1 For problems la through 1.c, assume that the length of a population of fish is normally distributed with population mean u = 63 cm and population standard deviation o = 9 cm. 1.a What proportion of the individual fish are longer than 76 cm? 1.b What proportion of the fish are between 42 and 84 cm long? 2 For problem 2.a through 2.c, assume that a population of automobile engines has a population mean useful life u = 120,000 miles and population standard deviation o = 8400 miles. Also assume the data is normally distributed. 2.a What proportion of the engines last more than 140,000 miles? b What proportion of the engines last between 110,400 to 130,600 miles? 2.c The manufacturer wants to write a warranty so that only 0.8% (0.008) of the engines fail while under warranty. For how long should the warranty be written? 3. Use the following data for problems 3.a, 3.b, 3.c, and 3.d The time required to wait for service at a government office is normally distributed. It has population mean p = 40 minutes with population standard deviation o = 8 minutes. 3.a What proportion of wait times are more than 54 minutes? 3.b What proportion of the wait times are less than 36 minutes? 3.c What proportion of the wait times are between 34 to 46 minutes? 3.d What is the 80th percentile (P80)? You are encouraged (but not required) to round to the nearest whole percentile. 4 A sociology professor finds that his student's scores on an exam are normally distributed with population mean u = 80 and population standard deviation o = 6. Find the 40th percentile. 5. Use the following data for problems 5.a and 5.b. Assume the height of 10 year old boys in the US is normally distributed. Also assume that the population mean is u = 54 inches and the population standard deviation is o = 1.6 inches. 5.a What proportion of 10-year-old boys are between 51 inches to 57 inches tall? 5.b Find the 25th percentile. 6 Use the following data for problems 6.a and 6.b. The incubation time (time to hatch) for chicken eggs is normally distributed with population mean u = 21 days with a population standard deviation o = 1 day. 6.a What proportion of eggs require more than 22.4 days to hatch? 6.b What proportion of eggs hatch between 18.5 to 23.5 days?7 A manufacturer of high intensity lamps finds that the useful life of the lamps is normally distributed with population mean u = 70 months and population standard deviation s = 12 months. The manufacturer wants to write a warranty so that only 1.5% (0.015) of the lamps fail while still under warranty. For how long should the warranty be written? 8 The time required for laboratory rats to complete a maze is normally distributed with population mean H = 45 minutes with population standard deviation o = 5.4 minutes. What proportion of the rats complete the maze with time between 37 to 53 minutes? 9 The useful life of a type of a type of washing machines is normally distributed. (Useful life means me the time before the machines needs to be repaired.) The population mean u = 84 months and population standard deviation o = 8 months. The manufacturer wants to write the warranty so only 2% (0.02) of the machines will need to be repaired while still under warranty. For how long should the warranty be written? 10 Extra credit: Use the following data for problems the two extra credit problems below. A community college instructor finds that his students score on an exam is normally distributed with a population mean H = 83 and population standard deviation o = 5. Each problem below is worth 5 extra credit points. 10.a The instructor wants to pass 95% of the class. What should be the minimum passing grade? Hint: what proportion of students will not pass? Where on the normal curve will these students be located? It's a really good idea to draw a picture of the normal curve, and label the portions of the curve where the passes and the not-passes are located. 10.b The instructor wants to give A's to 30% of his students. What should be the minimum grade for an A? Hint: Fill in the following table before you do this problem. What proportion of students will not get As? Where on the normal curve will the students who do not get As located? Left or right portion of the curve? Where on the curve are the students who do get As located