 4,932 Hispanic/Latino people, 10,680 Japanese American people, and 9,877 White people. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 Black people, 1,419 Native Hawaiian people, 1,406 Hispanic/Latino people, 4,715 Japanese American people, and 6,062 White people. Of the people smoking at least 31 cigarettes per day, there were 759 Black people, 788 Native Hawaiian people, 800 Latino people, 2,305 Japanese American people, and 3,970 White people.

Is this a right-tailed, left-tailed, or two-tailed test? Explain why.

State the decision and conclusion (in a complete sentence) for the following preconceived levels of α.

α = 0.05

a. Decision: ___________________

b. Reason for the decision: ___________________

c. Conclusion (write out in a complete sentence): ___________________

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.

df = _______

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.

What is the p-value?

Use the following information to answer the next three exercises: An archer’s standard deviation for hits is six (data is measured in distance from the center of the target). An observer claims the standard deviation is less.

What type of test should be used?

Use the following information to answer the next three exercises: An archer’s standard deviation for hits is six (data is measured in distance from the center of the target). An observer claims the standard deviation is less.

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.

State the null and alternative hypotheses.

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.

What type of test should be used?

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.

What is the p-value?

Decide whether the following statements are true or false.

As the number of degrees of freedom increases, the graph of the chi-square distribution looks more and more symmetrical.

Decide whether the following statements are true or false.

The mean and the median of the chi-square distribution are the same if df = 24.

For each problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

The marital status distribution of the population of U.S. men, ages 15 and older, is as shown in Table 11.35.

Suppose that a random sample of 400 U.S. young adults, 18 to 24 years old, yielded the following frequency distribution. We are interested in whether this age group fits the distribution of the U.S. adult population. Calculate the frequency one would expect when surveying 400 people. Fill in Table 11.35, rounding to two decimal places.

Use the following information to answer the next two exercises: The columns in Table 11.37 contain the Race/Ethnicity of U.S. Public Schools for a recent year, the percentages for students using mass transit to get to school, and the Overall Student Population. Suppose the right column contains the result of a survey of 1,000 local students from that year who used mass transit to get to school.

Perform a goodness-of-fit test to determine whether the local results follow the distribution of those that use mass transit, based on ethnicity.

Use the following information to answer the next two exercises: UCLA conducted a survey of more than 263,000 first-year students 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 women and 5,000 graduating men was done as a follow-up last year to determine what their actual majors were. The results are shown in the tables for Exercise 11.77 and Exercise 11.78. The second column in each table does not add to 100% because of rounding.

Conduct a goodness-of-fit test to determine if the actual college majors of graduating women fit the distribution of their expected majors.

Read the statement and decide whether it is true or false.

In a goodness-of-fit test, the expected values are the values we would expect if the null hypothesis were true.

Read the statement and decide whether it is true or false.

Use a goodness-of-fit test to determine if high school principals believe that students are absent equally during the week or not.

Read the statement and decide whether it is true or false.

In a goodness-of fit test, if the p-value is 0.0113, in general, do not reject the null hypothesis.

Read the statement and decide whether it is true or false.

Table 11.42 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. Perform a hypothesis test at the 5% significance level to determine whether the survey participants are a representative sample of the USA obese population.

For each problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chi-square solution sheet. Round expected frequency to two decimal places.

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 Table 11.44. Conduct a test of independence.

For each problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chi-square solution sheet. Round expected frequency to two decimal places.

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 given in Table 11.46. Conduct a test of independence.

For each problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chi-square solution sheet. Round expected frequency to two decimal places.

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 Table 11.48. Conduct a test of independence.

For each problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chi-square solution sheet. Round expected frequency to two decimal places.

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.

Read the statement and decide whether it is true or false.

The test for independence uses tables of observed and expected data values.

Read the statement and decide whether it is true or false.

In a test of independence, the expected number is equal to the row total multiplied by the column total divided by the total surveyed.

Read the statement and decide whether it is true or false.

Table 11.52 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.

For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

A psychologist is interested in testing whether there is a difference in the distribution of personality types for business majors and social science majors. The results of the study are shown in Table 11.54. Conduct a test of homogeneity. Test at a 5% level of significance.

For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

A fish and wildlife technician is interested in whether the distribution of fish caught in Green Valley Lake is the same as the distribution of fish caught in Echo Lake. Of the 191 randomly selected fish caught in Green Valley Lake, 105 were rainbow trout, 27 were other trout, 35 were bass, and 24 were catfish. Of the 293 randomly selected fish caught in Echo Lake, 115 were rainbow trout, 58 were other trout, 67 were bass, and 53 were catfish. Perform a test for homogeneity at a 5% level of significance.

For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

When looking at energy consumption, we are often interested in detecting trends over time and how they correlate among different countries. The information in Table 11.57 shows the average energy use (in units of kg of oil equivalent per capita) in the USA and the joint European Union countries (EU) for a specific six-year period. Do the energy use values in these two areas come from the same distribution? Perform the analysis at the 5% significance level.

For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

Is there a difference between the distribution of community college statistics students and the distribution of university statistics students in what technology they use on their homework? Of some randomly selected community college students, 43 used a computer, 102 used a calculator with built in statistics functions, and 65 used a table from the textbook. Of some randomly selected university students, 28 used a computer, 33 used a calculator with built in statistics functions, and 40 used a table from the textbook. Conduct an appropriate hypothesis test using a 0.05 level of significance.

Use the following information to answer the next twelve exercises: Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

A sample standard deviation of 15 minutes is the same as a sample variance of __________ minutes.

Use the following information to answer the next twelve exercises: Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

H₀: __________

Use the following information to answer the next twelve exercises: Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

chi-square test statistic = ________

Use the following information to answer the next twelve exercises: Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade the p-value.

Use the following information to answer the next twelve exercises: Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

How did you know to test the variance instead of the mean?

Use the following information to answer the next twelve exercises: Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

If an additional test were done on the claim of the average delay, but 45 flights were surveyed, which distribution would you use?

For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

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.

For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

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.

For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

According to an avid aquarist, the average number of fish in a 20-gallon tank is 10, with a standard deviation of two. His friend, also an aquarist, does not believe that the standard deviation is two. They count the number of fish in 15 other 20-gallon tanks. Based on the results that follow, do you think that the standard deviation is different from two? Data: 11; 10; 9; 10; 10; 11; 11; 10; 12; 9; 7; 9; 11; 10; 11

For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E Solution Sheets for the chi-square solution sheet. Round expected frequency to two decimal places.

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 very narrow standard deviation of $25. 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 standard deviation than claimed by the manufacturer? Use the 5% significance level. As a potential buyer, what would be the practical conclusion from your analysis?

a. Explain why a goodness-of-fit test and a test of independence are generally right-tailed tests.

b. If you did a left-tailed test, what would you be testing?