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Math 308 Spring 2009 Study Sheet and Sample Problems for Test 3 Text for TEST 3 Assigned Other material 7.1 Point estimation 5 7.2 Interval 17 estimation Hypothesis testing: Understand the hypothesis-testing five-step process. Know the hypotheses for tests about one mean, two means and more than two means. Know which test statistics to use based on the assumptions of the test statistic. Know how to relate the alternate hypothesis to the decision rule (rejection region). Know how to calculate the test statistic from a formula sheet. 7.3 Tests of Assigned Class Handout: Hypotheses Statistical Testing 7.4 Null hypotheses 7.30, 7.31, 7.35, Class Handout: and tests of 7.37,7.38 Statistical Testing hypotheses 7.5 Hypotheses about 7.39, 7.43, 7.48 Class Handout: one mean Statistical Testing 7.8 Inference 7.68, 72 Class Handout: concerning the Statistical Testing difference between two means 9.1-9.3 4, 11, 22, 27 12.1,12.2 ANOVA: Example on Online class notes: One-way ANOVA Handout Analysis of (Completely Variance randomized design) 11.1-11. Simple Example on Online class notes: Linear Regression Handout Regression Old Test Questions 1. A random sample of the number of MotoCars made at the Electric MotoCar Factory yielded the numbers in the table. The graph is a histogram of the number of MotoCars made for the twenty randomly chosen days. The mean daily number of cars was 36.75 and the standard deviation was 6.71115. The owners of Electric MotoCar want to know whether the actual (population) mean is greater than 34 cars/day. Fill in the five steps of the hypothesis-testing process for testing this hypothesis. Use a significance level of 0.05. MotoCars 8 36.00 28.00 49.00 44.00 6 35.00 37.00 Frequency 36.00 35.00 4 37.00 36.00 33.00 39.00 2 40.00 32.00 Mean = 36.75 31.00 Std. Dev. = 6.71115 N = 20 41.00 0 20.00 30.00 40.00 50.00 42.00 MotoCars 30.00 23.00 51.00 (a) Hypotheses[3 points]: (b) Significance level and test statistic (give your reasons for choosing the test statistic) [7]: (Question 1 continued) (c) Decision rule (rejection region) 4]: (d) Computed sample statistic [8]: (e) Decision (include reason for decision)[2]: 2. [8] Refer to problem 1. What is the probability of a Type II error if 35.45 MotoCars daily? 3. [10] Refer again to problem 1. Give the 95% confidence interval for the actual number of MotoCars produced daily. 4. [5] If we want to determine the average mechanical aptitude of a large group of workers, how large a random sample will we need to be able to assert with probability 0.95 that the sample mean will not differ from the true mean by more than 4 points? Assume that it is known from past experience that 25.0 5. [25] The number of cars manufactured at Factory A was collected for 16 randomly selected days. The number of cars manufactured by Factory B was collected for 19 randomly selected days. The table presents sample statistics for the two random samples from the two competing factories. The histograms present the sample frequency distribution of Factory A and Factory B. Factory A Factory B Mean = 206 Mean = 209 Std Dev Std Dev =1.21106 =2.000 n=16 n=19 Histogram Histogram for Factory= A for Factory= B 6 5 5 4 4 Frequency Frequency 3 3 2 2 1 1 Mean = 206.00 Mean = 209.00 Std. Dev. = 1.21106 Std. Dev. = 2.00 0 N = 16 0 N = 19 204.00 205.00 206.00 207.00 208.00 206.00 208.00 210.00 212.00 Cars Cars Perform a complete five-step hypothesis-testing procedure (including your conclusion) for the null hypothesis H 0 : A B . Let .05 . Number your steps. Explain your answers where appropriate. (Use problem 1 as a reference.) 6. [28] Given the following observations from four independent random samples, complete the five-step hypothesis-testing procedure for comparing the four means corresponding to the four Treatments. Number your steps and use problem 1 as a general reference for an appropriate answer. Treatment 1 Treatment 2 Treatment 3 Treatment 4 6 13 7 3 4 10 9 6 5 13 11 1 12 4 1 Complete the following table as part of your answer. Source Sum of Degrees of Mean Square F Squares freedom Treatment Error OTHER SAMPLE PROBLEMS FOR TEST 3: 1. The following figures are the number of widgets filled per day at two competing plants. Plant A Plant B 19 23 17 19 21 21 15 15 15 17 12 11 17 12 12 10 24 16 16 20 12 15 15 12 19 20 11 13 14 18 13 17 20 14 14 16 18 22 21 18 Mean = 17.85 Mean = 14.85 Std Dev = 3.50 Std Dev = 3.15 With the information you have been given so far, how would you verify that each sample is from a normal population. How would you find the probability of a sample mean of 17.85 or larger for Plant A if you assume that the actual mean for Plant A is 15? 