Exam 10: Chi-Square Tests and the F-Distribution

Each side of a standard six-sided die should appear approximately $\frac { 1 } { 6 }$ oof the time when the die is rolled. A player suspects that a certain die is loaded. The suspected die is rolled 90 times. The results are shown below. Calculate the chi-square test statistic χ $x ^ { 2 }$ to test the playerʹs claim. Number 1 2 3 4 5 6 Frequency 15 12 16 19 17 11

Four different types of fertilizers are used on raspberry plants. The number of raspberries on each randomly selected plant is given below. Find the critical value F0 to test the claim that the type of fertilizer makes no Difference in the mean number of raspberries per plant. Use $\alpha = 0.01$ Fertilizer 1 Fertilizer 2 Fertilizer 3 Fertilizer 4 6 8 6 3 5 5 3 5 6 5 3 3 7 5 4 4 7 5 2 4 6 6 3 5

A sociologist believes that the levels of educational attainment of homeless persons are not uniformly distributed. To test this claim, you randomly survey 100 homeless persons and record the educational attainment of each. The results are shown in the following table. Find the critical value χ $\chi _ { 0 } { } ^ { 2 }$ to test the sociologistʹs claim. Use $\alpha$ = 0.10. Response Frequency, f Less than high school 38 High school graduate/G.E.D. 34 More than high school 28

Find the critical value $\mathrm { F } _ { 0 }$ for a two-tailed test using α $\alpha$ = 0.05, d.f.N = 5, and d.f.D = 10.

Each side of a standard six-sided die should appear approximately $\frac { 1 } { 6 }$ of the time when the die is rolled. A player suspects that a certain die is loaded. The suspected die is rolled 90 times. The results are shown below. Test the playerʹs claim. $\text { Use } \alpha = 0.10 \text {. }$ Number 1 2 3 4 5 6 Frequency 19 12 15 17 16 11

Perform the indicated hypothesis test. Be sure to do the following: identify the claim and state the null and alternative hypotheses. Determine the critical value and rejection region. Calculate the test statistic. Decide to reject or to fail to reject the null hypothesis and interpret the decision in the context of the original claim. Assume that the samples are independent and that each population has a normal distribution. -A statistics teacher wants to see whether there is a significant difference between the variance of the ages of day students and the variance of the ages of night students. A random sample of 31 students is selected from each group. The data are given below. Test the claim that there is no difference between the variances of the two groups. Use $\alpha = 0.05$ $\text { Day Students}$ 22 24 24 23 19 19 23 22 18 21 21 18 18 25 29 24 23 22 22 21 20 20 20 27 17 19 18 21 20 23 26 $\text {Evening Students}$ 18 23 25 23 21 21 23 24 27 31 34 20 20 23 19 25 24 27 23 20 20 21 25 24 23 28 20 19 23 24 20

Perform the indicated hypothesis test. Be sure to do the following: identify the claim and state the null and alternative hypotheses. Determine the critical value and rejection region. Calculate the test statistic. Decide to reject or to fail to reject the null hypothesis and interpret the decision in the context of the original claim. Assume that the samples are independent and that each population has a normal distribution. -A study was conducted to determine if the variances of elementary school teacher salaries from two neighboring districts were equal. A sample of 25 teachers from each district was selected. The first district had a standard deviation of $\mathrm { s } _ { 1 }$ = $4830, and the second district had a standard deviation $s _ { 2 }$ = $4410. Test the claim that the variances of the salaries from both districts are equal. Use $\alpha = 0.05$

Find the critical value $\mathrm { F } _ { 0 }$ for a one-tailed test using α $\alpha$ = 0.05, d.f.N = 6, and d.f.D = 16.

Perform a homogeneity of proportions test to test whether the population proportions are equal. The hypotheses to be tested are: H₀: The proportions are equal H_a: At least one of the proportions is different from the others. -A random sample of 100 students from 5 different colleges was randomly selected, and the number who smoke was recorded. The results are listed below. Perform a homogeneity of proportions test to test the claim that the proportion of students who smoke is the same in all 5 colleges. Use $\alpha = 0.01$ Colleges 1 2 3 4 5 Smoker 18 25 12 33 22 Nonsmoker 82 75 88 67 78

Perform the indicated hypothesis test. Be sure to do the following: identify the claim and state the null and alternative hypotheses. Determine the critical value and rejection region. Calculate the test statistic. Decide to reject or to fail to reject the null hypothesis and interpret the decision in the context of the original claim. Assume that the samples are independent and that each population has a normal distribution. -Test the claim that $\sigma_ { 2 }^ { 1 } = \sigma_ { 2 } ^{ 2 }$ Two samples are randomly selected from populations that are normal. The sample statistics are given below. Use $\alpha = 0.05 .$ =25 =30 =5.776 =3.6

