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Microsoft Word - F Practice 201530.docx p. 1/6 STAT 1124 Practice Exam Time: 2 hrs. 1. An apple farmer wishes to estimate the mean yield per tree in her orchard. She wishes to do this by taking a random sample of 100 trees. The orchard is divided into several quarter-acre plots. a. If the farmer treats each plot as a cluster, what is she assuming about the plots? [1 mark] b. If the farmer treats each plot as a stratum, what is she assuming about the plots? [1 mark] c. What sampling method would you recommend to the farmer? Justify your answer. [2 marks] 2. In each of the following situations, identify the sources of bias, if any [2 marks each]. a. To estimate the attitude of women towards gender relations a researcher distributes questionnaires to members of several women organizations. b. To measure public opinion towards the police, random interviews are conducted. The interviews are done by a uniformed police officer. 3. For each of the following situations, i) state the appropriate graphical display [1 mark], and ii) state, if any, statistical procedure that we have covered in the course that may be applied to the situation. [1 mark] a. A study to investigate whether the distribution of student enrolment across various faculties (natural sciences, social sciences, or humanities) differs among Genders (females, males, and others). b. A study to investigate the relationship between average precipitation and average temperatures for North American cities. c. A study to test if the average age of Canadians have changed from 1981, when it was reported to be 34.8 years. A random sample of 100 Canadians was collected. 4. Answer each of the following questions with true or false. Justify your answer concisely. a. In a certain data set, the mean is equal to the median is equal to the mode. We can then conclude that the histogram of the data set is bell shaped. [2 marks b. In a large company, the average age of accounts receivable is 40 days, and its standard deviation is 25 days. We can therefore conclude that the age of accounts receivable is not Normally distributed. [2 marks] c. A 99% confidence interval is always preferable to a 90% confidence interval because it has a higher level of confidence. [2 marks] d. The 95% confidence interval for the average amount of mortgage approved by a local credit union is (130,000, 200,000). Therefore we can conclude that approximately 95% of all the mortgages approved by the credit union is between $130,000 and $200,000. [2 marks] p. 2/6 5. A data set that contains the price and per capita consumption of cigarettes over the last 23 years reveals an r value of 0.086. a. What graphical display we must examine before we can interpret the above r value? Give one reason why. [2 marks] b. What can you say about the (population) correlation coefficient between the price and consumption of cigarettes? [2 marks] c. Can you say that there is no relationship between the price and the level of consumption of cigarettes? Briefly explain your answer. [2 marks] 6. A pollster surveyed a random sample of adults about their reading of magazines. One question asked: “If you could read just one magazine, would it be one that emphasized Sports, News, or Entertainment?” The responses to this question are cross tabulated with education level as shown in the following table. Magazine Education Sports News Entertainment High School or less 22 24 44 Some Post Secondary 35 34 45 Post Secondary Degree 33 62 61 a. If one adult is to be randomly selected from the above sample, what is the probability that he/she has a post secondary degree and would choose a news magazine. [2 marks] b. If one adult is to be randomly selected from the above sample, what is the probability that he/she has a post secondary degree or would choose a news magazine. [2 marks] c. If one adult is to be randomly selected from the above sample, are the events "getting someone who has a post secondary degree" and "getting someone who would choose a news magazine" mutually exclusive? Explain your answer. [2 marks] 7. Based on the sample shown in #6, estimate the proportion of adults in this population who have post secondary education degrees. Calculate your estimate so you are 82% confident that it is correct.) [3 marks] 8. A gambling game works as follows: you flip a fair coin and roll a fair six-sided die. You will be paid $2 if you roll a 6 and $1 if you get a Head and an odd number of spots. Otherwise, you have to pay $2.50. a. What is the probability of winning money from this game? [2 marks] b. Construct the probability distribution of the amount won when you play this game once. (Losing money is equivalent to winning a negative amount.) [3 marks] c. In the long run, how much would you win (or lose) per game? [2 marks] p. 3/6 9. A study was conducted to investigate if the level of education (elementary, secondary, or post secondary) is related to awareness of safe sex practices (not aware, somewhat aware, or very aware). a. What statistical technique is the most appropriate for the study? [1 mark] b. If the Chi-square statistic is equal to 15.2, what conclusion should be made? [2 marks] c. What condition(s) must be satisfied for the above analysis to be valid? [1 mark] 10. Previous studies have concluded that home owners live in their homes for an average of 8.4 years before selling them. A realty company wishes to investigate if this figure has changed. A sample of 60 families who have recently sold their house was selected. The mean time that these families had lived in their homes was 8.94 years. The standard deviation was 3.49 years. a. What can be concluded from these data? Test the appropriate hypotheses at α = .05. Report the p- value. [5 marks] b. Suppose it has come to light that it is crucially important that if the average time people own their house before selling it has changed, to detect this change. Given this information, should we lower or raise the significance level of the test. Explain your answer. [2 marks] p. 4/6 Solution 1a. The mean yield per tree is approximately the same from plot to plot b. The mean yield per tree is quite different from plot to plot c. Stratified random sample. The mean yield per tree is probably different from plot to plot due to different soil and micro climatic conditions. 2a. There might be some selection bias. Women who do not belong to any women organizations, who probably have different opinions about gender relations, are systematically excluded from the sample. b. There might be a response bias. People will be discouraged to express negative opinions towards the police if they are interviewed by a uniformed police officer. 3a. Graph: block diagram (2 qualitative variables). Analysis: Chi-square analysis b. Graph: scatter plot (2 quantitative variables). Analysis: correlation/regression analysis c. Graph: Histogram (1 quantitative variable). Analysis: hypothesis test for population mean �. 4a. False. The uniform distribution, for example, has the same mean. median, and mode, but is not bell shaped. It is box shaped. b. True. Because, if it was normally distributed, approximately 2.5% of the data would lie to the left of (mean - 2 SD), ie to the left of -10, which is clearly impossible here. c. False. If the sample sizes are equal, a 90% C.I. will be narrower, and thus more precise, than a 99% C.I. d. False. The C.I. says that we can be 95% confident that the mean of all the approved mortgages will lie in the calculated interval. It says nothing about the proportion of approved mortgages that lie inside the interval. 5a. Scatter plot. To see if there are any outliers that might influence r. Also, a scatter plot will detect any non-linear pattern. b. The cut off point for r for n = 24 is 0.404. Therefore, the cut of point for n=23 must be larger. Our r = 0.086 is therefore smaller than the cut of point. Therefore we have to assume that the population correlation coefficient is 0. c. No we can't. It is possible that there is a linear relationship but the sample fails to detect it. (Type II Error occurred). Also, it is possible that there is a non-linear relationship that the correlation coefficient r, is not designed to detect. p. 5/6 6. To facilitate easier calculation we first add the column and row totals to the table of observed frequencies. Education Sports News Entertainment Total High School or less 22 24 44 90 Some Post Secondary 35 34 45 114 Post Secondary Degree 33 62 61 156 Total 90 120 150 360 a. Out of the total of 360 people, 62 have post secondary degrees and would choose news magazine. The desired probability is therefore !" #!$ = 0.1722 b. Since there are