Exam 10: Associations Between Categorical Variables

Suppose a random sample of 1,220 U.S. adults were asked about their opinion regarding federal spending on infrastructure (i.e. roads and bridges). Respondents were asked whether federal spending on infrastructure was (a) too low, (b) adequate, or (c) too high. Respondents were classified by income level. If we wanted to test whether there was an association between the Response to the question and income level, would this be a test of homogeneity or of independence?

Use the following information to answer the question. The data in the top row of the table shows the number of days for which the respondent participated in an outdoor activity for at least thirty minutes in the previous 60 days. The respondents also reported their level of happiness; Very H means very happy, and so on. $\text { Days of outdoor activity ( } 30 \text { minutes or more) }$ Level of Happiness 0 1 2 3 4 5+ Very H 8 12 17 19 21 24 Fairly H 4 5 25 14 25 30 Not very H 4 6 6 3 0 1 Not H 7 5 6 1 0 1 -The following table shows the data after merging categories so that there are two column categories (0-2 days of outdoor activity of at least thirty minutes and 3 or more days of outdoor activity of at least thirty minutes), and two row categories (happy and unhappy). Expected values for each cell are also shown in parenthesis. Test the hypothesis that there is an association between happiness and with level of daily outdoor activity in the last 60 days, using a significance level of 0.05. State the value of the test statistic rounded to two decimal places, state whether the p-value is closer to zero or one. 0-2 days of outdoor activity 3+ days of outdoor activity Happy 71 133 (87.787) (116.21) Unhappy 34 6 (17.213) (22.787)

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of heights than males. Men Women Expressed a fear of heights 68 109 Did not express a fear of heights 94 89 -Find the expected number of men who should express a fear of heights, if the variables are independent. Round to the nearest whole number.

The following table shows the results from a study to see if a home remedy ointment for mosquito bites worked better than a placebo. Each participant was randomly assigned to receive the home remedy ointment or the placebo ointment. "Improvement" means there were no symptoms of itching after three minutes. Home Remedy Placebo Total No Improvement 2 4 6 Improvement 6 5 11 Total 8 9 17 The alternative hypothesis is that the home remedy ointment leads to improvement (in this case, less itching). The p-value for a one-tailed Fisher's exact test with these data is $0.618$ . Suppose the study had turned out differently, as in the following table. Home No Improvement 6 0 6 Improvement 2 9 11 Total 8 9 17 Would Fisher's Exact Test have led to a p-value larger or smaller than 0.618?

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of public speaking then males. Men Women Expressed a fear of public speaking 112 79 Did not express a fear of public speaking 91 82 -Describe the categorical variables that are summarized in the table. Describe one association that could be tested based on the information from the table.

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. Correctly identified location of ground water Incorrectly identified location of ground water Observed 21 57 Expected 15.6 62.4 GOF-Test =2.336538462 p=.1263700849 d.f. =1 CNTRB =\{1.86923\ldots -Test the hypothesis that the dowsing rods worked better at locating ground water than guessing. Using a goodness-of-fit test and a 0.05 level of significance, choose the correct decision regarding the null hypothesis and conclusion statement.

The table below shows the gender and the percentage of each gender that spent different amounts at a local hardware store. The data was taken from a random sample of single shoppers collected over five consecutive Saturdays at the hardware store. Choose the reason(s) why you cannot do a Chi-square test with this data. \ 0-\ 20 >\ 20-\ 50 >\ 50 Male shopper 27\% 42\% 31\% Female shopper 46\% 35\% 19\%

The following table shows the results from a study to see if a home remedy ointment for mosquito bites worked better than a placebo. Each participant was randomly assigned to receive the home remedy ointment or the placebo ointment. "Improvement" means no symptoms of itching after three minutes. Home Remedy Placebo Total No Improvement 2 4 6 Improvement 6 5 11 Total 8 9 17 The alternative hypothesis is that the home remedy ointment leads to improvement (in this case, less itching). The p-value for a one-tailed Fisher's exact test with these data is $0.618$ . Suppose the study had turned out differently, as in the following table. Home Remedy Placebo Total No Improvement 0 6 6 Improvement 8 3 11 Total 8 9 17 Would Fisher's Exact Test have led to a p-value larger or smaller than 0.618?

