Exam 11: Chi-Square and Analysis of Variance

The following are the hypotheses for a test of the claim that college graduation statue and cola preference are independent. $H _ { 0 }$ : College graduation status and cola preference are independent. $H _ { 1 }$ : College graduation status and cola preference are dependent. If the test statistic: $\chi ^ { 2 } = 0.579$ and the critical value is $\chi ^ { 2 } = 5.991$ , what is your conclusion about the null hypothesis and about the claim?

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

Explain the computation of expected values for contingency tables in terms of probabilities. Refer to the assumptions of the null hypothesis as part of your explanation. You might give a brief example to illustrate.

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and chiropractor)to relieve back pain. If we test the claim at a 5% level of significance that success is independent of the method used, technology provides a P-value of 0.0655. What does the P-value tell us about the claim?

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and chiropractor)to relieve back pain. If we test the claim at a 5% level of significance that success is independent of the method used, technology provides a P-value of 0.0355. What does the P-value tell us about the claim?

Use a $\chi ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are independent. Tests for adverse reactions to a new drug yielded the results given in the table. At the $0.05$ significance level, test the claim that the treatment (drug or placebo) is independent of the reaction (whether or not headaches were experienced). Drug Placebo Headaches 11 7 No headaches 73 91

Use a $\chi ^ { 2 }$ test to test the claim that in the given contingency table, the row variable and the column variable are independent. 160 students who were majoring in either math or English were asked a test question, and the researcher recorded whether they answered the question correctly. The sample results are given below. At the $0.10$ significance level, test the claim that response and major are independent. Correct Incorrect Math 27 53 English 43 37

In conducting a goodness-of-fit test, a requirement is that __________________________.

Perform the indicated goodness-of-fit test. You roll a die 48 times with the following results. Number 1 2 3 4 5 6 Frequency 4 13 2 14 13 2 Use a significance level of 0.05 to test the claim that the die is fair.

According digits that follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. Leading Digit 1 2 3 4 5 6 7 8 9 Benford's law: distribution of leading digits 30.1\% 17.6\% 12.5\% 9.7\% 7.9\% 6.7\% 5.8\% 5.1\% 4.6\% When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be $0,18,0,79,476,180,8,23$ , and 0 , and those digits correspond to the leading digits of $1,2,3,4,5,6,7,8$ , and 9 , respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's Law, the check amounts appear to result from fraud. Use a $0.05$ significance level to test for goodness-of-fit with Benford's Law. Does it appear that the checks are the result of fraud?

A survey of students at a college was asked if they lived at home with their parents, rented an apartment, or owned their own home. The results are shown in the table below sorted by gender. At $\alpha = 0.05$ , test the claim that living accommodations are independent of the gender of the student. Live with Parent Rent Apartment Own Home Male 20 26 19 Female 18 28 26

Among the four northwestern states, Washington has 51% of the total population, Oregon has 30%, Idaho has 11%, and Montana has 8%. A market researcher selects a sample of 1000 subjects, with 450 in Washington, 340 in Oregon, 150 in Idaho, and 60 in Montana. At the 0.05 significance level, test the claim that the sample of 1000 subjects has a distribution that agrees with the distribution of state populations.

Describe the test of homogeneity. What characteristic distinguishes a test of homogeneity from a test of independence?

A researcher wishes to test the effectiveness of a flu vaccination. 150 people are vaccinated, 180 people are vaccinated with a placebo, and 100 people are not vaccinated. The number in each group who later caught the flu was recorded. The results are shown below. Vaccinated Placebo Control Caught the flu 8 19 21 Did not catch the flu 142 161 79

For a recent year, the following are the numbers of homicides that occurred each month in New York City: 38, 30, 46, 40, 46, 49, 47, 50, 50, 42, 37, 37. Use a 0.05 significance level to test the claim that homicides in New York City are equally likely for each of the 12 months. State your Conclusion about the claim.

According to Benford's Law, a variety of different data sets include numbers with leading (first)digits that follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. Leading Digit 1 2 3 4 5 6 7 8 9 Benford's law: distribution of leading digits 30.1\% 17.6\% 12.5\% 9.7\% 7.9\% 6.7\% 5.8\% 5.1\% 4.6\% When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 18, 0, 79, 476, 180, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's Law, the check amounts appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit with Benford's Law. Does it appear that the checks are the result of fraud?

Responses to a survey question about color preference for a candy are broken down according to gender in the table given below. At the 0.05 significance level, test the claim that candy Color preference and gender are independent. Red Blue Yellow Male 25 50 15 Female 20 30 10 What is your conclusion about the null hypothesis and about the claim?

Using the data below and a 0.05 significance level, test the claim that the responses occur with percentages of 15%, 20%, 25%, 25%, and 15% respectively. Response A B C D E Frequency 12 15 16 18 19

A survey of students at a college was asked if they lived at home with their parents, rented an apartment, or owned their own home. The results are shown in the table below sorted by gender. At $\alpha = 0.05$ , test the claim that living accommodations are independent of the gender of the student. Live with Parent Rent Apartment Own Home Male 20 26 19 Female 18 28 30

In studying the occurrence of genetic characteristics, the following sample data were obtained. You would like to test the claim that the characteristics occur with the same Frequency at the 0.05 significance level. Characteristic A B C D E F Frequency 28 30 45 48 39 39 What is the expected value for D?