Multinomial Experiments (15 problems)

A personnel director believes that the distribution of the reasons workers leave their jobs is different from percentages that were given in a USA Today article. The article stated 41% leave their job due to limited advancement potential, 25% for lack of recognition, 15% for low salary and benefits, 10% are unhappy with management, and 9% are bored and/or do not have a reason. The director randomly selects 200 workers who recently left their jobs and asks each his or her reason for doing so. The results are shown below. Using the guidelines on page 512 in our textbook (goodness of fit test), test the claim that the distribution is different. Use a = 0.01.

A local firm wanted to know if any local station has a significantly larger proportion of the TV viewers of the evening news broadcasts. The firm randomly selected 1000 people and asked them to specify which station’s news program they prefer. The responses were as follows:

Test whether or not the viewers prefer equally all the stations (with)

A researcher is interested in testing whether car color choice is indicative of a driver’s propensity to speed. The following data was collected by a local police department on the number of speeding tickets issued last month according to car color (n = 505). Using this data, test whether all car colors are equally implicated in speeding incidents. Assume that the police caught these speeders at random. Use the p-value approach, and state your hypotheses.

Article states that 53% of adult shoppers prefer to pay cash for purchases, 30% use checks, 16% use credit cards, and 1% have no preference. Store randomly selected 800 shoppers and asked their payment preferences. Results were that 400 paid cash, 210 paid by check, 170 paid with credit card, 20 had no preference. AT a=0.01, test the claim that the owner’s customers have the same preferences as those surveyed.

A poll asked adults whether they felt genetically modified food was safe to eat. 35% felt it was safe, 52% felt it was not safe, and 13% had no opinion. Random sample of 120 adults was asked the same question at a fair. 40 people felt that genetically modified food was safe, 60 felt that it was not safe, and 20 had no opinion. At the 0.01 level of significance, is there sufficient evidence to conclude that the proportions differ from those reported in the survey?

A company had 36 absences on Monday, 26 on Tuesday, 10 on Wednesday, 20 on Thursday, and 28 on Friday. At the .05 level, is there a difference in the number of absence per day?

State the hypotheses and identify the claim, find the critical value(s), compute the test value, make the decision, summarize the results.

Manager believes that 50% of her customers purchase word processing programs, 25% purchase spreadsheet programs, and 25% purchase database programming. A sample of purchases shows the following distribution. At a=0.05, is her assumption correct? Use the P-value method.

Program word processing spreadsheet database

No. of

Purchases 38 23 19

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The chief of security for the Mall of the Dakotas was directed to study the problem of missing goods. He selected a sample of 100 boxes that had been tampered with and ascertained that for 60 of the boxes, the missing pants, shoes, and so on were attributed to shoplifting. For 30 other boxes employees had stolen the goods, and for the remaining 10 boxes he blamed poor inventory control. In his report to the mall management, can he say that shoplifting is twice as likely to be the cause of the loss as compared with either employee theft or poor inventory control and that employee theft and poor inventory control are equally likely? Use the .02 significance level.

In “good news-bad news” settings, researchers report that 83% of women in the 21-34 age group say they prefer to hear the bad news first. This compares to 50% for the 35-44 group, 53% for the 45-54 group, and 70% for the 55 or over group.

For the “good news-bad news” situation and age groups described above, the corresponding percentages of men who say they prefer to hear the bad news first was reported as 70%, 75%, 76% and 10%. Assuming independent sample, each with n=100, use the 0.05 level in examining whether the population percentages could be equal for men in these four age groups.

Benford’s law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability , much greater than the expected 11.1% (i.e., one digit out of 9). More specifically, the empirical distribution of probabilities is

Using the suggested categories, we have the following table:

We want to test the following:

So we have now the table with the expected values (under the null hypothesis):

We compute now the Chi-Square statistics:

The critical value for 2 df and is 5.991. The test statistics is way beyond the critical value, so we reject the null hypothesis. This supports the claim the check amounts are the result of fraud.

Staff member wishes to determine whether the number of accidents is equally distributed during the week. A week was selected at random, and following data were obtained. Is there evidence to reject the hypothesis that the number of accidents is equally distributed throughout the week, at a = 0.05?

Poll shows that 74% of respondents felt other motorists were driving more aggressively than they did five years ago, 23% felt that other motorists were driving the same way they did 5 years ago, and 3% felt other motorists were driving less aggressively than they were driving 5 years ago. A sample survey of 180 senior drivers found that 125 felt that other motorists were driving more aggressively than they did 5 years ago, 36 felt that other motorists were driving about the same as they did 5 years ago, and 19 felt that other motorists were driving less aggressively than they did 5 years ago. At a significance level of .10 test the claim that senior drivers feel the same way as those who surveyed in the poll. State hypotheses and identify claim, find critical values, compute test value, make decision, summarize results.

Populations of state prisons nationwide by serious offenses are the following: violent offenses, 29.5%; property offenses, 29%; drug offenses, 30.2%; public order offenses-weapons, 10.6%; other, 0.7%. A state prison official wants to check how this compares to her state. She surveys 1000 inmates and finds 298 are violent offenders, 275 are property offenders, 344 are drug offenders, 80 are public order offenders, and 3 have other offenses. Can she conclude that the percentages for her prison are the same as national statistics? Use a=0.05.

Instant oatmeal mix is considering adding flavors to its mix. 200 people tested the flavors and gave preferences. Is there a preference for flavor at the .05 level? State hypotheses and identify claim, find critical value(s) compute test value, make the decision, and summarize results.

Plain 20

Cinnamon 58

Apple 48

Maple 22

Peach 52

In a particular market there are three commercial television stations, each with its own evening news program from 6:00 to 6:30 P.M. According to a report in this morning's local newspaper, a random sample of 150 viewers last night revealed 53 watched the news on WNAE (channel 5), 64 watched on WRRN (channel 11), and 33 on WSPD (channel 13). At the .05 significance level, is there a difference in the proportion of viewers watching the three channels?