Business Mathematics I
Homework 9
Prepared for
Stephen Reyes
Math 115a, Section
University of Arizona
By
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Submitted on
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We, the undersigned, affirm that all of us participated fully and equally in the completion of this
assignment and that the work contained herein is original. Furthermore, we understand that
sanctions will be imposed individually as well as jointly if any part of this work is found to
violate the Student Code of Conduct, the Code of Academic Integrity, or the policies and
procedures established for this course.
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1. (Exercise 1 from Random Sampling) Make 16 one-inch square slips of paper, labeled with
the following numbers.
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7
Let X be the random variable that is the number on a slip, chosen at random from your pieces
of paper. (i) Assuming that you are equally likely to draw any given slip, use what you know
about probability to compute the actual value of E ( X ) . A random sample of observations of
X can be formed by successively drawing slips at random from the pieces of paper. In order
for the samples to each observe X, you must replace each slip that you draw before making
the next draw. (ii) Select a random sample of size 5, and compute its mean. (iii) Collect
nine more random samples and compute their means. How do the sample means compare
with the actual expected value, E ( X ) , of X?
Solution.
2. (Exercise 3 from Random Sampling) A random sample of ten starting salaries in your field
yields $31,600, $22,900, $28,200, $28,500, $34,000, $30,000, $25,900, $26,400, $27,100,
and $29,700. Let S be the random variable giving the starting salary of a randomly selected
person in your field. (i) Use the sample to estimate  S . (ii) What does this number mean in
the real world of jobs and salaries?
Solution.
3. (Exercise 4 from Random Sampling) Combine the 50 observations of the random variable X
that you made in Problem 1 into one large sample. (i) Use this data to estimate E ( X ) . (ii)
How well does this estimate compare with the actual expected value of X?
Solution.
4. (Exercise 10 from Random Sampling) In previous exercises we have considered records of
daily sales kept by a small business. These data are given in the sheet Raw Data of Daily
Sales.xls. Sorting and Copy/Paste can be used to separate the records by the day of the week.
Let GM be the random variable giving the gross sales on a Monday. We will treat GM as a
continuous random variable, and assume that the 200 observations of GM are a random
sample. Use suitable grouping of the sample data to plot bar and line graph approximations
for the p.d.f., f GM , of GM .
Solution.
5. (Exercise 12 from Random Sampling) (i) Use the random sample from A that is provided by
your team’s data to plot an approximation for the p.d.f., f A . (ii) Create a line graph that
displays both your approximation of f A and the p.d.f. for the exponential random variable
with the same mean as your random sample from A.
Solution.
6. (Exercise 13 from Random Sampling) Redo Problem 5 using the random variable B instead
of A.
Solution.
7. (Exercise 14 from Random Sampling) Use the random sample from S that is provided by
your team’s data to plot an approximation for the p.d.f., f S .
Solution.
8. (Exercise 20 from Random Sampling) The business whose daily sales data are shown in the
Excel file Daily Sales.xls, will be eligible to apply for a federal assistance program if its
gross sales on 8 randomly selected business days are all under $7,000. (i) Use the functions
RANDBETWEEN, VLOOKUP, and MAX in the sheet Raw Data to simulate 3,000 sets of
eight-day gross sales records. (ii) Use the COUNTIF function to estimate the probability
that the business will be eligible for the federal program.
Solution.
9. (Exercise 26 from Random Sampling) (i) Create an Excel file that generates and displays
bootstrap random samples of any size from 1 to 300 from your team’s data for S. Use
formatting and conditional formatting as in the worksheet Random Sampling in Queue
Focus.xls. (ii) Create cells that calculate the minimum, mean, and maximum of your
samples.
Solution.