Solve each of the problems to the best of your ability. Show all your work: If work done on paper, show all calculations If work done on programmable calculator (TI-83/84/89), show copy of program with calculations If work done using Excel, SPSS (or PSPP), Minitab, or StatCrunch: ensure copy of file is submitted with quiz For sample sizes under 30, assume normal comparison distributions unless otherwise specifically stated in problem. The final four problems present a hypothesis testing scenario or a correlation/regression scenario. You get to decide which hypothesis testing procedure you want to use to determine whether the result of the study is statistically significant! Be sure to answer all the questions for each problem. Access to z-tables, t-tables, chi-square tables, sample correlation coefficient r tables, and F-distribution tables is found in Content > Course Resources > Statistical Resources in our LEO classroom. 1) Given the following box plot: 0 a. Identify: 4 Q 1= 6 10 14 Q3= M = b. Which quarter has the smallest spread of data? What is that spread? c. Calculate IQR: __________________. d. Is/Are there any potential outlier(s)? If so, identify it/them. In either case, support your answer mathematically (use formulas) 1 e. Are there more data in the interval 4 - 6 or in the interval 6 - 8? Support your answer. 2) The following full IQ scores were obtained by testing a group of children who live near a lead smelter and are exposed to high concentrations of lead in the environment: 82 88 93 104 a. 85 88 75 83 85 104 80 96 101 76 89 80 Complete the following frequency table: IQ Interval 67.5 - 72.5 Frequency Relative Frequency 72.5 - 77.5 77.5 - 82.5 82.5 - 87.5 87.5 - 92.5 92.5 - 97.5 97.5 - 102.5 102.5 - 107.5 107.5 - 112.5 b. What percentage of observed IQ scores were lower than 92.5? c. In which IQ interval does the median lie? Explain why. 2 80 79 94 75 3) You're invited to play a game! You pay $10 to then pick one envelope out of a drum full of 100 identical envelopes. Each of the envelopes contains a coupon for a free gift: Four of the coupons are for a free gift worth $40 Six of the coupons are for a free gift worth $12 Eighty (80) of the coupons are for a free gift worth $8 Ten of the coupons are for a free gift worth $6. a. What is the probability that a single randomly-selected envelope contains a coupon for a free gift worth $12 or more? b. What is the expected value of this game? Is it financially worth playing? c. Find the standard deviation of x (coupon value) 4) Channel 94 TV in Metropolis recently conducted a \"straw poll\" of 1500 randomly-selected Metropolis voters to determine their current mayoral candidate preferences. Voters were also asked their age in years (18 to 39 or 40+). Following table was obtained. Mayoral Candidate Preferred Democrat Independent Libertarian Republican Total Voter Age 18 - 39 227 113 71 115 526 Voter Age 40 + 273 176 134 391 974 Total 500 289 205 506 1500 (Report your answers as fractions or as decimal values rounded to the nearest hundredth.) Find the probability that a randomly-selected Metropolis voter preferred: (a) the Democratic candidate and is age 18 - 39. Answer: ______________ (b) the Democratic candidate or is age 18 - 39. Answer: ______________ (c) the Democratic candidate given that viewer is age 18 - 39. Answer: ______________ (d) Are the events \"Libertarian\" and \"voter age 40+\" independent? Explain mathematically why or why not. 3 5) Suppose adults in a particular town drive a mean of 65 mph with a standard deviation of 15 mph. Using the standard normal distribution: a What percentage of adults drive below 35 mph? b What percentage of adults drive between 56 and 80 mph? c At what speed does someone need to drive to be above the 90th percentile (in the top 10%)? 