STAT 400 Lecture AL1 Practice Problems 7, 8 (Part 1) Spring 2015 Dalpiaz 1. In a random sample of 100 Hawk & Hummingbird Airline (HHA) direct flights from New York to Boston, the average number of passengers was 56.3, with sample standard deviation 11.33. a) Construct a 95% confidence interval for the overall average number of passengers on this route. b) To make a reasonable profit, HHA must average at least 58 passengers per flight on this route. The president of the airline is concerned that the average number of passengers is less than 58. Perform the appropriate test at = 0.05. c) Suppose the actual overall average number of passengers on this flight is 57. Then in part (b) a ( Type I Error, Type II Error, correct decision ) was made. d) Find the p-value of the test in part (b). e) Using the P-value obtained in part (d), state your decision ( Accept H 0 or Reject H 0 ) for = 0.10. f) Describe in the context of the problem what happens if Type I Error occurs. g) Describe in the context of the problem what happens if Type II Error occurs. 2. Hawk & Hummingbird Airline wants to determine the proportion of passengers that bring only carry-on luggage to the flight from New York to Boston. a) In a random sample of 200 passengers, 44 passengers had only carry-on luggage. Is there enough evidence to conclude that more than 20% of all passengers have only carry-on luggage? Use = 0.10. b) Find the p-value of the test in part (a). c) Find a 90% confidence interval for the overall proportion of passengers who have only carry-on luggage. d) Find the minimum sample size required in order to estimate the proportion of passengers who have only carry-on luggage to within 2% with 90% confidence, if it is known that this proportion is at most 0.30.3. In a random sample of 25 direct flights from New York to Boston by Hawk & Hummingbird Airline, the sample mean flight time was 56 minutes and the sample standard deviation was 8 minutes. (Assume the flight times are approximately normally distributed.) a) Construct a 98% confidence interval for the overall mean flight time on this route. b) Test whether the actual overall mean flight time on this route is 1 hour (60 minutes) or it is different at 5% significance level. c) Find the P-value of the test in part (b). d) Using the P-value obtained in part (c), state your decision ( Accept H 0 or Reject H 0 ) for = 0.01. e) Suppose that the actual overall mean flight time on this route is 58 minutes. Then in part (b) a ( Type I Error, Type II Error, correct decision ) was made. 4. The manager of a department store is thinking about establishing a new billing system for the store's credit customers. After a thorough financial analysis, she determines that the new system will be cost effective only if the mean monthly account is more than $170. A random sample of 169 monthly accounts is drawn for which the sample mean is $177 and the sample standard deviation is $65. a) Can the manager conclude from this that the new system will be cost effective? Perform the appropriate test using a 5% level of significance. b) Suppose that the true value of the overall mean monthly account is $175. Then in part (a), ( Type I Error, Type II Error, correct decision ) was made. c) What is the p-value of the test in part (a) ? d) Using the p-value from part (c), state your decision (Accept H 0 or Reject H 0 ) for = 0.10. e) Describe in the context of the problem what happens when a Type I Error occurs. f) Describe in the context of the problem what happens when a Type II Error occurs. g) Construct a 95% confidence interval for the overall mean monthly account. h) How large should the sample be if the manager wants to estimate the overall mean monthly account to within $5 with 95% confidence? (Use the sample standard deviation as an estimate of the population standard deviation.)5. Suppose that the lengths of the trout fry in a pond at the fish hatchery have the overall standard deviation of 0.8 inch. A random sample of 49 fry will be netted and their lengths measured. a) What is the probability that the sample mean will be within 0.2 inch of the population mean ? b) Suppose the sample mean of the 49 fry netted is x = 3.4 inches. Construct a 95% confidence interval for the overall mean lengths of the trout fry in the pond. c) What is the minimum sample size required for estimating the overall mean lengths of the trout fry in the pond to within 0.2 inch with 95% confidence? d) Construct a 90% confidence interval for the overall mean lengths of the trout fry in the pond. e) Construct a 94% confidence interval for the overall mean lengths of the trout fry in the pond. 6. Mercury makes a 2.4 liter V-6 engine. The company’s engineers believe that the standard deviation of power delivered by the engine is 8 HP. a) A potential buyer wants to estimate the average power and intends to sample 100 engines (each engine to be run a single time). What is the probability that the sample mean will differ from the average power delivered by the engine by more than 2 HP? b) How many engines need to be tested if we want to estimate the average power to within 2 HP with 90% confidence? 7. A manufacturer of video games wants to install some machines in shopping malls. In a pilot study on the potential profitability of this enterprise, games were placed for one week in 10 randomly chosen shopping malls. The weekly profits in dollars were as follows: 110.80 67.90 141.20 93.60 75.80 131.30 106.40 87.80 94.10 98.00 Assume the population distribution is normal. Construct a 90% confidence interval for the mean weekly profit for these games in all shopping malls. [Hint: x = 1,006.90, x2 = 106,059.39.]8. In a random sample of 20 homeowners in Anytown, the average monthly electric bill during June was $64. The sample standard deviation was $11. Assume electric bills are approximately normally distributed. The manager of the Anytown Power Plant believes that the variance of June monthly electric bills is 100 ($ squared). Test whether the variance of June monthly electric bills is 100 or it is different at a 10% level of significance. 9. Several employees of Bob’s Computer Warehouse copmlain that the variance of the
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