Perform the indicated hypothesis test. Be sure to do the following: identify the claim and state the null and alternative hypotheses. Determine the critical value and rejection region. Calculate the test statistic. Decide to reject or to fail to reject the null hypothesis and interpret the decision in the context of the original claim. Assume that the samples are independent and that each population has a normal distribution. -A statistics teacher believes that the variance of test scores of students in her evening statistics class is lower than the variance of test scores of students in her day class. The results of an exam, given to the day and evening students, are shown below. Can the teacher conclude that the scores of evening students have a lower variance? Use $\alpha = 0.01$ Day Students Evening Students =31 =41 =9.8 =5.3

Calculate the test statistic F to test the claim that $\sigma _ { 1 } ^ { 2 } = \sigma _{ 2 }^ { 2 } .$ Two samples are randomly selected from populations that are normal. The sample statistics are given below. =13 =12 =17.248 =13.75

Calculate the test statistic F to test the claim that does not equalsigma start subscript 2 end subscript squared Two samples are randomly selected from populations that are normal. The sample statistics are given below. =11 =18 =1.156 =0.52

A sociologist believes that the levels of educational attainment of homeless persons are not uniformly distributed. To test this claim, you randomly survey 100 homeless persons and record the educational attainment of each. The results are shown in the following table. At α $\alpha$ = 0.10, is there evidence to support the sociologistʹs claim that the distribution is not uniform? Response Frequency, Less than high school 38 High school graduate/G.E.D. 34 More than high school 28

A sports researcher is interested in determining if there is a relationship between the type of sport and type of team winning (home team versus visiting team). A random sample of 526 games is selected and the results are Given below. Calculate the chi-square test statistic χ $\chi ^ { 2 }$ to test the claim that the type of team winning is Independent of the sport. Football Basketball Soccer Baseball Home team wins 39 156 25 83 Visiting team wins 31 98 19 75

Perform the indicated one-way ANOVA test. Be sure to do the following: identify the claim and state the null and alternative hypotheses. Determine the critical value and rejection region. Calculate the test statistic F. Decide to reject or to fail to reject the null hypothesis and interpret the decision in the context of the original claim. Assume that each population is normally distributed and that the population variances are equal. -A realtor wishes to compare the square footage of houses of similar prices in 4 different cities. The data are listed below. Can the realtor conclude that the mean square footage is the same in all four cities? Use $\alpha = 0.01$ City \#1 City \#2 City \#3 City \#4 2150 1780 1530 2400 1980 1540 1670 2350 2000 1690 1580 2600 2210 1650 1750 2150 1900 1500 2000 1600 2200 2350

Find the indicated expected frequency. -A researcher wants to determine whether the number of minutes adults spend online per day is related to gender. A random sample of 315 adults was selected and the results are shown below. Find the expected Frequency for the cell $\mathrm { E } _ { 2,2 }$ to determine if there is enough evidence to conclude that the number of minutes Spent online per day is related to gender. Round to the nearest tenth if necessary. Minutes spent online per day Gender 0-30 30-60 60-90 90+ Male 23 35 76 46 Female 31 42 46 16

Perform the indicated one-way ANOVA test. Be sure to do the following: identify the claim and state the null and alternative hypotheses. Determine the critical value and rejection region. Calculate the test statistic F. Decide to reject or to fail to reject the null hypothesis and interpret the decision in the context of the original claim. Assume that each population is normally distributed and that the population variances are equal. -Four different types of fertilizers are used on raspberry plants. The number of raspberries on each randomly selected plant is given below. Perform a Scheffe´ Test to determine which means have a significance difference. Use $\alpha = 0.01 .$ Fertilizer 1 Fertilizer 2 Fertilizer 3 Fertilizer 4 6 5 6 3 7 8 3 5 5 5 4 3 6 5 3 4 7 5 2 5 6 6 3 4

Perform a homogeneity of proportions test to test whether the population proportions are equal. The hypotheses to be tested are: H₀: The proportions are equal H_a: At least one of the proportions is different from the others. -Random samples of 400 men and 400 women were obtained and participants were asked whether they planned to vote in the next election. The results are listed below. Perform a homogeneity of proportions test 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. Use $\alpha$ = 0.05. Men Women Plan to vote 230 255 Do not plan to vote 170 145

A medical researcher wishes to try three different techniques to lower blood pressure of patients with high blood pressure. The subjects are randomly selected and assigned to one of three groups. Group 1 is given Medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six Weeks, each subjectʹs blood pressure is recorded. Find the critical value $\mathrm { F } _ { 0 }$ to test the claim that there is no Difference among the means. Use $\alpha$ = 0.05. Group 1 Group 2 Group 3 9 8 4 12 2 12 11 3 4 15 5 8 13 4 9 8 0 6