6) A psychology professor of a large class became curious as to whether the students who turned in tests first scored differently from the population mean score on the test. The overall test population standard deviation was 10; the scores were approximately normally distributed. The mean score for the first 20 students to turn in tests was 78. a. Should you use normal or Student-t distribution tables for this scenario? Be sure to include the reason for your choice. b. Use a 90% confidence level to calculate error bound EBM = _________________ c. State the 90% confidence interval for the true population mean score of the test. CI = ____________________________ 4 7) A health psychologist knew that corporate executives in general have an average score of 80 with a standard deviation of 12 on a stress inventory and that the scores are normally distributed. In order to learn whether corporate executives who exercise regularly have lower stress scores, the psychologist measured the stress of 20 exercising executives and found them to have a mean score of 72. Is this difference significant at p < 0.01? a. b. c. d. e. f. State the alternate/research hypothesis and the null hypothesis. Identify the characteristics of the comparison distribution. Identify the type of hypothesis testing procedure you will use, and why you choose it. Determine either or the cutoff score(s)/statistic. Determine either the p-value or the sample score/statistic. Determine whether to reject/fail to reject the null hypothesis, AND state the result of the test (either HA supported, or study is inconclusive). g. Determine the confidence interval for the 99% confidence level. 5 8) A company planning to open a coffee house for college students wants to know if their customers will drink more coffee if the coffee house is decorated in a Paris motif or in a San Francisco motif. The marketing analysis team sets up two similar rooms with the two motifs. Eight students spend an afternoon in each room, drinking all the coffee they like. The order in which they sit in the rooms is rotated so that half spend their first afternoon in the Paris room and half in the San Francisco room. The amount of coffee each participant drinks in each room is shown below. Using p < 0.05, is there a significant difference between the numbers of cups of coffee consumed in the two rooms? Cups of Coffee Participant Paris A 8.5 B 4.3 C 2.0 D 7.8 E 7.0 F 9.1 G 3.3 H 3.5 San Francisco 8.4 4.6 1.7 7.3 7.2 7.4 3.0 3.5 a. b. c. d. e. f. State the alternate/research hypothesis and the null hypothesis. Identify the characteristics of the comparison distribution. Identify the type of hypothesis testing procedure you will use, and why you choose it. Determine either or the cutoff score(s)/statistic. Determine either the p-value or the sample score/statistic. Determine whether to reject/fail to reject the null hypothesis, AND state the result of the test (either research hypothesis supported, or study is inconclusive). g. Depending on the hypothesis test you use, either Calculate the effect size of your study, OR determine the confidence interval for a 95% confidence level. 6 9) The expected percentage of number of pets students in the USA have in their homes, and the results of a random sample of 1000 students in the Eastern USA on the number of pets in their home provided the following data: Number of Pets 0 1 2 3 4 Expected Percentage (%) 18 25 30 18 9 Observed Frequency 210 240 320 140 90 At a 0.01 significance level, does it appear that the distribution \"number of pets\" in the Eastern USA is different from the distribution of the USA student population as a whole? a. b. c. d. e. f. State the alternate/research hypothesis and the null hypothesis. Identify the characteristics of the comparison distribution. Identify the type of hypothesis testing procedure you will use, and why you choose it. Determine either or the cutoff score(s)/statistic. Determine either the p-value or the sample score/statistic. Determine whether to reject/fail to reject the null hypothesis, AND state the result of the test (either research hypothesis supported, or study is inconclusive). 7 10) SCUBA divers have maximum dive times they cannot exceed when going to different depths. The following table shows different depths with the maximum dive times in minutes: Depth (feet) Max Dive Time (minutes) 60 55 70 45 80 35 90 25 100 22 a. Calculate the least squares regression line (line of best fit) b. Calculate correlation coefficient r and identify type of correlation (strong positive, weak negative, etc.) between independent variable depth (x) and dependent variable maximum dive time (y) c. Predict the maximum dive time for 110 feet. 8 Surname 1 Name Professor Course Date Analysis in excel The final four problems present a hypothesis testing scenario or a correlation/regression scenario. You get to decide which hypothesis testing procedure you want to use to determine whether the result of the study is statistically significant! Be sure to answer all the questions for each problem. Access to z-tables, t-tables, chi-square tables, sample correlation coefficient r tables, and F-distribution tables is found in Content > Course Resources > Statistical Resources in our LEO classroom. 