Work Sheet 7: Chapter 13

STAT 113 Week 11 (Mar 18)

Work Sheet 7: Chapter 13

**On all the problems with calculations, please also answer whether the problems are forward or backward and whether they are >, <, or “between” problems.**

Ø Forward/Backward Normal Distribution Problem

1. Rachel wonders how much television kids watch these days. She figures children television usage in a day is Normally distributed with a mean of 2 hours and a standard deviation of 0.3 hours.

(a) What is the range for the bottom 19% of children television usage?

Backward and “” problem

Closest Standardized Score = -0.9 => -0.9*0.3 + 2 = 1.73 hours,

the range for the bottom 19% is 1.73 hours or less

(b) Mark the mean, the answer you got from part (a) and shade the appropriate area that corresponds to the 19% in part (a). Also sketch a Normal curve for part (a) in terms of Z. (Sketch Z = 0, the standardized score, and shade the appropriate area that corresponds to the 19% in part (a).)

Show the mean 2 in the middle, draw a line at 1.73 with shading to the left of that line and labeled with shaded area = 19%.

Show the mean Z=0 in the middle, draw a line at -0.9 with shading to the left of that line and labeled with shaded area =19%.

Closest Standardized Score = 0.7 so 0.7*0.3 + 2 = 2.21 hours

Backward and “” problem

the range for the top 24% is 2.21 hours or more

(d) Mark the mean, the answer you got from part (c) in the curve below and shade in the appropriate area that corresponds to the 24% referred to in part (c). Also sketch a Normal curve for part (c) in terms of Z. (Sketch Z = 0, the standardized score, and shade the appropriate area that corresponds to the 24% in part (c).)

Show the mean 2 in the middle, draw a line at 2.21 with shading to the right of that line and labeled with shaded area = 24%.

Show the mean Z=0 in the middle, draw a line at 0.7 with shading to the right of that line and labeled with shaded area =24%.

2. The annual rate of return on stock indexes (which combine many individual stocks) is very roughly Normal. Since 1945, the Standard & Poor’s 500 index has had a mean yearly return of 12%, with a standard deviation of 16.5%. Take this Normal distribution to be the distribution of yearly returns over a long period.

(a) In what range do the middle 95% of all yearly returns lie?

Backward and “between” problem

Use Empirical Rule:

95%=>Mean ± 2 standard deviation = (12-2*16.5, 12+2*16.5)=(-21%, 45%)

(b) Mark the mean, the answer you got from part (a) in the curve below and shade in the appropriate area that corresponds to the 95% referred to in part (a). Also sketch a Normal curve for part (a) in terms of Z. (Sketch Z = 0, the standardized score, and shade the appropriate area that corresponds to the 95% in part (a).)

Show the mean 12% in the middle, draw lines at -21% and 45% with shading between two lines and labeled with shaded area = 95%.

Show the mean Z=0 in the middle, draw lines at -2 and 2 with shading between two lines and labeled with shaded area = 95%.

Backward and “” problem

Use Empirical Rule: Mean - 1 standard deviations= 12-16.5= - 4.5%

If we do this as a backward problem using the Normal table the answer will be the same: Closest Standardized Score = -1.0 so -1.0*16.5 + 12 = -4.5%

(d) The market is down for the year if the return on the index is less than zero. In what proportion of years is the market down?

Forward and “” problem

X=~N (μ=12%, σ=16.5%)

P(X<0) = P(Z < 0-1216.5) = P(Z < -0.7)=24.20%

(e) In what proportion of years does the index gain 25% or more?

Forward and “” problem

P(X > 25%) = 1-P(X < 25%)

=1-P(Z < 25-1216.5) =1-P(Z < 0.8)=100%-78.81%=21.19%

(f) How high the return is in the 90th percentile?

Backward and “” problem

The closest standardized score: 1.3

1.3*16.5+12=33.45(%)

(g) How high the return is in the highest 5% of all yearly returns?

Backward and “” problem

The closest standardized score: 1.6

1.6*16.5+12=38.4(%)