Exam 11: Goodness-Of-Fit and Contingency Tables

True or False: For a test of independence, the population that the data has come from must be normally distributed.

Describe a goodness-of-fit test.

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?

Describe the null hypothesis for the test of independence. List the assumptions for the $x ^ { 2 }$ test of independence.

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?

Use a 0.01 significance level to test the claim that the proportion of men who plan to vote in the next election is the same as the proportion of women who plan to vote. 300 men and 300 women were randomly selected and asked whether they planned to vote in the next election. The results are shown below. Men Women Plan to vote 170 185 Do not plan to vote 130 115

While conducting a goodness-of-fit test if the observed and expected values are close, you would expect which of the following:

The following table represents the number of absences on various days of the week at an elementary school. Monday Tuesday Wednesday Thursday Friday 45 32 17 25 38 Identify the number of degrees of freedom for a goodness-of-fit test (for a uniform distribution), assuming a 0.05 significance level.

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

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

Define categorical data and give an example

The following table shows the number of employees who called in sick at a business for different days of a particular week. Day Sun Mon Tues Wed Thurs Fri Sat Number sick 8 12 7 11 9 11 12 I) At the 0.05 level of significance, test the claim that sick days occur with equal frequency on the different days of the week. II) Test the claim after changing the frequency for Saturday to 152 . Describe the effect of this outlier on the test.

Which statement is not true for goodness-of -fit tests?

The following are the hypotheses for a test of the claim that college graduation statue and cola preference are independent. H₀: College graduation status and cola preference are independent. H₁: 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?

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

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 Use a 0.05 significance level to test the claim that the proportion of people catching the flu is the same in all three groups.

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 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 $a = 0.05$ 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

Perform the indicated goodness-of-fit test. A company manager wishes to test a union leader's claim that absences occur on the different week days with the same frequencies. Test this claim at the 0.05 level of significance if the following sample data have been compiled. Day Mon Tue Wed Thurs Fri Absences 37 15 12 23 43

Select the null hypothesis for a test of independence