BASIC PRACTICE OF STATS-LL W/SAPLINGPLU
BASIC PRACTICE OF STATS-LL W/SAPLINGPLU
8th Edition
ISBN: 9781319216245
Author: Moore
Publisher: MAC HIGHER
Question
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Chapter 3, Problem 3.47E

(a)

To determine

To obtain: The proportion of months with returns greater than 0 and the proportion of months with returns greater than 4%.

(a)

Expert Solution
Check Mark

Answer to Problem 3.47E

The proportion of months with returns greater than 0 is 0.5832 and proportion of months with returns greater than 4% is 0.2206.

Explanation of Solution

Given info:

The distribution of the 369 monthly returns follows a normal distribution with mean of 0.84% and standard deviation of 4.097%.

Calculation:

For proportion of months with returns greater than 0:

Define the random variable x as percentage monthly returns.

The formula for the standardized score is,

z=xμσ

The months with returns greater than 0 is denoted as x>0 .

Subtract the mean and then divide by the standard deviation to transform the value of x into standard normal z.

x0.844.097>00.844.097>0.844.097z>0.21

Where, the standardized score z=x0.844.097

The proportion of months with returns greater than 0, is obtained by finding the area to the right of –0.21 but, the Table A: Standard normal cumulative proportions apply only for cumulative areas from the left.

Use Table A: Standard normal cumulative proportions to find the area to the left of –0.21.

Procedure:

  • Locate –0.2 in the left column of the A-2 Table.
  • Obtain the value in the corresponding row below 0.01.

That is, P(z<0.21)=0.4168

The area to the right of –0.21 is,

P(z>0.21)=1P(z<0.21)=10.4168=0.5832

Thus, the proportion of months with returns greater than 0 is 58.32%.

For proportion of months with returns greater than 4%:

The months with returns greater than 4% is denoted as x>4 .

Subtract the mean and then divide by the standard deviation to transform the value of x into standard normal z.

x0.844.097>40.844.097>3.164.097z>0.77

Where, the standardized score z=x0.844.097

The proportion of months with returns greater than 4% is obtained by finding the area to the right of 0.77. But, the Table A: Standard normal cumulative proportions apply only for cumulative areas from the left.

Use Table A: Standard normal cumulative proportions to find the area to the left of 0.77.

Procedure:

  • Locate 0.7 in the left column of the A-2 Table.
  • Obtain the value in the corresponding row below 0.07.

That is, P(z<0.77)=0.7794

The area to the right of 0.77 is,

P(z>0.77)=1P(z<0.77)=10.7794=0.2206

Thus, the proportion of months with returns greater than 4% is 22.06%.

(b)

To determine

To obtain: The proportion of actual returns greater than 0 and the proportion of actual

returns greater than 4%.

To check: The whether the results suggest that N(0.84, 4.097) provides a good approximation to the distribution of returns over this period.

(b)

Expert Solution
Check Mark

Answer to Problem 3.47E

The proportion of actual returns greater than 0 is 0.6264 and the proportion of actual returns greater than 4% is 0.2213.

Yes, results suggest that N(0.84, 4.097) provides a good approximation to the distribution of returns over this period.

Explanation of Solution

Given info:

The data shows the percentage of returns on common stocks. From the data, the total number of returns is 348, the actual returns greater than 0 is 218 and the actual returns greater than 4% is 77.

Calculation:

For proportion of actual returns greater than 0:

The formula to find the proportion of actual returns greater than 0 is,

Proportion of actual returns greater than 0  = Actual returns greater than 0Total number of returns

Substitute 218 for ‘Actual returns greater than 0’, 348 for ‘Total number of returns’.

Proportion of actual returns greater than 0 = 218348=0.6264

Thus, the proportion of actual returns greater than 0 is 0.6264.

For proportion of actual returns greater than 4%:

The formula to find the proportion of actual returns greater than 4% is,

Proportion of actual returns greater than 4%  = Actual returns greater than 4%Total number of returns

Substitute 77 for ‘Actual returns greater than 4%’, 348 for ‘Total number of returns’.

Proportion of actual returns greater than 4% = 77348=0.2213

Thus, the proportion of actual returns greater than 4% is 0.2213.

Comparison:

The percentage of months with returns greater than 0 is 58.32% and the percentage of months with returns greater than 4% is 22.06%.

Using normal distribution, the percentage of months with returns greater than 0 is 62.64% and the percentage of months with returns greater than 4% is 22.13%. Therefore, the percentages are approximately equal.

Thus, the results suggest that N(0.84, 4.097) provides a good approximation to the distribution of returns over this period.

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