Statistics

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HOMEWORK WORKSHEETS section by section Assume all populations normal except for the last chp on non-parametric statistics. 9.2 Z Test for Mean 1. A random sample of 50 medical school applicants at a university has a mean raw score of 31 on the multiple-choice portions of the Medical College Admission Test (MCAT). A student claims that the mean raw score for the school’s applicants is more than 30. Assume the population standard deviation is 2.5. At alpha = .01, test the claim. Assume population normally distributed. 2. A consumer group claims that the mean annual consumption of cheddar cheese by a person in the United States is at most 10.3 pounds. A random sample of 100 people in the United States has a mean annual cheddar cheese consumption of 9.9 pounds. Assume the population standard deviation is 2.1 pounds. At alpha = .05, test the claim. Assume population normally distributed. 3. The lengths of time (in years) it took a random sample of 32 former smokers to quit smoking permanently are listed. Assume the population standard deviation is 6.2 years. At alpha = .05, test the claim that the mean time it takes for smokers to quit smoking permanently is 15 years. Assume population normally distributed. 15.7 13.2 22.6 13.0 10.7 18.1 14.7 7.0 17.3 7.5 21.8 12.3 19.8 13.8 16.0 15.5 13.1 20.7 15.5 9.8 11.9 16.9 7.0 19.3 13.2 14.6 20.9 15.4 13.3 11.6 10.9 21.6 9.3 T Test for Mean 1. A county is considering raising the speed limit on a road because they claim that the mean speed of vehicles is greater than 45 miles per hour. A random sample of 25 vehicles has a mean speed of 48 miles per hour and a standard deviation of 5.4 miles per hour. At alpha = .10, test the claim. Assume population normally distributed. 2. An oceanographer claims that the mean dive depth of a North Atlantic right whale is 115 meters. A random of 34 dive depths has a mean of 121.2 meters and a standard deviation of 24.2 meters. At alpha = .10 test the claim. Assume population normally distributed. 3. You receive a brochure from a large university. The brochure claims that the mean class size for full-time faculty is fewer than 32 students. You randomly select 18 classes taught by full-time faculty and determine the class size of each. The results are shown in the table below. At alpha = .05 test the claim. Assume population normally distributed.

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Class Sizes 35 28 29 33 32 40 26 25 29 28 30 36 33 29 27 30 28 25 9.4 1-Prop Z Test 1. A medical researcher claims that less than 25% of U.S. adults are smokers. In a random sample of 200 U.S. adults, 19.3% say that they are smokers. At alpha = .05, test the claim. 2. A research center claims that at most 75% of U.S. adults think that drivers are safer using hands-free cell phones instead of using hand-held cell phones. In a random sample of 150 U.S. adults, 77% think that drivers are safer using hands-free cell phones instead of hand-held cell phones. At alpha = .01, test the claim. 3. A research center claims that more than 80% of females, ages 20-29 are taller than 62 inches. In a random sample of 150 females ages 20-29, 79% are taller than 62 inches. At alpha = .10 test the claim. 4. A humane society claims that less than 35% of U.S. households own a dog. In a random sample of 400 U.S. households, 156 say they own a dog. At alpha = .10, test the claim. 11.1 and 10.1 2 Sample Z Test and 2 Sample Z Interval 1. To compare the braking distance for two types of tires, a safety engineer conducts 35 braking tests for each type. The mean braking distance for Type A is 42 feet. Assume the population standard deviation is 4.7 feet. The mean braking distance for Type B is 45 feet. Assume the population standard deviation is 4.3 feet. At alpha = .10 test the claim that the mean braking distances are different for the two types of tires. Assume populations are normally distributed. 2. An energy company wants to choose between two regions in a state to install energy- producing wind turbines. A researcher claims that the wind speed in Region A is less than the wind speed in Region B. To test the regions, the average wind speed is calculated for 60 days in each region. The mean wind speed in Region A is 14.0 miles per hour. Assume the population standard deviation is 2.9 miles per hour. The mean wind speed in Region B is 15.1 miles per hour. Assume the population standard deviation is 3.3 miles per hour. At alpha = .05, test the claim. Assume populations are normally distributed. 3. The mean ACT score for 43 male high school students is 21.1. Assume the population standard deviation is 5.0. The mean Act score for 56 female high school students is 20.9. Assume the population standard deviation is 4.7. At alpha = .01 test the claim that male and female high school students have equal ACT scores. Assume populations are normally distributed.

