Exam 12: Inference on Categorical Data

Perform a Test for Independence -A medical researcher is interested in determining if there is a relationship between adults over 50 who exercise regularly and low, moderate, and high blood pressure. A random sample of 236 adults over 50 is selected and the results are given below. Find the critical value $x _ { 0 } ^ { 2 }$ to test the claim that regular exercise and low, moderate, and high blood pressure are independent. Use $\alpha = 0.01$ . Blood Pressure Low Moderate High Reg. Exercise 35 62 25 No Reg. Exercise 21 65 28

A random sample of 160 car purchases are selected and categorized by age. The results are listed below. The age distribution of drivers for the given categories is $18 \%$ for the under 26 group, $39 \%$ for the 26-45 group, 31\% for the 46-65 group, and $12 \%$ for the group over 65 . Test the claim that all ages have purchase rates proportional to their driving rates. Use $\alpha = 0.05$ .

Perform a Test for Independence -The contingency table below shows the results of a random sample of 200 registered voters that was conducted to see whether their opinions on a bill are related to their party affiliation. Use $\alpha = 0.05$ . Find the critical value $x _ { 0 } ^ { 2 }$ , to test the claim of independence.

Perform a Test for Homogeneity of Proportions -A random sample of 100 employees from 5 different companies was randomly selected, and the number who take public transportation to work was recorded. The results are listed below. Perform a homogeneity of proportions test to test the claim that the proportion who take public transportation to work is the same in all 5 companies. Use $\alpha = 0.01$ .

Perform a Test for Independence -The contingency table below shows the results of a random sample of 200 registered voters that was conducted to see whether their opinions on a bill are related to their party affiliation. Find the chi-square test statistic, $\chi ^ { 2 }$ , to test the claim of independence.

A multinomial experiment with $\mathrm { k } = 4$ cells and $\mathrm { n } = 300$ produced the data shown in the following table. Do these data provide sufficient evidence to contradict the null hypothesis that $\mathrm { p } _ { 1 } = 0.20 , \mathrm { p } _ { 2 } = 0.20$ , $p _ { 3 } = 0.30$ , and $p _ { 4 } = 0.30$ ? Test using $\alpha = 0.05$ .

Perform a Test for Homogeneity of Proportions -A random sample of 400 men and 400 women was randomly selected and asked whether they planned to attend a concert in the next month. The results are listed below. Perform a homogeneity of proportions test to test the claim that the proportion of men who plan to attend a concert in the next month is the same as the proportion of women who plan to attend a concert in the next month. Use $\alpha = 0.05$ . Men Women Plan to attend concert 230 255 Don't plan to attend concert 170 145

The results of a recent national survey reported that $70 \%$ of Americans recycle at least some of the time. As part of their final project in statistics class, Nayla and Roberto survey 5 random students on campus and ask them if they recycle at least some of the time. They then repeat this experiment 1000 times. The results of their research are shown below. (number who recycle out of 5) 0 1 2 3 4 5 Frequency 2 25 139 306 356 172 Is there evidence to support the belief that the random variable $X$ , the number of students out of 5 who recycle at least some of the time, is a binomial random variable with $\mathrm { p } = 0.7$ at the $\alpha = 0.05$ level?

A __________test is an inferential procedure used to determine whether a frequency distribution follows a defined distribution.

Perform a Test for Independence -A researcher wants to determine if the number of minutes spent watching television per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Find the critical value $x _ { 0 } ^ { 2 }$ to determine if there is enough evidence to conclude that the number of minutes spent watching television per day is related to gender. Use $\alpha = 0.05$ .

A teacher figures that final grades in the chemistry department are distributed as: $\mathrm { A } , 25 \% ; \mathrm { B } , 25 \% ; \mathrm { C } , 40 \%$ ; D, $5 \% ; F , 5 \%$ . At the end of a randomly selected semester, the following number of grades were recorded. Calculate the chi-square test statistic $\chi ^ { 2 }$ to determine if the grade distribution for the department is different than expected. Use $\alpha = 0.01$ . Grade A B C D F Number 36 42 60 8 14

Many track hurdlers believe that they have a better chance of winning if they start in the inside lane that is closest to the field. For the data below, the lane closest to the field is Lane 1 , the next lane is Lane 2, and so on until the outermost lane, Lane 6. The data lists the number of wins for track hurdlers in the different starting positions. Find the critical value $\chi _ { 0 } ^ { 2 }$ to test the claim that the probabilities of winning are the same in the different positions. Use $\alpha = 0.05$ . The results are based on 240 wins. Starting Position 1 2 3 4 5 6 Number of Wins 32 45 50 36 33 44

Perform a Test for Independence -A sports statistician is interested in determining if there is a relationship between the number of home team and visiting team losses and different sports. A random sample of 526 games is selected and the results are given below. Find the critical value $\chi _ { 0 } ^ { 2 }$ to test the claim that the number of home team and visiting team losses is independent of the sport. Use $\alpha = 0.01$ . Football Basketball Soccer Baseball Home team losses 39 156 25 83 Visiting team losses 31 98 19 75

Perform a Test for Homogeneity of Proportions -In a chi-square test of homogeneity of proportions we test the claims that

Perform a Test for Independence -Test the null hypothesis of independence of the two classifications, A and B, of the $3 \times 3$ contingency table shown below. Test using $\alpha = 0.01$ .

Perform a Test for Independence -A researcher wants to determine if the number of minutes spent watching television per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Is there enough evidence to conclude that the number of minutes spent watching television per day is related to gender? Use $\alpha$ $= 0.05$ .

Determine the expected counts for each outcome. $n=300$ pi 0.30 0.25 0.20 0.25 Expected Counts

Perform a Test for Independence -A medical researcher is interested in determining if there is a relationship between adults over 50 who exercise regularly and low, moderate, and high blood pressure. A random sample of 236 adults over 50 is selected and the results are given below. Test the claim that regular exercise and low, moderate, and high blood pressure are independent. Use $\alpha = 0.01$ . Blood Pressure Low Moderate High Reg. Exercise 35 62 25 No Reg. Exercise 21 65 28

A random sample of 160 car purchases are selected and categorized by age. The results are listed below. The age distribution of drivers for the given categories is $18 \%$ for the under 26 group, $39 \%$ for the $26 - 45$ group, $31 \%$ for the 45-65 group, and $12 \%$ for the group over 65 . Calculate the chi -square test statistic $\chi ^ { 2 }$ to test the claim that all ages have purchase rates proportional to their driving rates. Use $\alpha = 0.05$ . Age Under 26 26-45 46-65 Over 65 Purchases 66 39 25 30

Perform a Test for Independence -A researcher wants to determine if the number of minutes spent watching television per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Calculate the chi -square statistic $\chi ^ { 2 }$ to determine if there is enough evidence to conclude that the number of minutes spent watching television per day is related to gender. Use $\alpha = 0.05$ .