1) Given the following box plot: 0 a. Identify: 4 Q1= 6 10 14 Q3= M = 6 b. Which quarter has the smallest spread of data? Q1 What is that spread? 6-4=2 c. Calculate IQR: _____10-4=6_____________. d. Is/Are there any potential outlier(s)? If so, identify it/them. In either case, support your answer mathematically (use formulas) 1.5IQR=1.5*6=9 There are no potential outliers 1 Surname 2 e. Are there more data in the interval 4 - 6 or in the interval 6 - 8? Support your answer. There are relatively data. This is because from 4 to 6 we have one data point in between the two data points. 2) The following full IQ scores were obtained by testing a group of children who live near a lead smelter and are exposed to high concentrations of lead in the environment: 82 88 93 104 a. 85 88 75 83 85 104 80 96 101 76 89 80 Complete the following frequency table: IQ Interval 67.5 - 72.5 Frequency 0 Relative Frequency 0/20=0.00 72.5 - 77.5 3 3/20=0.15 77.5 - 82.5 5 5/20=0.25 82.5 - 87.5 3 3/20=0.15 87.5 - 92.5 3 3/20=0.15 92.5 - 97.5 3 3/20=0.15 97.5 - 102.5 1 1/20=0.05 102.5 - 107.5 2 2/20=0.10 107.5 - 112.5 0 0/20=0.00 b. What percentage of observed IQ scores were lower than 92.5? 0.15+0.15+0.25+0.15=0.70=70% c. In which IQ interval does the median lie? Explain why. 87.5 - 92.5 This is because it is the middle interval 2 80 79 94 75 Surname 3 3 Surname 4 3) You're invited to play a game! You pay $10 to then pick one envelope out of a drum full of 100 identical envelopes. Each of the envelopes contains a coupon for a free gift: Four of the coupons are for a free gift worth $40 Six of the coupons are for a free gift worth $12 Eighty (80) of the coupons are for a free gift worth $8 Ten of the coupons are for a free gift worth $6. a. What is the probability that a single randomly-selected envelope contains a coupon for a free gift worth $12 or more? P=6/(4+6+80+10)=6/100=0.06 b. What is the expected value of this game? Is it financially worth playing? Expected value=932/100=$93.2 Yes it is c. Find the standard deviation of x (coupon value) S=93.2*0.06=$5.592 4) Channel 94 TV in Metropolis recently conducted a \"straw poll\" of 1500 randomly-selected Metropolis voters to determine their current mayoral candidate preferences. Voters were also asked their age in years (18 to 39 or 40+). Following table was obtained. Mayoral Candidate Preferred Democrat Independent Libertarian Republican Total Voter Age 18 - 39 227 113 71 115 526 Voter Age 40 + 273 176 134 391 974 Total 500 289 205 506 1500 (Report your answers as fractions or as decimal values rounded to the nearest hundredth.) Find the probability that a randomly-selected Metropolis voter preferred: (a) the Democratic candidate and is age 18 - 39. Answer: ______227/500=0.454________ (b) the Democratic candidate or is age 18 - 39. Answer: _____227/526=0.432_________ (c) the Democratic candidate given that viewer is age 18 - 39.Answer: ____0/500=0_________ (d) Are the events \"Libertarian\" and \"voter age 40+\" independent? Explain mathematically why or why not. 4 Surname 5 134/205=0.654 134/9740.138 They are not independent 5) Suppose adults in a particular town drive a mean of 65 mph with a standard deviation of 15 mph. Using the standard normal distribution: a What percentage of adults drive below 35 mph? Z=(35-65)/15=-2 Percentage=0.0228*100=2.28% b What percentage of adults drive between 56 and 80 mph? Z=(80-65)/15=1 P=0.8413 Z=(56-65)/15=-0.6 P=0.2743 percentage=.8413-.2743=0.567*100=56.7% c At what speed does someone need to drive to be above the 90th percentile (in the top 10%)? P=10/100=0.90 1.64=(x-65)/15 X=65+24.6 X=89.6mph 6) A psychology professor of a large class became curious as to whether the students who turned in tests first scored differently from the population mean score on the test. The overall test population standard