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4. A sociologist claims that children ages 6-17 spent more time watching television in 1981 than children 6-17 do today. A study was conducted in 1981 to find the time that children ages 6-17 spent watching television on weekdays. The results (in hours per weekday) are shown below. Assume the population standard deviation is .6 hour. Assume populations are normally distributed. 2.0 2.5 2.1 2.3 2.1 1.6 2.6 2.1 2.1 2.4 2.1 2.1 1.5 1.7 2.1 2.3 2.5 3.3 2.2 2.9 1.5 1.9 2.4 2.2 1.2 3.0 1.0 2.1 1.9 2.2 Recently, a similar study was conducted. The results are shown below. Assume the population standard deviation is .5 hour. 2.9 1.8 0.9 1.6 2.0 1.7 2.5 1.1 1.6 2.0 1.4 1.7 1.7 1.9 1.6 1.7 1.2 2.0 2.6 1.6 1.5 2.5 1.6 2.1 1.7 1.8 1.1 1.4 1.2 2.3 At alpha = .05, test the claim. 5. Construct a 95% confidence interval for the difference between the mean annual salaries of microbiologists in Maryland and California using the following data: Assume populations are normally distributed. Microbiologists in Maryland:

Microbiologists in California:

2 Sample T Test and 2 Sample T Interval 1. A marine biologist claims that the mean length of mature female pink seaperch is different in fall and winter. A sample of 26 mature female pink seaperch collected in fall has a mean length of 127 millimeters and a standard deviation of 14 millimeters. A sample of 31 mature female pink seaperch collected in winter has a mean length of 117 millimeters and a standard deviation of 9 millimeters. At alpha = .01, test the claim. Assume the population variances are equal. Assume populations are normally distributed.

X1 = $102,650 n1 = 42 σ1 = $8795

X2 = $85,430 n2 = 38 σ 2 = $9250

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2. A personnel director from Pennsylvania claims that the mean household income is greater in Allegheny County than it is in Erie County. In Allegheny County, a sample of 19 residents has a mean household income of $49,700 and a standard deviation of $8800. In Erie County, a sample of 15 residents has a mean household income of $42,000 and a standard deviation of $5100. At alpha = .05 test the claim. Assume the population variances are not equal. Assume populations are normally distributed. 3. The tensile strength of a metal is a measure of its ability to resist tearing when it is pulled lengthwise. A new experimental type of treatment produced steel bars with the tensile strengths (in newtons per square millimeter) listed below. Assume populations are normally distributed. Experimental Method: 391 383 333 378 368 401 339 376 366 348 The old method produced steel bars with the tensile strengths (in newtons per square millimeter) listed below. Old Method: 362 382 368 398 381 391 400 410 396 411 385 385 395 At alpha = .01, test the claim that the new treatment makes a difference in the tensile strength of steel bars. Assume the population variances are equal. Assume populations are normally distributed. 4. To compare the mean times spent waiting for a kidney transplant for two age groups, you randomly select several people in each age group who have had a kidney transplant. The results are shown below. Construct a 95% confidence interval for the difference in mean times spent waiting for a kidney transplant for the two age groups. Assume populations are normally distributed.

Sample Statistics for Kidney Transplants 35-49 50-64

X1 =1805days X2 =1629days s1 =166days s2 = 204days n1 = 21 n2 =11

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11.2 and 10.2 Dependent Samples (Before/After) T Test and T Interval

1. A researcher claims that a post-lunch nap decreases the amount of time it takes males to sprint 20 meters after a night with only 4 hours of sleep. The table shows the amounts of time (in seconds) it took for 10 males to sprint 20 meters after a night with only 4 hours of sleep, when they did not take a post-lunch nap and when they did take a post-lunch nap. At alpha = .01, is there enough evidence to support the researcher’s claim?

