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

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. Find the critical value χ $x _ { 0 } ^ { 2 }$ to test the playerʹs claim. Use $\alpha = 0.10 .$ Number 1 2 3 4 5 6 Frequency 11 15 12 16 17 19

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. =25 =30 =6.498 =4.05

The frequency distribution shows the ages for a sample of 100 employees. Are the ages of employees normally distributed? Use $\alpha$ = 0.05. Class boundaries Frequency, 29.5-39.5 14 39.5-49.5 29 49.5-59.5 31 59.5-69.5 18 69.5-79.5 8

Find the critical value $\mathrm { F } _ { 0 }$ to test the claim that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text {. }$ Two samples are randomly selected from populations that are normal. The sample statistics are given below. Use $\alpha = 0.05$ =25 =30 =3.61 =2.25

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. Calculate the chi-square test statistic χ $\chi^ { 2 }$ to test the sociologistʹs claim. 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 $\alpha$ = 0.02, d.f.N = 5, and d.f.D = 10.

Find the critical value $\mathrm { F } _ { 0 }$ 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. Use $\alpha = 0.02$ =13 =12 =7.84 =6.25

Find the test statistic F to test the claim that the populations have the same mean. Brand 1 Brand 2 Brand 3 =8 =8 =8 =3.0 =2.6 =2.6 =0.50 =0.60 =0.55

A random sample of 160 car crashes 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. Find the critical value χ $\chi _ { 0 } ^ { 2 }$ to test the claim that all ages have Crash rates proportional to their driving rates. Use $\alpha = 0.05$ Age Under 26 26-45 46-65 Over 65 Drivers 66 39 25 30

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 $\sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 }^ { 2 }$ Two samples are randomly selected from populations that are normal. The sample statistics are given below. Use $\alpha = 0.02$ =11 =18 =2.89 =1.3

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 critical value $\chi _ { 0 } ^ { 2 }$ to determine if there is enough evidence to conclude that the number of minutes spent online per day is Related to gender. Use $\alpha$ = 0.05. Minutes spent online per day Gender 0-30 30-60 60-90 90+ Male 25 35 75 45 Female 30 45 45 15

A coffeehouse wishes to see if customers have any preference among 5 different brands of coffee. A sample of 200 customers provided the data below. Find the critical value χ $\chi _ { 0 } ^ { 2 }$ to test the claim that the distribution is Uniform. $\text { Use } \alpha = 0.01 \text {. }$ Brand 1 2 3 4 5 Customers 30 32 55 65 18

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. -Test the claim that the populations have the same mean. Use $\alpha = 0.05$ Brand 1 Brand 2 Brand 3 =8 =8 =8 =3.0 =2.6 =2.6 =0.50 =0.60 =0.55

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. -The weights of a random sample of 121 women between the ages of 25 and 34 had a standard deviation of 28 pounds. The weights of 121 women between the ages of 55 and 64 had a standard deviation 21 pounds. Test the claim that the older women are from a population with a standard deviation less than that for women in the 25 to 34 age group. Use $\alpha = 0.05$

The contingency table below shows the results of a random sample of 200 state representatives that was conducted to see whether their opinions on a bill are related to their party affiliation. Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Find the chi-square test statistic, χ $\chi ^ { 2 }$ to test the claim of independence.

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 test statistic F to test the claim that there is no Difference among the means. Group 1 Group 2 Group 3 13 8 6 12 5 12 11 3 4 15 2 8 9 4 9 8 0 4 949 804

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

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 $\sigma _ { 1 } ^ { 2 } > \sigma _{ 2 }^ { 2 }$ Two samples are randomly selected from populations that are normal. The sample statistics are given below. Use $\alpha = 0.01 .$ =16 =13 =3200 =1250

Calculate the test statistic F to test the claim that $\sigma _ { 1 } ^ { 2 } \leq \sigma _ { 2 } ^ { 2 }$ Two samples are randomly selected from populations that are normal. The sample statistics are given below. =16 =15 =12.615 =11.76

A teacher figures that final grades in the statistics 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 $\alpha = 0.01$ Grade A B C D F Number 36 42 60 14 8