STAT-UB.0103 SPRING 2012
Homework Set 4
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1. If Z is a standard normal random variable, find the probability that
a.
0  Z  1.24
d.
-0.88 < Z < 0.14
b.
Z > -0.45
e.
-1.13 < Z  -0.92
c.
| Z |  0.80
f.
| Z |  0.70
2. If Z is a standard normal random variable, find the values of w that satisfy the
following. These need not be done by interpolation; just use the closest table value.
a.
b.
P[ Z > w ] = 0.32
P[ Z  w ] = 0.71
c.
d.
P[ Z  w ] = 0.14
P[ -w  Z  w ] = 0.60
You can use Minitab as well. You’ll need Calc  Probability Distributions 
Normal and the Inverse cumulate probability feature.
3. The diameters of apples from Happy Mac Orchard have diameters which are
approximately normally distributed with mean  = 2.8 inches and standard deviation
 = 0.3 inch. Apples can be size-sorted by being made to roll over a mesh screens. At
this farm, the steps are done sequentially.
First, the apples are rolled over a screen with mesh size 2.5 inches. This separates
out all apples with diameters < 2.5 inches.
Second, the remaining apples are rolled over a screen with mesh size 3.3 inches.
This separates out all apples with diameters between 2.5 and 3.3 inches.
All the apples will now be separated into three groups.
a.
Find the proportion of apples with diameter < 2.5 inches.
b.
Find the proportion of apples with diameters between 2.5 and 3.3 inches.
c.
Find the proportion of apples with diameters greater than 3.3 inches.
HINT: If X represents the diameter of a random apple, and if the screen has a mesh
size m, then P[X < m] represents the proportion of apples which will fall through.
4. The chocolate chip cookies that are produced at Perry’s Cookie Emporium have
weights which are approximately normally distributed with mean weight 180 grams and
with standard deviation 20 grams. The cookies, however, are sold by count, not by
weight. This is a high-markup business, and Perry wants to improve his image. He
decides to set aside lightest 20% of the cookies to be packaged and sold separately. What
cookie weight will divide the lightest 20% from the heaviest 80% ?
5. Clyde’s Deli is situated inside a large industrial park. The weekday gross sales at
Clyde’s average $2,480, with a standard deviation of $360. Find the probability that the
average over the next 50 weekdays will exceed $2,400. Please note the assumptions that
are used in making the calculation.
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STAT-UB.0103 SPRING 2012
Homework Set 4
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6. Examine the file CHS\GEYSER1.MTP. You can get this from the Web at
www.stern.nyu.edu/~gsimon/statdata. Look for the CHS folder.
Column C2 gives the duration, in minutes, of eruptions of the Old Faithful Geyser in
Yellowstone National Park. Column C3 gives the interval, in minutes, until the following
eruption. The concern here is whether the data in these columns follow a normal
distribution. Here we’ll just examine C2. (Column C3 is qualitatively very similar to
C2.)
Use Graph  Probability Plot to decide whether C2 follows a normal distribution.
What conclusion do you reach? You might also try Graph  Histogram to check
whether there is a simple description.
7. You are about to take a sample from a population in order to use X to estimate .
Here the population consists of items with monetary values. You would like the error
of estimate to be governed by the condition P[ | X -  |  $5 ]  0.80. If you think that
, the population standard deviation, could be as large as $40, find the smallest sample
size n which will allow you to satisfy the condition.
HINT: Assume that  = $40. If  is really smaller, then you will still satisfy
the condition.
HINT: You will need to find a normal table point w so that P[ | Z |  w ] = 0.80.
 z 
HINT: The formula n    / 2  will be useful.
 E 
2
8. A population of daily sales figures is approximately normally distributed with a mean
of $14,000 and a standard deviation of $3,000.
(a)
You’d like to predict tomorrow’s sales. (Yes, this is an inference
question.) Give an interval (a, b) for which you are will to say that the
probability is about 95% that tomorrow’s sales will be between a and b.
(b)
You’d like to predict the average sales over the next 15 business days.
(This is another inference question.) Give an interval (c, d) for which you
are will to say that the probability is about 95% that the sales over the next
15 business days will be between c and d.
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STAT-UB.0103 SPRING 2012
Homework Set 4
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9. You have been tracking the “cash reserve” of a very large mutual fund, hoping to find
clues about future behavior. The cash reserve is currently at 4.32. The units here are
millions of dollars. You believe that the daily changes to this reserve are normally
distributed, with mean -0.02 and with standard deviation 0.06. Find the probability that
the reserves will be below 3.90 after 25 days.
10. The Vindicator Mutual Fund family has a fund that tracks industrial metals. This is
currently trading at $14.20 per share. It is believed that the daily price follows a
lognormal random walk with drift  = +0.02 and with volatility expressed through
 = 0.15. Find the probability that the price after thirty trading days will exceed $16.00.
HINT: Recall that the lognormal random walk is governed through
Pn = P0 e X1  X 2 ... X n , where the Xi’s are normal with mean  and standard
deviation .
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