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.**
 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%.
(c) What is the range for the top 24% of children television usage?
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%.
(c) The bottom 16% of all yearly returns will be lower than what percent?
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 <
𝟎−𝟏𝟐
𝟏𝟔.𝟓
) = 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 <
𝟐𝟓−𝟏𝟐
𝟏𝟔.𝟓
) =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(%)