In the first quarter of 2017, a group of mutual funds had a...

In the first quarter of 2017, a group of mutual funds had a mean return of 6.4% and a standard deviation of 2.6%. The returns were well-described by a Normal model. According to the Normal model N(6.4%,2.6%) determine what percentage of this group of funds you would expect to have the following returns.\ a) Over 6.8%?\ c) More than 1%?\ b) Between 0% and 7.6%?\ d) Less than 0%?\ a) The expected percentage of returns that are over 6.8% is %.\(Type an integer or a decimal rounded to one decimal place as needed.)

Transcript

00:01 For this exercise, we consider some random variable, let's call it x, that's normally distributed with a mean of 6 .9 % and a standard deviation of 2 .6%.

00:13 And for part a, we are asked what percentage are over 6 .8%.

00:21 So this is the probability that x is more than 6 .8.

00:28 Graphically, if this represents our normal distribution for the returns, it has a mean of 6.

00:34 0 .9%, which is right in the middle, and the standard deviation of 2 .6%.

00:40 6 .8 is somewhere around here.

00:44 The probability that x is bigger than 6 .8 is equal to the area under the curve and to the right of 6 .8.

00:52 So that corresponds to the area of this blue -shaded region.

00:58 Since the total area under the curve adds up to 1, the area of the blue -shaded region is equal to 1 minus the area under the curve to the left of 6 .8, which is 1 minus the area of the red -shaded.

01:10 Shaded region.

01:12 So we can re -express this as 1 minus probability that x is less than or equal to 6 .8.

01:21 Now let's use excel to solve this problem.

01:24 So we have two terms.

01:26 We have 1 minus the probability that x is at most of 6 .8.

01:30 So if we go to excel, start a computation with an equal sign.

01:34 We have 1 minus, and then we want to use the normal distribution function, which is highlighted here in blue.

01:39 We select that.

01:40 For the first argument, we enter 6 .8, and then we enter the mean and the standard deviation of the normal distribution.

01:49 We hit true for the cumulative argument because we want the probability that x is anything up to 6 .8, and we hit enter, and we get .5153.

02:08 And then for b, we want the probability that a return is between 0 and 7 .6%.

02:13 Actually, for a, we should express this in percent, since that's how the question is worded.

02:17 So that's 51 .53%.

02:22 For b, we want the probability that x is between 0 and 7 .6.

02:33 And this can be re -expressed as the probability that x is less than equal to 7 .6, minus the probability that x is less than 0.

02:50 So the first term can be thought of as the total area under the curve to the left of 7 .6...

Question

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