Male 1 2 3 4 5 Sprint Time (w/o nap)

4.07 3.94 3.92 3.97 3.92

Sprint Time With nap

3.93 3.87 3.85 3.92 3.90

Male 6 7 8 9 10 Sprint Time (w/o nap)

3.96 4.07 3.93 3.99 4.02

Sprint Time With nap

3.85 3.92 3.80 3.89 3.89

2. A physical therapist claims that one 600-milligram dose of Vitamin C will increase muscular endurance. The table below shows the numbers of repetitions 15 males made on a hand dynamometer (measures grip strength) until the grip strengths in three consecutive trials were 50% of their maximum grip strength. At alpha = .05 test the claim. Assume populations are normally distributed. Participant 1 2 3 4 5 6 7 8 Repetitions (using placebo)

417 279 678 636 170 699 372 582

Repetitions (using Vitamin C)

145 185 387 593 248 245 349 902

Participant 9 10 11 12 13 14 15 Repetitions (using placebo)

363 258 288 526 180 172 278

Repetitions (using Vitamin C)

159 122 264 1052 218 117 185

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3. A company claims that its consumer product ratings (0 -10) have changed from last year to this year. The table below shows the company’s product ratings from the same eight consumers for last year and this year. At alpha = .05, test the claim. Assume populations are normally distributed. Consumer 1 2 3 4 5 6 7 8 Rating (last year)

5 7 2 3 9 10 8 7

Rating (this year)

5 9 4 6 9 9 9 8

4. A sleep disorder specialist wants to test the effectiveness of a new drug that is reported to increase the number of hours of sleep patients get during the night. To do so, the specialist randomly selects 16 patients and records the number of hours of sleep each gets with and without the new drug. The table below shows the results of the two-night study. Construct a 90% confidence interval for . Assume populations are normally distributed. Patient 1 2 3 4 5 6 7 8 Hours of sleep (without the drug)

1.8 2.0 3.4 3.5 3.7 3.8 3.9 3.9

Hours of sleep Using the drug)

3.0 3.6 4.0 4.4 4.5 5.2 5.5 5.7

Patient 9 10 11 12 13 14 15 16 Hours of sleep (without the drug)

4.0 4.9 5.1 5.2 5.0 4.5 4.2 4.7

Hours of sleep Using the drug)

6.2 6.3 6.6 7.8 7.2 6.5 5.6 5.9

µd

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11.3 and 10.3 2 Prop Z Test and 2 Prop Z Interval 1) In a 4-week study about the effectiveness of using magnetic insoles to treat plantar heel pain, 54 subjects wore magnetic insoles and 41 subjects wore nonmagnetic insoles. The results are shown below. At alpha = .01, test the claim that there is a difference in the proportion of subjects who feel all or mostly better between the two groups.

Do you feel all or mostly better? Magnetic Insoles Nonmagnetic

Insoles Yes 17 18 No 37 23 2) In a survey of 200 males ages 18 to 24, 39% were enrolled in college. In a survey of 220 females ages 18 to 24, 45% were enrolled in college. At alpha = .05, test the claim that the proportion of males ages 18 to 24 who enrolled in college is less than the proportion of females ages 18 to 24 enrolled in college. 3) In a survey of 480 drivers from the South, 408 wear a seatbelt. In a survey of 360 drivers from the Northeast, 288 wear a seat belt. At alpha = .05, test the claim that the proportion of drivers who wear seat belts is greater in the South than in the Northeast. 4) In a survey of 10,000 students taking the SAT, 6% were planning to study education in college. In another study of 8000 students taken 10 years before, 9% were planning to study education in college. Construct a 95% confidence interval for p1 – p2 where p1 is the proportion from the recent study and p2 is the proportion from the survey taken 10 years ago.

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11.4 2 Sample F Test 1. The table below shows a sample of the waiting times (in days) for a heart transplant for two age groups. At alpha = .05, test the claim that the variances of the waiting times differ between the two age groups.

