EM561 HW2 1. Given a set of 9 sample observations: 79, 100, 75, 83, 82, 85, 84, 79, 90. a) Calculate the sample mean; b) Calculate the sample variance and sample standard deviation; c) Calculate the sample median; d) Find the 95th percentile by Excel; e) Find the 1st and 3rd quartiles by Excel. 2. Suppose two stocks, A and B, have the following returns for each of the last five years. What are the sample covariance and sample correlation coefficient for the returns of these two stocks? How would you describe the relation between Stocks A and B? Returns Year Stock A Stock B 1 5.00% -1.00% 2 7.00% 0.00% 3 2.00% 5.00% 4 -5.00% -1.00% 5 3.00% 2.50% 3. A synthetic fiber used in manufacturing carpet has tensile strength that is normally distributed with mean 75 psi and standard deviation 3.5 psi. A random sample of 25 fiber specimens is taken. a) What is the probability distribution for the sample mean tensile strength of 25 fiber specimens? (Hint: if the parent population is NORMALLY distributed, then the sampling distribution of sample means follow a normal distribution regardless of the sample size.) b) Sketch the probably distribution of the sample mean tensile strength of 25 fiber specimens, in comparison with that of the tensile strength of a single fiber specimen. c) What is the probability that a random sample of 25 fiber specimens will have sample mean tensile strength that exceeds 75.75 psi? d) How would your answer in c) change if the sample size increases to 49? 4. The life in hours of a 75- watt light bulb is known to be normally distributed with =15 hours. A random sample of n=20 bulbs has a mean life of = 1015 hours. a) Construct a 95% two- sided confidence interval on the mean life. b) Construct a 99% two- sided confidence interval on the mean life. c) How do answers a) and b) compare to each other? Why? d) Suppose we wanted the error in estimating the mean life for the 95% two-sided confidence interval to be 5 hours, what sample size should be used? e) Suppose we wanted the error in estimating the mean life for the 99% two-sided confidence interval to be 5 hours, what sample size should be used? f) How do answers d) and e) compare to each other? Why? 5. A machine produces metal rods used in an automobile suspension system. A random sample of 15 rods is selected, and the diameter is measured. The resulting data (in millimeters) are as follows: 8.25 8.22 8.24 8.26 8.27 8.23 8.20 8.26 8.19 8.23 8.20 8.28 8.24 8.25 8.24 a) Calculate a 95% two-sided confidence interval on the mean rod diameter. b) Calculate a 95% upper confidence bound on the mean. Compare this bound with the upper bound of the two-sided confidence interval and discuss why they are different. 6. An article in Cancer Research \"Analysis of litter-matched time-to response data, with modifications for recovery of interlitter information,\" (1977, Vol. 37, pp. 3863-3868) tested the tumorigenesis of a drug. Rats were randomly selected from litters and given the drug. The times of tumor appearance were recorded as follows: 102, 105, 105, 78, 90, 88, 104, 96, 82, 70, 89, 91, 39, 103, 93, 85, 104, 104, 81, 67, 104, 104, 104, 87, 104, 89, 78, 104, 86, 76, 103, 102, 80, 45, 94, 104, 104, 76, 80, 72, 73 Calculate a 95% confidence interval on the standard deviation of time until a tumor appearance. 