Acertain stock market had a mean return of 2.4% in a recent year. assume that the returns for stocks on the market were distributed normally, with a mean of 2.4 and a standard deviation of 10. complete parts (a) through (g) below. a. if you select an individual stock from this population, what is the probability that it would have a percentage return less than 0 (that is, a loss)? . (round to four decimal places as needed.) b. if you select an individual stock from this population, what is the probability that it would have a percentage return between negative 12 and negative 19? (round to four decimal places as needed.) c. if you select an individual stock from this population, what is the probability that it would have a percentage return greater than negative 4? (round to four decimal places as needed.) d. if you select a random sample of four stocks from this population, what is the probability that the sample would have a mean percentage return less than 0 (a loss)?

a) 0.4052; b) 0.0587; c) 0.7389; d) 0.3156

Step-by-step explanation:

We use z scores to find these probabilities.  The formula for the z score of an individual value is

Our mean, μ, is 2.4 and our standard deviation, σ, is 10.

For part a,

We want P(X < 0).  Plugging this value into our formula for a z score, we have

z = (0-2.4)/10 = -2.4/10 = -0.24

Using a z table, we see that the area under the curve to the left of this value is 0.4052.  This is the probability x is less than 0.

For part b,

We want P(-19 < X < -12).  We find the z score of both endpoints and then subtract their areas:

z = (-19-2.4)/10 = -21.4/10 = -2.14

z = (-12-2.4)/10 = -14.4/10 = -1.44

The area under the curve to the left of z = -2.14 is 0.0749.  The area under the curve to the left of z = -1.44 is 0.0162.  This means the area between them is

0.0749-0.0162 = 0.0587.

For part c,

We want P(X > -4).  Since the z table gives us the area under the curve to the left of each value, we will find P(X < -4) and then subtract that from 1:

z = (-4-2.4)/10 = -6.4/10 = -0.64

The area under the curve tot he left of this is 0.2611.  This makes P(X > -4)

1 - 0.2611 = 0.7389.

For part d,

We will use the formula for a z-score of a sample mean:

We have the same mean and standard deviation as in all the other parts.  The sample size is 4.  

For P(X < 0),

z = (0-2.4)/(10÷√4) = -2.4/(10÷2) = -2.4/5 = -0.48

The area under the curve to the left of this is 0.3156.

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