18-34 35-49 158 170 212 209 213 173 162 194 196 200 169 210 2.A state school administrator claims that the standard deviations of science assessment test scores for eighth-grade students are the same in Districts 1and 2. A sample of 12 test scores from District 1 has a standard deviation of 36.8 points, and a sample of 14 test scores from District 2 has a standard deviation of 32.5 points. At alpha = .10, test the administrator’s claim. 3. An employment information service claims that the standard deviation of the annual salaries for actuaries is greater in New York than in California. You select a sample of actuaries from each state. The results of each survey are shown below. At alpha = .05, test the claim. Actuaries in New York Actuaries in California S1 = $39, 700 S2= $29,000 N1 = 41 N2=61 4.1, 4.2 13.1 Linear Correlation and Regression

1. Two variables have a positive linear correlation. Does the dependent variable increase or decrease as the independent variable increases?

2. Two variables have a negative linear correlation. Does the dependent variable increase or decrease as the independent variable increases?

3. Describe the range of values for the correlation coefficient. 4. What the sample correlation coefficient r measure? Which value indicates a

stronger correlation r = .918 or r = -.932? Explain your reasoning. 5. Discuss the difference between r and . 6. In your own words, what does it mean to say “correlation does not imply causation” ?

ρ

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In the following Exercises (a) display the data in a scatter plot, (b) calculate the sample correlation r, and (c) describe the type of correlation and interpret the correlation in the context of the data. 15. The ages (in years) of 10 men and their systolic blood pressures (in millimeters of mercury): Age, x 16 25 39 45 49 64 70 29 57 22 Systolic blood pressure, y

109 122 143 132 199 185 199 130 175 118

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16. The maximum weights (in kilograms) for which one repetition of a half squat can be performed and the times (in seconds) to run a 10-meter sprint for 12 international soccer players:

Max weight, x

175 180 155 210 150 190 185 160 190 180 160 170

Time, y 1.80 1.77 2.05 1.42 2.04 1.61 1.70 1.91 1.60 1.63 1.98 1.90 17. The earnings per share (in dollars) and the dividends per share (in dollars) for 6 medical supply companies in a recent year: Earnings per share, x

2.79 5.10 4.53 3.06 3.70 2.20

Dividends per share, y

.52 2.40 1.46 .88 1.04 .22

18. The weights (in pounds) of eight vehicles and the variability of their braking distances (in feet) when stopping on a dry surface are shown below in the table. At alpha = .01, test the claim that there is a significant linear correlation between vehicle weight and variability in braking distance on a dry surface.

Weight, x 5940 5340 6500 5100 5850 4800 5600 5890 Variability, y

1.78 1.93 1.91 1.59 1.66 1.50 1.61 1.70

Regression In the exercises below, find the equation of the regression line for the data. Then construct a scatter plot of the data and draw the regression line. {Each pair of data has a significant correlation.} Then use the regression equation to predict the value of y for each of the x-values, if meaningful. If the x-value is not meaningful to predict the value of y, explain why not. 1. The height (in feet) and the numbers of stories of nine notable buildings in Atlanta. Height, x

869 820 771 696 692 676 656 492 486

Stories, y

60 50 50 52 40 47 41 39 26

(a) x = 800 feet (b) x = 750 feet (c) x = 400 feet (d) x = 625 feet

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2. The number of hours 9 students spent studying for a test and their scores on that test. Hours spent studying, x

0 2 4 5 5 5 6 7 8

Test scores, y

40 51 64 69 73 75 93 90 95

(a) x = 3 hours (b) x = 6.5 hours (c) x = 13 hours (d) x = 4.5 hours 3. The heart rates (in beats per minute) and QT intervals (in milliseconds) for 13 males (the figure below shows the QT interval of a heartbeat in an electrocardiogram).

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Heart rate, x

60 75 62 68 84 97 66 65 86 78 93 75 88

QT interval, y

403 363 381 367 341 317 401 384 342 377 329 377 349

(a) x = 120 beats per min (b) x = 67 beats per min (c) x = 90 beats per min (d) x = 83 beats per min Coefficient of Determination In the exercises below, use the value of the correlation coefficient to calculate the coefficient of determination . What does this tell you about the explained variation of the data about the regression line? About the unexplained variation? 1. r = .465 2. r = -.957 13.3 Multiple Regression In the exercises below, use the multiple regression equation to predict the y-values for the values of the independent variable.