deviation was 10; the scores were approximately normally distributed. The mean score for the first 20 students to turn in tests was 78. a. Should you use normal or Student-t distribution tables for this scenario? Be sure to include the reason for your choice. We will use a normal distribution. This is because n<30 and the population variance is known. b. Use a 90% confidence level to calculate error bound 5 Surname 6 Confidence Interval Estimate for the Mean Data Population Standard Deviation 10 Sample Mean 78 Sample Size 20 Confidence Level 90% Intermediate Calculations Standard Error of the Mean 2.2361 Z Value 1.6449 Interval Half Width 3.6780 Confidence Interval Interval Lower Limit 74.32 Interval Upper Limit 81.68 EBM = _2.2361_______________ c. State the 90% confidence interval for the true population mean score of the test. CI = (74.32, 81.68) 6 Surname 7 7) A health psychologist knew that corporate executives in general have an average score of 80 with a standard deviation of 12 on a stress inventory and that the scores are normally distributed. In order to learn whether corporate executives who exercise regularly have lower stress scores, the psychologist measured the stress of 20 exercising executives and found them to have a mean score of 72. Is this difference significant at p < 0.01? Z Test of Hypothesis for the Mean Data Null Hypothesis = Level of Significance Population Standard Deviation Sample Size Sample Mean Intermediate Calculations Standard Error of the Mean Z Test Statistic Two-Tail Test Lower Critical Value Upper Critical Value p-Value Reject the null hypothesis 80 0.01 12 20 72 2.6833 2.9814 2.5758 2.5758 0.0029 a. State the alternate/research hypothesis and the null hypothesis. H 0 : 1= 2 H A : 1 2 b. Identify the characteristics of the comparison distribution. The distribution follows a normal distribution c. Identify the type of hypothesis testing procedure you will use, and why you choose it. We will use a normal distribution. This is because n<30 and the population variance is known. d. determine either or cutoff score(s) statistic.alpha =0.01 e. p-value sample score f. whether to reject fail nullhypothesis, state result of test (either hasupported, study inconclusive). hypothesis conclude that alternative has been supported. g. confidence interval for 99% level. 7 surname 8 ci =70 2.5812> Course Resources > Statistical Resources in our LEO classroom. 1) Given the following box plot: 0 a. Identify: 4 Q1= 6 10 14 Q3= M = 6 b. Which quarter has the smallest spread of data? Q1 What is that spread? 6-4=2 c. Calculate IQR: _____10-4=6_____________. d. Is/Are there any potential outlier(s)? If so, identify it/them. In either case, support your answer mathematically (use formulas) 1.5IQR=1.5*6=9 There are no potential outliers 1 Surname 2 e. Are there more data in the interval 4 - 6 or in the interval 6 - 8? Support your answer. There are relatively data. This is because from 4 to 6 we have one data point in between the two data points. 2) The following full IQ scores were obtained by testing a group of children who live near a lead smelter and are exposed to high concentrations of lead in the environment: 82 88 93 104 a. 85 88 75 83 85 104 80 96 101 76 89 80 Complete the following frequency table: IQ Interval 67.5 - 72.5 Frequency 0 Relative Frequency 0/20=0.00 72.5 - 77.5 3 3/20=0.15 77.5 - 82.5 5 5/20=0.25 82.5 - 87.5 3 3/20=0.15 87.5 - 92.5 3 3/20=0.15 92.5 - 97.5 3 3/20=0.15 97.5 - 102.5 1 1/20=0.05 102.5 - 107.5 2 2/20=0.10 107.5 - 112.5 0 0/20=0.00 b. What percentage of observed IQ scores were lower than 92.5? 0.15+0.15+0.25+0.15=0.70=70% c. In which IQ interval does the median lie? Explain why. 87.5 - 92.5 This is because it is the middle interval 2 80 79 94 75 Surname 3 3 Surname 4 3) You're invited to play a game! You pay $10 to then pick one envelope out of a drum full of 100 identical envelopes. Each of the envelopes contains a coupon for a free gift: Four of the coupons are for a free gift worth $40 Six of the coupons are for a free gift worth $12 Eighty (80) of the coupons are for a free gift worth $8 Ten of the coupons are for a free gift worth $6. a. What is the probability that a single randomly-selected envelope contains a coupon for a free gift worth $12 or more? P=6/(4+6+80+10)=6/100=0.06 