2. A cola dispensing machine at XYZ company dispenses 9 ounce cups of cola. An employee of XYZ thinks that the machine is cheating consumers. A random sample of 50 cups has a mean of 8.7 ounces and a standard deviation of 0.5 ounces. If the mean volume of cola dispensed is actually 9 ounces, what is the probability of a mean this small or smaller? 5. The functional reach of adult men is a normally distributed random variable with mean 32.33 inches and standard deviation 1.63 inches. (a) What is the probability that the reach of a randomly selected man exceeds 34.5 inches? (b) A second man is randomly selected. What is the probability that both men have reaches exceeding 34.5 inches? (Assume independence.) (c) What is the probability that the reach of a randomly selected man is between 31.515 inches and 33.96 inches? 6. The Leakey Brothers offer a one-year warranty on radiator replacements. The time to failure of a Leakey radiator has an exponential distribution with mean process rate λ= 0.6 per year. (a) What is the probability of a radiator failing within the warranty period? (b) If the Leakey Brothers make a profit of $125 on every radiator that survives the warranty period and lose $50 on each radiator that fails under warranty, what is the expected profit per radiator sold? Some sample questions for Test 3: Note that there is no ANOVA question. 1. The following figures are the number of widgets produced per day at two competing plants. Plant A Plant B 19 23 17 19 21 21 15 15 15 17 12 11 17 12 12 10 24 16 16 20 12 15 15 12 19 20 11 13 14 18 13 17 20 14 14 16 18 22 21 18 Mean = 17.85 Mean = 14.85 Std Dev = 3.50 Std Dev = 3.15 A. With the information you have been given so far, how would you verify that each sample is from a normal population. Why is this “check” important? B. Find the 95% CI for the mean of the number of widgets made daily by Plant A. C. Test the hypothesis that Plant A makes more widgets daily than Plant B. 2. A cola dispensing machine at XYZ company dispenses 9 ounce cups of cola. An employee of XYZ thinks that the machine is cheating consumers. A random sample of 50 cups has a mean of 8.7 ounces and a standard deviation of 0.5 ounces. A. Construct a 95% confidence interval for the average amount of cola dispensed by this machine. B. Test the hypothesis that the actual mean volume of cola is less than 9 oz. 3. How would a 95% confidence interval compare to the 99% interval? a) wider b) same width c) narrower 4. The production foreman of Blue Ox Brewing Company is worried that the bottling process is overfilling. Each bottle of Blue Ox Beer is supposed to contain 12 oz of fluid. A random sample of 40 bottles is taken from bottles coming off the production line, and the contents of each bottle are carefully measured. It is found that the mean amount of beer for the sample of bottles is 12.1 oz with a standard deviation of 0.2 oz. A. What is the 99% CI for μ? B. Test the hypothesis that the mean amount of beer is greater than 12 oz. 5. Plant therapists believe that plants can reduce the stress levels of humans. At Christian Brothers University, a random sample of 10 undergraduates took part in a study to determine what effect the presence of a plant has on relaxation. Each student participated in two sessions – one with a plant and one without a plant. During each session, finger temperature was measured (increasing finger temperature indicates an increased level of relaxation). You are asked to analyze the data. Some of your results are presented here. d = Plant – NoPlant for an individual d = mean difference in finger temperature 95.0 % C.I. for d : (-1.975, 1.995) A. Based on the confidence interval results, can the researchers state that plants help college students to relax? Explain. B. You are asked to test the hypothesis that the presence of a plant increases relaxation. Explain how you would do the test. Give all five steps. 6. The caffeine counts (in mg) are given below for a dozen cans of King of Caffeine cola randomly selected from the production line. The sample has a mean of 33.05 and a standard deviation of 1.13. 34.2 33.7 31.9 34.3 31.6 32.7 33.1 35.2 31.6 32.9 33.0 32.4 Test the hypothesis that the mean is greater than 32 mg.