1. The equation used to predict the annual cauliflower yield (in pounds per acre) is

, where x1 is the number of acres planted and x2 is the number of acres harvested.

r2

y ^ = 24,791+ 4.508x1 − 4.723x2

(a)x1 = 36,500 x2 = 36,100 (b)x1 = 38,100 x2 = 37,800 (c)x1 = 39,000 x2 = 38,800 (d)x1 = 42,200 x2 = 42,100

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2.The volume (in cubic feet) of a black cherry tree can be modeled by the equation

, where x1 is the tree’s height (in feet) and x2 is the tree’s diameter (in inches).

12.1 Goodness of Fit 1. A researcher claims that the ages of people who go to movies at least once a month are distributed as shown in the figure below. You randomly select 1000 people who go to the movies at least once a month and record the age of each. The table shows the results. At alpha = .10, test the researcher’s claim.

Survey results

Age Frequency, f 2-17 240 18-24 214 25-39 183 40-49 156 50+ 207

y ^ = −52.2+ .3x1 + 4.5x2

(a)x1 = 70 x2 = 8.6 (b)x1 = 65 x2 =11.0 (c)x1 = 83 x2 =17.6 (d)x1 = 87 x2 =19.6

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2. A research firm claims that the distribution of the days of the week that people are most likely to order food for delivery is different from the distribution shown in the figure below. You randomly select 500 people and record which day of the week each is most likely to order food for delivery. The table below shows the results. At alpha = .01, test the research firm’s claim.

Survey results Day Frequency, f Sunday 43 Monday 16 Tuesday 25 Wednesday 49 Thursday 46 Friday 168 Saturday 153

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12.2 Tests of Independence 1. The contingency table below shows the results of a random sample of students by the location of school and the number of those students achieving basic skill levels in three subjects. At alpha = .01, test the hypothesis that the variables are independent. Subject Location of school Reading Math Science Urban 43 42 38 Suburban 63 66 65 2. The contingency table below shows the results of a random sample of former smokers by the number of times they tried to quit smoking before they were habit- free and gender. At alpha = .05, test the claim that the number of times they tried to quit before they were habit-free is related to gender. Number of times tried to quit before habit-free Gender 1 2-3 4 or more Male 271 257 149 Female 146 139 80 14.1 ANOVA 1. The table below shows the costs per ounce (in dollars) for a sample of toothpastes exhibiting very good stain removal, good stain removal, and fair stain removal. At alpha = .05, test the claim that at least one mean cost per ounce is difference from the others.

Very good .47 .49 .41 .37 .48 .51 Good .60 .64 .58 .75 .46 Fair .34 .46 .44 .60

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2. The well-being index is a way to measure how people are faring physically, emotionally, socially, and professionally, as well as to rate the overall quality of their lives and their outlooks for the future. The table below shows the well-being index scores for a sample of states from four regions of the United States. At alpha = .10, test the claim the mean scores is the same for all regions.

Northeast Midwest South West 67.6 66.6 64.2 66.1 67.3 67.6 64.1 67.4 68.4 65.6 65.8 69.7 66.2 68.9 66.1 68.5 66.5 65.5 62.7 65.2 68.6 68.5 68.0 66.7 67.4 63.6 67.1 68.0 65.2 68.8 65.2 67.7 64.0 66.6 15.1 Non-Parametric Statistics and the Sign Test

1. What is a nonparametric test? How does a nonparametric test differ from a parametric test? What are the advantages and disadvantages of using a nonparametric test?

2. When the sign test is used, what population parameter is being tested? 3. Describe the test statistic for the sign test when the sample size n is less than

or equal to 25 and when n is greater than 25. 4. In your own words, explain why the hypothesis test discussed in this section

is called the sign test. 5. Explain how to use the sign test to test a population median. 6. List the two conditions that must be met in order to use the paired-sample

sign test. 7. A financial service accountant claims that the median amount of new credit

card charges for the previous month was more than $300. You randomly select 12 credit card accounts and record the amount of new charges for each account for the previous month. The amounts (in dollars) are listed below. At alpha = .01, test the accountant’s claim.

