## Take Home Exam 3: Special Note Before Starting the Exam: If you scan your solutions to the exam and save it as a pdf or image file and put it on dropbox and I can not read it or open it, you will not receive credit for the exam. Furthermore, if you write the solutions up in word, latex ect. and give me a print out, which does not include all the pages you will not get credit for the missing pages. Also if your folder on dropbox is not clearly labeled and I can not find your exam then you will not get credit for the exam. Finally, please make sure you put your name on the exam!! Math 2100 Exam 3, Out of Class, Due by December 8th, 2015 at 5:00 pm. Name: Problem 1. (15 points) A random variable is said to have the (standard) Cauchy distribution if its PDF is given by f (x) = 1 π 1 1+ x2 , −∞< x <∞ This problem uses computer simulations to demonstrate that a) samples from this distribution often have extreme outliers (a consequence of the heavy tails of the distribution), and b) the sample mean is prone to the same type of outliers. Below is a graph of the pdf a) (5 points) The R commands x=rcauchy(500); summary(x) generate a random sample of size 500 from the Cauchy distribution and display the sample’s five number summary; Report the five number summary and the interquartile range, and comment on whether or not the smallest and largest numbers generated from this sample of 500 are outliers. Repeat this 10 times. b) (5 points) The R commands m=matrix(rcauchy(50000), nrow=500); xb=apply(m,1,mean);summary(xb) generate the matrix m that has 500 rows, each of which is a sample of size n=100 from the Cauchy distribution, compute the 500 sample means and store them in xb. and display the five number summary xb. Repeat these commands 10 times, and report the 10 sets of five number summaries. Compare with the 10 sets of five number summaries from part (a), and comment on whether or not the distribution of the averages seems to be more prone to extreme outliers as that of the individual observations. c) (5 points) Why does this happen? (hint: try to calculate E(X) and V(X) for this distribution) and does the LLN and CLT apply for samples from a Cauchy distribution? Hint: E(X) is undefined for this distribution unless you use the Cauchy Principle Value as such for the mean lim a→∞ xf (x)dx −a a∫ In addition x2 1+ x2 dx = x2 +1−1 1+ x2 dx = 1− 1 1+ x2 " # $ % & ' ∫ ∫ ∫ dx 1 1+ x2 dx = tan−1 ∫ x +C Problem 2. (5 points) A marketing expert for a pasta-making company believes that 40% of pasta lovers prefer lasagna. If 9 out of 20 pasta lovers choose lasagna over other pastas, what can be concluded about the expert's claim? Use a 0.05 level of significance. Problem 3. (10 points) A coin is tossed 20 times, resulting in 5 heads. Is this sufficient evidence to reject the hypothesis that the coin is balanced in favor of the alternative that heads occur less than 50% of the time (essentially is this significant evidence to claim that the coin is unbalanced in favor of tails)? Use a 0.05 level of significance. Problem 4. (25 points) Since the chemical benzene may cause cancer, the federal government has set the maximum allowable benzene concentration in the workplace at 1 part per million (1 ppm) Suppose that a steel manufacturing plant is under investigation for possible violations regarding benzene level. The Occupational Safety and Health Administration (OSHA) will analyze 14 air samples over a one-month period. Assume normality of the population from which the samples were drawn. a) (3 points) What is an appropriate null hypothesis for this scenario? (Give this in symbols) b) (3 points) What is an appropriate alternative hypothesis for this scenario? (Give this in symbols) c) (3 points) What kind of hypothesis test is this: left-tailed, right-tailed or two-tailed? Explain how you picked your answer. d) (3 points) Is this a one-sample t-test or a one-sample test using a normal distribution? Explain how you picked your answer. e) (4 points) If the test using this sample of size 14 is to be done at the 1% significance level, calculate the critical value(s) and describe the rejection region(s) for the test statistic. Show your work. f) (5 points) OHSA finds the following for their sample of size 14: a mean benzene level of 1.51 ppm and a standard deviation of 1.415 ppm. What should be concluded at the 1% significance level? Support your answer with calculation(s) and reasoning. g) (4 points) Calculate the p-value for this test and verify that this answer would lead to the same conclusion you made in part f. Problem 5. (15 points) A normally distributed random variable Y possesses a mean of μ = 20 and a standard deviation of σ = 5. A random sample of n = 31 observations is to be selected. Let X be the sample average. (X in this problem is really x _ ) a)(5 points) Describe the sampling distribution of X (i.e. describe the distribution of X and give μx, σx ) b) (5 points) Find the z-score of x = 22 c) (5 points) Find P(X ≥ 22) = Problem 6. (10 points) A restaurants receipts show that the cost of customers' dinners has a distribution with a mean of $54 and a standard deviation of $18. What is the probability that the next 100 customers will spend a total of at least $5800 on dinner? Problem 7. (10 points) The operations manager of a large production plant would like to estimate the mean amount of time a worker takes to assemble a new electronic component. Assume that the standard deviation of this assembly time is 3.6 minutes and is normally distributed. a) (3 points) After observing 120 workers assembling similar devices, the manager noticed that their average time was 16.2 minutes. Construct a 92% confidence interval for the mean assembly time. b) (2 points) How many workers should be involved in this study in order to have the mean assembly time estimated up to ± 15 seconds with 92% confidence? c) (5 points) Construct a 92% confidence interval if instead of observing 120 workers assembling similar devices, rather the manager observes 25 workers and notice their average time was 16.2 minutes with a standard deviation of 4.0 minutes. Problem 8. (10 points): A manufacturer of candy must monitor the temperature at which the candies are baked. Too much variation will cause inconsistency in the taste of the candy. Past records show that the standard deviation of the temperature has been 1.2oF . A random sample of 30 batches of candy is selected, and the sample standard deviation of the temperature is 2.1oF . a. (5 points) At the 0.05 level of significance, is there evidence that the population standard deviation has increased above 1.2oF ? b. (3 points) What assumption do you need to make in order to perform this test? c. (2 points) Compute the p-value in (a) and interpret its meaning.

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