# Practice question Of Statistical Inference

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### Practice question Of Statistical Inference

1. 1. Practice QuestionQ1. Two Population Mean z-testDo organic farming methods cause a change in produce size? A random sample of n1 = 89 organicallygrown tomatoes had sample mean weight 1x = 3.8 ounces. Another random sample of n2 = 75 tomatoesthat were not grown organically had sample mean weight 2x = 4.1 ounces. Previous studies show that1 0.9  ounce and 2 1.5  ounces. Does this indicate a difference between population mean weights oforganically grown tomatoes compared with those not grown organically? Use a 5% level of significance.Q2. Simple Linear RegressionThe editor of a major academic book publisher claims that a large part of the cost of books is the cost ofpaper. This implies that larger books will cost more money. As an experiment to analyze the claim, auniversity student visits the bookstore and records the number of pages and the selling price of twelverandomly selected books. These data are listed below.Σx = 8674 Σy = 592 Σx2= 6937938 Σy2= 30428 Σxy = 455595Book 1 2 3 4 5 6 7 8 9 10 11 12Number of Pages (x) 844 727 360 915 295 706 410 905 1058 865 677 912Selling Price (\$) (y) 55 50 35 60 30 50 40 53 65 54 42 58Formulate a simple linear regression equation.Q3. Inferences of Simple Linear RegressionThe editor of a major academic book publisher claims that a large part of the cost of books is the cost ofpaper. This implies that larger books will cost more money. As an experiment to analyze the claim, auniversity student visits the bookstore and records the number of pages and the selling price of twelverandomly selected books. These data are listed below.Σx = 8674 Σy = 592 Σx2= 6937938 Σy2= 30428 Σxy = 455595Book 1 2 3 4 5 6 7 8 9 10 11 12Number of Pages (x) 844 727 360 915 295 706 410 905 1058 865 677 912Selling Price (\$) (y) 55 50 35 60 30 50 40 53 65 54 42 58At 5% significance level can it be said that Price of the book can be estimated by Number of pages.Q4. Inferences of Simple Linear CorrelationThe editor of a major academic book publisher claims that a large part of the cost of books is the cost ofpaper. This implies that larger books will cost more money. As an experiment to analyze the claim, auniversity student visits the bookstore and records the number of pages and the selling price of twelverandomly selected books. These data are listed below.Σx = 8674 Σy = 592 Σx2= 6937938 Σy2= 30428 Σxy = 455595Book 1 2 3 4 5 6 7 8 9 10 11 12Number of Pages (x) 844 727 360 915 295 706 410 905 1058 865 677 912Selling Price (\$) (y) 55 50 35 60 30 50 40 53 65 54 42 58Using α = 0.01 infer that Price and Number of pages of the book are positively linearly correlated.
2. 2. Q5. Chi square Goodness of FitA recent study shows the proportions people answered to the question “Where does most of my stresscome from at work?”Demand Co-workers Boss Layoff Other54% 20% 10% 8% 8%A study of 800 random employees was conducted asking the same question. The following were theresults:Demand Co-workers Boss Layoff Other404 183 94 80 39Is there significant evidence to suggest that this distribution is different than the population study? Testat the 1% significance level.Q6. ANOVAOne important factor in selecting software for word processing and database management systems is thetime required to learn how to use a particular system. In order to evaluate three file managementsystems, a firm devised a test to see how many training hours were needed for five of its wordprocessing operators to become proficient in each of three systems. Σx2= 4682TotalSystem A 16 19 14 13 18 80System B 16 17 13 12 17 75System C 24 22 19 18 22 105Using a 5% level, is there any difference between the training time needed for the three systems?Q7. Paired t-testThe manager of a consulting firm in Lansing, Michigan, is trying to assess the effectiveness of computerskills training given to all new entry-level professionals. In an effort to make such an assessment, headministers a computer skills test immediately before and after the training program to each of 20randomly chosen employees. The pre-training and post-training scores of these 20 individuals areshown in the table below.Employee 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Score before 62 63 74 64 84 81 54 61 81 86 75 71 86 74 65 90 72 71 85 66Score after 77 77 83 88 80 80 83 88 80 88 93 78 82 84 86 89 81 90 86 92Using a 10% level of significance, do the given sample data support that the firm‟s training programs iseffective in increasing the new employee‟s working knowledge of computing?Q8. Pooled t-testDo children diagnosed with ADHD (attention deficit/hyperactivity disorder) have smaller brains thanchildren without this condition? Brain scans were completed for 152 children with ADHD and 139children without ADHD. Summary values are given below.n mean sChildren with ADHD 152 1059.4 117.5Children without ADHD 139 1104.5 111.3Is there evidence that the mean brain volume of children with ADHD is smaller than the mean forchildren without ADHD? Use at 0.05% significance level?