7. A random sample of 50 suspension helmets used by motorcycle riders and automobile racecar drivers was subjected to an impact test, and on 20 of these helmets some damage was observed. a) Find a 95% two-sided confidence interval on the true proportion of helmets of this type that would show damage from this test. b) Using the point estimate of p obtained from the preliminary sample of 50 helmets, how many helmets must be tested to be 95% confident that the error in estimating the true value of p is less than 0.02? c) How large must the sample be if we wish to be at least 95% confident that the error in estimating p is less than 0.02, regardless of the true value of p? 8. Cloud seeding has been studied for many decades as a weather modification procedure (for an interesting study of this subject, see the article in Technometrics, \"A Bayesian Analysis of a Multiplicative Treatment Effect in Weather Modification,\" Vol. 17, pp. 161-166). The rainfall in acre-feet from 20 clouds that were selected at random and seeded with silver nitrate follows: 18.0, 30.7, 19.8, 27.1, 22.3, 18.8, 31.8, 23.4, 21.2, 27.9, 31.9, 27.1, 25.0, 24.7, 26.9, 21.8, 29.2, 34.8, 26.7, and 31.6. a) Can you support a claim that mean rainfall from seeded clouds exceeds 25 acre-feet? Use = 0.01. Find the P-value. b) Check that rainfall is normally distributed. 9. If the standard deviation of hole diameter exceeds 0.01 millimeters, there is an unacceptably high probability that the rivet will not fit. Suppose that n = 15 and s = 0.008 millimeter. Is there strong evidence to indicate that the standard deviation of hole diameter exceeds 0.01 millimeter? Use = 0.01. State any necessary assumptions about the underlying distribution of the data. Find the P-value for this test. 10. An experiment with artillery shells yields the following data on the characteristics of lateral deflections and ranges. Would you conclude that deflection and range are independent? Use = 0.05. What is the P-value for this test? 11. An automobile manufacturer needs to buy aluminum sheets with an average thickness of 0.05 inch. The manufacturer collects a random sample of 40 sheets from a potential supplier. The thickness of each sheet in this sample is measured (in inches) and recorded below. Are these measurements normally distributed? Summarize your results. Intervals Frequency <0.03 1 (0.03, 0.04) 10 (0.04, 0.05) 13 (0.05, 0.06) 12 >0.06 4 12. Study the Case of SnowPea Online Grocery. Answer questions 2 and 3 from the management. Answer all questions using proper hypothesis tests. Case Study: SnowPea Online Grocery The SnowPea grocery is a Chicago-based grocery online sales and delivery chain store. Most of SnowPea's deliveries are within a 10-mile radius, but it occasionally delivers to customers more than 10 miles away. SnowPea employs a number of delivery people, four of whom are relatively new hires. The store has recently been receiving customer complaints about excessively long delivery times. Therefore, SnowPea has collected data on a random sample of deliveries by its four new delivery people during the peak dinner time. The data are in the file SnowPea.xlsx. The variables are as follows: Deliverer: which person made the delivery PrepTime: time from when order was place until delivery person started driving it to the customer TravelTime: time to drive from SnowPea to customer Distance: distance (miles) from SnowPea to customer Several questions are asked by the management. Using statistical analysis to address these questions and make reasonable recommendations to SnowPea management. 1. SnowPea is concerned that one or more of the new delivery people might be slower than others. 2. SnowPea would like to advertise that it can achieve a total delivery time of no more than M minutes for all customers within a 10-mile radius. On all orders that take more than M minutes, SnowPea will give the customers a $10 certificate on their next purchase. a) Try different values of M in analyzing the proper sample statistic. b) Suppose SnowPea makes 1000 deliveries within the 10-mile limit. What is the financial implications for the analysis in a)? c) What M do you recommend? 