Suppose a random sample of 1,105 adults were asked about their opinion regarding salaries of full-time employees at a local state funded university. Respondents were asked whether salaries at the university were (a) too low, (b) adequate, or (c) too high. Respondents were also classified by income level. State the correct hypothesis to test whether there is an association between the response to the questions and income level.

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of public speaking then males. Men Women Expressed a fear of public speaking 112 79 Did not express a fear of public speaking 91 82 -Suppose a goodness-of-fit test is used to test the claim that obesity rates in the elderly have changed since the time of the Egyptian mummies. The p-value is calculated to be 0.00023. Describe the value of the chi-square test statistic (is it likely to be large? small? ) and the decision regarding the null hypothesis that there is no difference in obesity rates for mummies and modern-day elderly people.

Use the following information to answer the question. Bellei Extravecchio Balsamico Tradizionale is a very expensive brand of balsamic vinegar. A twelve ounce bottle typically costs two-hundred dollars or more. In a blind taste test, a group of food experts tasted three premium balsamic vinegars, with the most expensive one being Bellei Extravecchio Balsamico Tradizionale. When asked to pick the most expensive balsamic vinegar, 92 got it right and 76 got it wrong. -Calculate the observed value of the chi-square statistic. Round to the nearest hundredth.

Use the following information to answer the question. Lambda olive oil is touted as the "World's most expensive olive oil." twelve ounce bottle typically costs fifty dollars or more. In a blind taste test, a group of food experts tasted three premium olive oils, one of which was Lambda olive oil. When asked to pick the Lambda olive oil, 84 got it right and 87 got it wrong. -If this group were just guessing, how many people (out of 171) would be expected to guess correctly?

Choose the statement that is not true about the chi-square distribution or choose (d) if all the statements are true.

Use the following information to answer the question. Bellei Extravecchio Balsamico Tradizionale is a very expensive brand of balsamic vinegar. A twelve ounce bottle typically costs two-hundred dollars or more. In a blind taste test, a group of food experts tasted three premium balsamic vinegars, with the most expensive one being Bellei Extravecchio Balsamico Tradizionale. When asked to pick the most expensive balsamic vinegar, 92 got it right and 76 got it wrong. -If this group were just guessing, how many people (out of 168) would be expected to guess correctly?

Choose the statement that is not true about Fisher's Exact Test or choose (d) if all the statements are true.

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. Correctly identified location of ground water Incorrectly identified location of ground water Observed 21 57 Expected 15.6 62.4 GOF-Test =2.336538462 p=1263700849 d.f. =1 CNTRB =\{1.86923\ldots -Choose the correct null and alternative hypothesis.

Suppose a researcher was interested in learning more about parents' concerns when their children start elementary school. The researcher asks the parents of 800 randomly selected first graders in rural school district and the parents of 950 randomly selected first graders in an urban school district to rate their level of concern with the following statement: We are (a) not at all concerned (b) somewhat concerned or (c) very concerned about the nutrition level of school lunches. If we Wanted to test whether there was an association between the response to the question and the type Of school district that the first grader was attending, would this be a test of homogeneity or of Independence?

Suppose a study was conducted to see whether there is an association between marital status and vehicle color. The table below shows the results from the study. Assume all conditions for testing have been met. Vehicle color red black/white/silver blue/green/other Married 45 22 19 Unmarried 35 11 18 This was an observational study of randomly chosen vehicle owners. Test the hypothesis that marital status and vehicle color are associated, using a significance level of $0.05$ . Choose the correct decision regarding the null hypothesis and conclusion. Refer to the computer output below. - Test =1.754774169 p=.4158681217 =2

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of public speaking then males. Men Women Expressed a fear of public speaking 112 79 Did not express a fear of public speaking 91 82 -Calculate the percentage of people in the study that did not express a fear of public speaking. Round to the nearest tenth of a percent.

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of flying than males. Men Women Expressed a fear of flying 42 112 Did not express a fear of flying 58 98 -What fraction represents the proportion of people in the study who did not express a fear of flying?