Question

A dishwasher has a mean life of 12 years with an estimated standard deviation of 1.25 years "Appliance life expectancy," 2013). Assume the life of a dishwasher is normally distributed. a.) True or False: The random variable x defines the life expectancy of a dishwasher b.) Find the probability that a dishwasher will last more than 15 years. c.) Find the probability that a dishwasher will last less than 8 years. d.) Find the probability that a dishwasher will last between 8 and 10 years. e) A manufacturer of dishwashers only wants to replace free of charge 5% of all dishwashers. How long should the manufacturer make the warranty period? For part (a): Enter TRUE or FALSE For parts (b) through (d): Enter answers in decimal form rounded to nearest ten-thousandth (4 places after decimal). Examples of correctly entered answers: 0.0000 0.0003 0.0070 0.0100 0.4000 0.6651 1.000 For part (e): Enter answer rounded to nearest whole number of yearls). Do not include units of time.

Answer 1

Given that: - Let X be a dishwasher (life) X n N (12, (1.25)^2) mu = 12 sigma = 1.25 TRUE P[X > 15] = P[X - mu/sigma > 15 - 12/1.25] = P[Z > 2.4] = 1 - P[Z < 2.4] using Z-table, = 1 - 0.9918 = 0.0082 P[X < 8] = P[X - mu/sigma < 8 - 12/1.25] = P[Z < -3.2] using Z-table = 0.0007 \r\n P[8 < X < 10] = P[8 -12/1.25 < X - mu/sigma < 10 - 12/1.25] = P[-3.2 < Z < -1.6] = P[Z < -1.6] - P[Z < -3.2] using Z-table, = 0.0548 - 0.0007 = 0.0541 P[X < X_1] = 0.05 P[Z < X_1 - 12/1.25] = 0.05 using Z-table X_1 - 12/1.25 = -1.645 X_1 = 12 - 2.0563 X_1 = 9.9438 X_1 = 10 (approx. nearest whole number)