346.71 382.59 255.03 202.17 309.80 265.88 299.41 270.38 296.54 318.46 245.92 309.47

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8. A real estate agent claims that the median sales price of new privately owned one-family homes sold in a recent month is $193,000 or less. The sales prices (in dollars) of 10 randomly selected homes are listed below. At alpha = .05, test the agent’s claim.

200,800 229,500 205,900 190,700 140,200 193,900 249,000 170,900 184,500 207,500 9. A physician claims that lower back pain intensity scores will decrease after receiving acupuncture treatment. The table below shows the lower back pain intensity scores for eight patients before and after receiving acupuncture for eight weeks. At alpha= .05, test the physician’s claim. Patient 1 2 3 4 5 6 7 8 Intensity (before)

59.2 46.3 65.4 74.0 79.3 81.6 44.4 59.1

Intensity (after)

12.4 22.5 18.6 59.3 70.1 70.2 13.2 25.9

10. A tutoring agency claims that by completing a special course, students will improve their critical reading SAT scores. In part of a study, 12 students take the critical reading part of the SAT, complete the special course, then take the critical reading part of the SAT again. The students’ scores are shown below. At alpha = .05, test the agency’s claim. Student 1 2 3 4 5 6 7 8 9 10 11 12 1st score

300 450 350 430 300 470 530 200 200 350 360 250

2nd score

300 520 400 410 300 480 700 250 390 350 480 300

15.2 and 15.3 Wilcoxon Signed Rank and Wilcoxon Rank Sum

1. How do you know whether to use a Wilcoxon signed-rank test or a Wilcoxon rank sum test.

2. In a study testing the effects of calcium supplements on blood pressure in

men, 12 men were randomly chosen and given supplements for 12 weeks. The table below shows the measurements for each subject’s diastolic blood pressure taken before and after the 12-week treatment period. At alpha = .01, test the claim that there was no reduction in diastolic blood pressure.

Patient 1 2 3 4 5 6 7 8 9 10 11 12 Before 108 109 120 129 112 111 117 135 124 118 130 115 After 99 115 105 116 115 117 108 122 120 126 128 106

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3. A college administrator claims that there is a difference in the earnings of people with bachelor’s degrees and those with advanced degrees. The table below shows the earnings (in thousands of dollars) of a random sample of 11 people with bachelor’s degrees and 10 people with advanced degrees. At alpha = .05, test the claim. Bachelor’s 56 52 65 78 72 52 46 58 62 54 56 Advanced 84 87 95 81 86 86 93 93 90 82

4. A teacher’s union representative claims that there is a difference in the salaries earned by teachers in Wisconsin and Michigan. The table below shows the salaries (in thousands of dollars) of a random sample of 11 teachers from Wisconsin and 12 teachers from Michigan. At alpha = .05, test the claim.

Wisconsin 55 59 49 56 51 61 55 61 53 47 52 Michigan 64 68 58 65 60 70 64 70 62 56 61 79 Kruskal-Wallis Test

1. What are the conditions for using a Kruskal - Wallis test?

2. The table below shows the annual premiums for a random sample of home insurance policies in Connecticut, Massachusetts, and Virginia. At alpha = .05 test the claim that the distribution of the annual premiums in at least one state is different from the others.

State Annual premium (in dollars) CONN 1053 848 1013 1163 1288 929 1070 MA 1132 1052 1007 1322 1137 916 784 VA 885 800 616 695 982 688 605 3. The table below shows the annual salaries for a random sample of private industry workers in Kentucky, North Carolina, South Carolina, and West Virginia. At alpha = .10, test the claim that the distribution of the annual salaries of private industry workers in at least one state is different from the others. State Hourly pay rate (in dollars) KT 35.3 37.0 45.9 57.5 33.7 28.3 35.3 NC 43.5 41.9 36.6 54.3 35.5 39.6 43.5 SC 29.8 37.4 43.5 42.9 34.7 36.1 29.8 WV 31.6 42.7 33.4 41.9 47.1 34.9 31.6