b. What is the expected value of this game? Is it financially worth playing? Expected value=932/100=$93.2 Yes it is c. Find the standard deviation of x (coupon value) S=93.2*0.06=$5.592 4) Channel 94 TV in Metropolis recently conducted a \"straw poll\" of 1500 randomly-selected Metropolis voters to determine their current mayoral candidate preferences. Voters were also asked their age in years (18 to 39 or 40+). Following table was obtained. Mayoral Candidate Preferred Democrat Independent Libertarian Republican Total Voter Age 18 - 39 227 113 71 115 526 Voter Age 40 + 273 176 134 391 974 Total 500 289 205 506 1500 (Report your answers as fractions or as decimal values rounded to the nearest hundredth.) Find the probability that a randomly-selected Metropolis voter preferred: (a) the Democratic candidate and is age 18 - 39. Answer: ______227/500=0.454________ (b) the Democratic candidate or is age 18 - 39. Answer: _____227/526=0.432_________ (c) the Democratic candidate given that viewer is age 18 - 39.Answer: ____0/500=0_________ (d) Are the events \"Libertarian\" and \"voter age 40+\" independent? Explain mathematically why or why not. 4 Surname 5 134/205=0.654 134/9740.138 They are not independent 5) Suppose adults in a particular town drive a mean of 65 mph with a standard deviation of 15 mph. Using the standard normal distribution: a What percentage of adults drive below 35 mph? Z=(35-65)/15=-2 Percentage=0.0228*100=2.28% b What percentage of adults drive between 56 and 80 mph? Z=(80-65)/15=1 P=0.8413 Z=(56-65)/15=-0.6 P=0.2743 percentage=.8413-.2743=0.567*100=56.7% c At what speed does someone need to drive to be above the 90th percentile (in the top 10%)? P=10/100=0.90 1.64=(x-65)/15 X=65+24.6 X=89.6mph 6) A psychology professor of a large class became curious as to whether the students who turned in tests first scored differently from the population mean score on the test. The overall test population standard deviation was 10; the scores were approximately normally distributed. The mean score for the first 20 students to turn in tests was 78. a. Should you use normal or Student-t distribution tables for this scenario? Be sure to include the reason for your choice. We will use a normal distribution. This is because n<30 and the population variance is known. b. Use a 90% confidence level to calculate error bound 5 Surname 6 Confidence Interval Estimate for the Mean Data Population Standard Deviation 10 Sample Mean 78 Sample Size 20 Confidence Level 90% Intermediate Calculations Standard Error of the Mean 2.2361 Z Value 1.6449 Interval Half Width 3.6780 Confidence Interval Interval Lower Limit 74.32 Interval Upper Limit 81.68 EBM = _2.2361_______________ c. State the 90% confidence interval for the true population mean score of the test. CI = (74.32, 81.68) 6 Surname 7 7) A health psychologist knew that corporate executives in general have an average score of 80 with a standard deviation of 12 on a stress inventory and that the scores are normally distributed. In order to learn whether corporate executives who exercise regularly have lower stress scores, the psychologist measured the stress of 20 exercising executives and found them to have a mean score of 72. Is this difference significant at p < 0.01? Z Test of Hypothesis for the Mean Data Null Hypothesis = Level of Significance Population Standard Deviation Sample Size Sample Mean Intermediate Calculations Standard Error of the Mean Z Test Statistic Two-Tail Test Lower Critical Value Upper Critical Value p-Value Reject the null hypothesis 80 0.01 12 20 72 2.6833 2.9814 2.5758 2.5758 0.0029 a. State the alternate/research hypothesis and the null hypothesis. H 0 : 1= 2 H A : 1 2 b. Identify the characteristics of the comparison distribution. The distribution follows a normal distribution c. Identify the type of hypothesis testing procedure you will use, and why you choose it. We will use a normal distribution. This is because n<30 and the population variance is known. d. determine either or cutoff score(s) statistic.alpha =0.01 e. p-value sample score f. whether to reject fail nullhypothesis, state result of test (either hasupported, study inconclusive). hypothesis conclude that alternative has been supported. g. confidence interval for 99% level. 7 surname 8 ci =70 2.5812>