3. 3. Q9. F-testA study was performed to determine whether men and women differ in their repeatability in assemblingcomponents on printed circuit boards. Two samples of 26 men and 21 women were selected and eachsubject assembled the units. The two sample standard deviations of assembly time were smen=0.98 minand swomen=1.02 min. Is there evidence to support the claim that men and women differ in repeatabilityfor this assembly task? Use the significance level 0.01.Q10. Mann WhitneyBecause of the rising costs of industrial accidents, many chemical, mining, and manufacturing firmshave instituted safety courses. Employees are encouraged to take these courses designed to heightensafety awareness. A company is trying to decide which one of two courses to institute. To help make adecision eight employees take course 1 and another eight take course 2 (both are independent). Eachemployee takes a test, which is graded out of a possible 25. The safety test results are shown below.Assume that the scores are not normally distributed. Do these data provide sufficient evidence at the 5%level of significance to infer that the marks from course 1 are lower than those of course 2Course 1 14 21 17 14 17 19 20 16Course 2 20 18 22 15 23 21 19 15Q11. Pooled t-testTwo different methods of teaching French vocabulary were tried on two independent groups. Method 1was used with a group of 40 students selected at random. Method 2 was used on another group of 42students selected at random. After one month the same vocabulary test was given to both groups. Theaverage score (out of 100) for the group using method 1 was 87 with standard deviation 4. The averagescore for the group using method 2 was 89 with standard deviation 4.2. Test the claim that studentstaught with method 2 perform better on the vocabulary test. Use a 1% significance level. Assumepopulation standard deviations are equal.Q12. Magnets are often used by people to treat a variety of disorders. Researchers recently treated a group ofpatients with magnets and another group of patients with a fake magnet treatment. The results are givenbelow. Use non-pooled t-test to test the claim that the magnet treatment is more effective at loweringpain in arthritis patients:Treatment Group Placebo GroupSample size 35 35Mean Pain Reduction 0.49 0.44Standard Deviation 0.96 1.4Q13. Mann WhitneyA Study of Wood reports the following data on burn time (hours) for samples of oak and pine. Testat level 0.01 to see whether there is any difference in true average burn time for the two types of wood.Oak 1.80 0.75 1.63 1.64 4.50 1.31 1.85 0.56Pine 1.06 1.48 1.41 1.60 0.81 1.28Q14. It has been suggested that British office workers are not taking their full lunch breaks, but spend part ofthem working at their desks; the median lunch „hour‟ now is being 34 minutes. An office supervisor in a
4. 4. large purchasing department, intrigued by this, noted the time spent away from their desks at lunchtimeby 10 randomly chosen staff, without their knowledge. The data were (minutes):55 20 31 12 18 35 28 16 14 32She was shocked when she looked at the data. Does it suggest, at the 5% level, that her staff is takingeven less than 34 minutes away from their desks at lunchtime? (Use wilcoxon sign rank test)Q15. Chi squareIn a study in which the subjects were 15 patients suffering from pulmonary sarcoid disease, blood gasdeterminations were made. The variance of the sample was 450. Test the hypothesis that the populationvariance is less than 250. Use α = 0.05.Q16. z-test of one meanThe scores on an aptitude test required for entry into a certain job position have a mean of 500 and astandard deviation of 120. If a random sample of 36 applicants has a mean of 546, is there evidence thattheir mean score is different from the mean that is expected from all applicants? Use α = 0.01.Q17. z-test of one proportionIt has been reported that 40% of the adult population participates in computer hobbies during theirleisure time. A random sample of 180 adults found that 65 engaged in computer hobbies.At  = 0.01, is there