The average return for large-cap domestic stock funds over the three years 2009-2011 was 14.4%.† Assume the three-year returns were normally distributed acro: with a standard deviation of 4.4%. (a) What is the probability an individual large-cap domestic stock fund had a three-year return of at least 18%? (b) What is the probability an individual large-cap domestic stock fund had a three-year return of 15% or less? (c) How big does the return have to be to put a domestic stock fund in the top 20% for the three-year period? Step 1 (a) What is the probability an individual large-cap domestic stock fund had a three-year return of at least 18%? It is given that the average return for large-cap domestic stock funds over a three-year time span was 14.4% with a standard deviation of 4.4%. Let x represent an individual large-cap domestic stock fund. The probability that this is at least 18% results in the probability statement P(x >✔ Step 2 Before finding this probability, the random variable x needs to be converted to the normal random variable z so that a table of probabilities can be used. Recall the 1 to convert an x value to the normal random variable, z, where x is the value that needs to be converted, u is the population mean, and is the population standard. deviation. z = The mean was given to be 14.4% and the standard deviation was given to be 4.4%. Find the value of the z-statistic corresponding to x = 18, rounding the result to decimal places. = = x-μ J 18 14.4 0.818✔✔ x-μ 4.4 Z 0.82 Thus, the new probability statement using the normal random variable z is P(Z > 0.818 ✔ 14.4 0.00 Step 3 The probability statement was determined to be P(z ≥ 0.82). Recall that the normal probability table below gives the area under the curve to the left of a given z-va the entire area under this curve is 1. Here, we want the area to the right of z = 0.82, so we can subtract the area to the left of z = 0.82 from 1. 0.01 0.02 = 0.2061 ✔ 0.03 0.04 0.05 0.7939 0.06 0.2061 0.82 ). 0.07 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 0.08 ≥ 18). 0.09 Use the table excerpt above to find the probability that an individual large-cap domestic stock fund has a three-year return of at least 18%, rounding to four decima P(Z ≥ 0.82) = 1 - 0.7939 0.8106 0.8133