3. The policy in the previous problem is simple to state and simple to administer. However, it is somewhat unfair to customers who live close to SnowPea - they will never get $10 certificates! A fairer, but more complex, policy is the following. SnowPea first analyzes the data and finds that total delivery times can be predicted fairly well with the equation: Predicted Delivery Time = 14.8 + 2.06Distance (This is based on regression analysis, the topic of Chapters 11) Also, most of these predictions are within 5 minutes of the actual delivery times. Therefore, whenever SnowPea receives an order over the phone, it looks up the customer's address in its computerized geographical database to find distance, calculates the predicted delivery time based on this equation, rounds this to the nearest minute, adds 5 minutes, and guarantees this delivery time or else a $10 certificate. It does this for all customers, even those beyond the 10-mile limit. a) Assuming again that the delivery people in the sample are representative of all of SnowPea's delivery people, conduct proper statistical analysis to study the impact of this proposed policy. b) Suppose SnowPea makes 1000 deliveries. What's the financial implication of a)? SnowPea Delivery Times Order Deliverer PrepTime TravelTime Distance TotalTime Speed 1 3 13.0 30.0 13.3 43.0 0.443333 2 1 10.9 17.8 8.6 28.7 0.483146 3 1 9.2 9.2 4.7 18.4 0.51087 4 3 7.2 14.7 4.3 21.9 0.292517 5 2 14.5 21.9 12.2 36.4 0.557078 6 2 9.5 8.1 3.6 17.6 0.444444 7 2 7.1 31.6 11.3 38.7 0.357595 8 3 8.5 29.0 10.3 37.5 0.355172 9 2 9.8 4.9 2.6 14.7 0.530612 10 1 14.3 13.4 5.3 27.7 0.395522 11 1 12.0 12.9 4.5 24.9 0.348837 12 2 13.0 8.8 4.1 21.8 0.465909 13 1 14.9 8.7 4.2 23.6 0.482759 14 1 9.7 29.9 11.7 39.6 0.391304 15 2 10.0 18.2 9.5 28.2 0.521978 16 3 10.8 22.9 8.8 33.7 0.384279 17 1 12.1 5.0 1.7 17.1 0.34 18 1 9.9 9.9 4.5 19.8 0.454545 19 1 7.9 9.0 3.7 16.9 0.411111 20 2 17.2 6.5 3.4 23.7 0.523077 21 4 10.2 29.3 3.4 39.5 0.116041 22 3 15.0 24.0 9.8 39.0 0.408333 23 4 12.5 19.5 5.4 32.0 0.276923 24 1 11.0 10.5 5.0 21.5 0.47619 25 4 15.7 29.0 7.9 44.7 0.272414 26 2 10.1 2.6 1.4 12.7 0.538462 27 3 8.2 19.0 7.5 27.2 0.394737 28 1 9.2 13.4 7.0 22.6 0.522388 29 2 18.7 8.9 5.6 27.6 0.629213 30 4 11.0 9.3 2.2 20.3 0.236559 31 1 9.3 22.8 8.2 32.1 0.359649 32 2 10.1 14.5 7.7 24.6 0.531034 33 1 11.4 26.1 12.9 37.5 0.494253 34 3 18.0 10.8 3.8 28.8 0.351852 35 4 14.0 28.8 11.7 42.8 0.40625 36 2 15.4 21.2 10.7 36.6 0.504717 37 4 6.1 6.9 2.8 13.0 0.405797 38 3 13.9 21.9 8.4 35.8 0.383562 39 4 9.5 17.8 4.6 27.3 0.258427 40 1 10.3 9.5 4.3 19.8 0.452632 41 3 14.4 29.7 6.9 44.1 0.232323 42 2 9.4 23.5 12.2 32.9 0.519149 43 3 6.6 17.0 6.3 23.6 0.370588 44 4 15.6 18.5 5.9 34.1 0.318919 45 2 9.7 13.8 9.2 23.5 0.666667 46 1 14.0 5.3 2.5 19.3 0.471698 47 1 11.7 3.3 1.4 15.0 0.424242 48 4 12.7 0.7 0.2 13.4 0.285714 49 1 14.6 1.9 0.6 16.5 0.315789 50 1 9.5 12.6 6.5 22.1 0.515873 51 1 7.1 7.2 3.3 14.3 0.458333 52 3 8.5 7.3 3.0 15.8 0.410959 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 4 4 4 1 4 4 4 1 3 3 3 4 3 2 2 3 2 2 1 3 1 1 1 2 9.9 14.3 12.0 13.0 16.2 10.3 8.5 9.5 13.1 8.4 17.0 14.2 8.8 12.9 10.9 9.7 13.4 8.7 17.3 7.0 9.0 13.1 11.0 12.0 26.2 28.7 23.1 12.6 28.3 9.4 29.8 21.6 10.1 24.1 6.0 18.6 13.6 29.5 17.8 23.8 18.9 7.0 16.6 31.3 3.5 11.9 11.6 7.5 5.2 8.9 7.7 7.2 8.2 4.1 13.0 11.6 3.1 5.8 2.8 2.4 4.6 14.0 6.9 6.4 10.1 2.9 9.3 8.6 1.5 4.4 3.6 3.3 36.1 43.0 35.1 25.6 44.5 19.7 38.3 31.1 23.2 32.5 23.0 32.8 22.4 42.4 28.7 33.5 32.3 15.7 33.9 38.3 12.5 25.0 22.6 19.5 0.198473 0.310105 0.333333 0.571429 0.289753 0.43617 0.436242 0.537037 0.306931 0.240664 0.466667 0.129032 0.338235 0.474576 0.38764 0.268908 0.534392 0.414286 0.560241 0.27476 0.428571 0.369748 0.310345 0.44