sufficient evidence to conclude that the proportion differs from 40%?Q18. Wilcoxon sign rank testIt is recommended that women should not consume more than 70g of fat per day. A random sample of13 student nurses at St Clares were asked to estimate as carefully as possible how much fat they ate onone particular day. The results were ( measured in grams)85 120 45 95 100 50 65 85 105 125 65 49Is there evidence at the 5% level that student nurses at St Clares are consuming more fat than theyshould?Q19. Chi square test of IndependenceThe demand for an MBA program‟s optional courses and majors is quite variable year over year. Theresearch hypothesis is that the academic background of the students (i.e. their undergrad degrees) affectstheir choice of major. A random sample of data on last year‟s MBA students was collected andsummarized in a contingency tableMBA MajorUndergraduateDegreeHumanResource Finance Marketing TotalBA 31 13 16 60B.E 8 16 7 31BBA 12 10 17 39Other 10 5 7 22Total 61 44 47 152
5. 5. Q20. Paired Wilcoxon sign rank testAssume that an English comprehension test is given to a random sample of 12 students before and afterthey complete an English course. Their scores are shown below before the course and after the course.Complete an hypothesis test to determine if the course improved the test score.After 115 115 119 117 114 114 109 110 106 112 115 111Before 111 108 114 105 102 106 100 103 101 105 107 107Using α = 0.05.Q21. Chi square Goodness of FitLast year the labor union bargaining team listed five items and asked each employee to identify the onemost important to him or her. The items and the corresponding percentages of most important responsesare given in the table below. The bargaining team needs to determine if the distribution of responsesfrom this year‟s survey “fits” last year‟s distribution.ItemsPercent of FavorableResponses Last YearTallied Responses Fromthis yearVacation Time 4% 30Salary 65% 290Safety Regulations 13% 70Health & RetirementBenefits 12% 70Overtime Policy & Pay 6% 40Is there evidence that employee attitudes on job benefits are the same this year as last year? Use 0.01significance level.Q22. The director of admissions at a state college is interested in seeing if admissions status (admitted, waiting list,denied admission) at his college is independent of the type of community in which an applicant resides. He takesa sample of recent admissions decisions and forms the following table:Admitted Wait List Denied TotalUrban 45 21 17 83Rural 33 13 24 70Suburban 34 12 39 85Total 112 46 80 238He will use this table to do a chi-square test of independence with a level of significance of 0.01.Q23. Chi square test of IndependenceRecent studies have found that American children are more obese than in the past. The amount of time childrenspend watching television has received much of the blame. A survey of 100 ten-year-olds revealed the followingwith regards to weights and average number of hours a day spent watching television. We are interested in testingwhether the average number of hours spent watching TV and weights are independent at 1% level of significance.WeightsTV HoursTotal0-3 3-6 6+More than 10 lbs. overweight 1 9 20 30Within 10 lbs. of normal weight 20 15 15 50More than 10 lbs. underweight 10 5 5 20Total 31 29 40 100
6. 6. Q24. Chi square Goodness of FitA researcher wishes to see of the number of adults who do not have health insurance is equally distributed amongthree categories:Category Less than 12 years 12 years More than 12 yearsFrequency 29 20 11Is there enough evidence to suggest that the number of people who do not have health insurance is not equallydistributed over the three categories? Use α = 0.05.Q25. t-test Confidence intervalAn industrial designer wants to determine the average amount of time it takes an adult to assemble an “easy toassemble” toy. A sample of 16 times yielded an average time of 19.92 minutes, with a sample standard deviationof 5.73 minutes. Assuming normality of assembly times, provide a 95% confidence interval for the meanassembly time.Q26. z-test