Exam 12: Inference on Categorical Data

The results of a recent national survey reported that 70% of Americans recycle at least some of the time. As partof their final project in statistics class, Nayla and Roberto survey 5 random students on campus and ask them ifthey recycle at least some of the time. They then repeat this experiment 1000 times. The results of their researchare shown below. X (number who recycle out of 5) 0 1 2 3 4 5 Frequency 2 27 139 305 358 169 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 company wants to determine if its employees have any preference among 5 different health plans which it offers to them. A sample of 200 employees provided the data below. Test the claim that the probabilities show no preference. Use α=0.01 . Plan 1 2 3 4 5 Employees 32 18 55 65 30

A spinner is mounted on a piece of cardboard divided into six areas of equal size. Each of the six areas is a different color (blue, yellow, red, green, white and orange). When the spinner is spun, each color should be selected by the spinner approximately $\frac { 1 } { 6 }$ of the time. A student suspects that a certain spinner is defective. The suspected spinner is spun 90 times. The results are shown below. Calculate the chi -square test statistic $x ^ { 2 }$ to test the student's claim. Use $\alpha$ =0.10 . Color Blue Yellow Red Green White Orange Frequency 15 12 11 17 19 16

A random sample of 100 employees from 5 different companies was randomly selected, and the number whotake public transportation to work was recorded. The results are listed below. Perform a homogeneity ofproportions test to test the claim that the proportion who take public transportation to work is the same in all 5companies. Use α = 0.01. Companies 1 2 3 4 5 Use Public Trans. 18 25 12 33 22 Don't Use Public Trans. 82 75 88 67 78

The degrees of freedom for a $x ^ { 2 }$ goodness-of-fit test when there are 6 categories and a sample of size 1200 is

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

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

As the number of the degrees of freedom increases, the χ2 distribution

A sports statistician is interested in determining if there is a relationship between the number of home teamand visiting team losses and different sports. A random sample of 526 games is selected and the results aregiven below. Test the claim that the number of home team and visiting team losses is independent of the sport.Use α = 0.01. Football Basketball Soccer Baseball Home team losses 39 156 25 83 Visiting team losses 31 98 19 75

A teacher figures that final grades in the chemistry department are distributed as: A, 25 % ; B, 25 % ; C, 40 % ; D, 5 % ; F, 5 % . At the end of a randomly selected semester, the following number of grades were recorded. Determine if the grade distribution for the department is different than expected. Use α=0.01 . Grade A B C D F Number 36 42 60 14 8

A spinner is mounted on a piece of cardboard divided into six areas of equal size. Each of the six areas is a different color (blue, yellow, red, green, white and orange). When the spinner is spun, each color should be selected by the spinner approximately $\frac { 1 } { 6 }$ of the time. A student suspects that a certain spinner is defective. The suspected spinner is spun 90 times. The results are shown below. Test the student's claim. Use α=0.10 . Color Blue Yellow Red Green White Orange Frequency 12 11 17 16 19 15

A spinner is mounted on a piece of cardboard divided into six areas of equal size. Each of the six areas is a different color (blue, yellow, red, green, white and orange). When the spinner is spun, each color should be selected by the spinner approximately $\frac { 1 } { 6 }$ of the time. A student suspects that a certain spinner is defective. The suspected spinner is spun 90 times. The results are shown below. Find the critical value $\chi _ { 0 } ^ { 2 }$ to test the student's claim. Use $\alpha = 0.10$ . Color Blue Yellow Red Green White Orange Frequency 12 11 16 19 17 15

The contingency table below shows the results of a random sample of 200 registered voters that was conductedto see whether their opinions on a bill are related to their party affiliation. Use α = 0.05. Party Opinion Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Find the critical value , to test the claim of independence.

A multinomial experiment with k = 4 cells and n = 300 produced the data shown in the following table. Cell 1 2 3 4 65 69 80 86 Do these data provide sufficient evidence to contradict the null hypothesis that $\mathrm { p } _ { 1 } = 0.20$ , $\mathrm { p } _ { 2 } = 0.20$ , $\mathrm { p } _ { 3 } = 0.30$ , and $\mathrm { p } _ { 4 } = 0.30$ ? Test using $\alpha = 0.05$ .

A researcher wants to determine if the number of minutes spent watching television per day is independent ofgender. 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 televisior per day is related to gender. Use $\alpha = 0.05$ . $\text { Gender }$$\quad$$\text { Minutes spent watching TV per day }$ 0-30 30-60 60-90 90- over Male 25 35 75 45 Female 30 45 45 15

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

In a χ2 test of independence, the null hypothesis is that

A teacher figures that final grades in the chemistry department are distributed as: A, 25 % ; B, 25 % ; 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 $x ^ { 2 }$ to determine if the grade distribution for the department is different than expected. Use $\alpha$ =0.01 . Grade A B C D F Number 42 36 60 14 8