Confidence intervalWhat is the smallest sample size required to provide a 95% confidence interval for a mean, if itimportant that the interval be no longer than 1cm? You may assume that the population is normal withvariance 9cm2.Q27. z-test of one meanBatCo (The Battery Company) produces your typical consumer battery. The company claims that theirbatteries last at least 100 hours, on average. Your experience with the BatCo battery has been somewhatdifferent, so you decide to conduct a test to see if the companies claim is true. You believe that themean life is actually less than the 100 hours BatCo claims. You decide to collect data on the averagebattery life (in hours) is 98.5 with a population standard deviation of 15.54. Test the claim at 5%significance level.Q28. ANOVAAutomobile insurance appraisers examine cars that have been involved in accidental collisions andestimate the cost of repairs. An insurance executive claims that there are significant differences in theestimates from different appraisers. To support his claim he takes a random sample of six cars that haverecently been damaged in accidents. Three appraisers then estimate the repair costs of all six cars. Thedata are shown below. ∑x2= 16657000Estimated Repair CostCar Appraiser 1 Appraiser 2 Appraiser 3 Total1 650 600 750 20002 930 910 1010 28503 440 450 500 13904 750 710 810 22705 1190 1050 1250 34906 1560 1270 1450 4280Total 5520 4990 5770 16280
7. 7. Q29. Confidence Interval of One Mean (t-distribution)Suppose a large labor union wishes to estimate the mean number of hours per month a union member is absentfrom work. The union decides to sample 320 of its members at random and monitor their working time for 1month. At the end of the month, the total number of hours absent from work is recorded for each employee. If themean and standard deviation of the sample are x = 9.6 hours and s = 6.4 hours, find a 90% confidence interval forthe true mean number of hours absent per month per employee.Q30. Confidence Interval of One ProportionA representative of a consumer organization took a random sample of 250 egg cartons from the dairy section of avery large supermarket and found that 80 cartons had at least one broken egg. Find a 90% confidence interval forthe proportion of cartons in the population that had at least one broken egg in them.Q31. Confidence Interval of One ProportionSuppose that a new drug Xydenal is used to treat patients with lung cancer. The treatment was successful on 134of the 245 patients it was administered to. Assume that these patients are representative of the population ofindividuals who have lung cancer. Determine a 95% C.I. for the proportion successfully treated.Q32. Confidence Interval of One Mean (t-distribution)A random sample of 53 observations was taken. The average in the sample was 90 with a variance of 400.Construct a 99% confidence interval for .Q33. Confidence Interval of One Standard DeviationThe cholesterol concentration in the yolks of each of a sample of 18 randomly selected eggs laid bygenetically engineered chickens were found to have a mean value, x , of 9.38 mg/g of yolk and astandard deviation, s, of 1.62 mg/g. Use this information to construct a 95% confidence intervalestimate of the true variance and standard deviation of the cholesterol concentration in these egg yolks.Q34. Confidence Interval of One Standard DeviationFor a class project, some students are doing a survey to find out the average number of hours a Denisonstudent spends studying each week. They take a simple random sample of 91 Denison students andcompute a sample mean of 5.4 hours per week with a standard deviation of 1.2 hours. Find a 95%confidence interval for the population variance.Q35. Confidence Interval of One ProportionA university dean is interested in determining the proportion of students who receive some sort of financial aid.Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 ofthem are receiving financial aid. Use a 90% confidence interval to estimate the true proportion of students whoreceive financial aid.