Data Analysis with Microsoft Excel: Updated for Office 2007

218 Fundamentals of Statistics

1. Explore the following statistical concepts:

a. Defi ne the term random variable.

b. How is a random variable different

from an observation?

c. What is the distinction between x

and m?

2. A sample of the top 50 women-owned
   businesses in Wisconsin is undertaken.

Does this constitute a random sample?

Explain your reasoning. Can you make
any inferences about women-owned
businesses on the basis of this sample?

3. The administration counts the number
   of low-birth-weight babies born each
   week in a particular hospital. Assume,
   for the sake of simplicity, that the rate of
   low-birth-weight births is constant from
   week to week.
   a. Of the distributions that we have
   studied, which one is applicable
   here?
   b. If the average number of low-
   birth-weight babies is 5, what is the
   probability that no low-birth-weight
   babies will be born in a single week?
   c. The administration counts the low-
   birth- weight babies every week and
   then calculates the average count for
   the entire year. What is the approxi-
   mate distribution of the average?


4. The results of fl ipping a coin follow a
   probability distribution called the

Bernoulli distribution. A Bernoulli dis-
tribution has two possible outcomes,
which we’ll designate with the numeric
values 0 and 1. The probability function
for the Bernoulli distribution is

Bernoulli Distribution

P^1 Y 5125 p, P^1 Y 502512 p 0 ,p, 1

where p is between 0 and 1. For exam-

ple, if we tossed an unbiased coin and

indicated the value of a head with 1 and

a tail with 0, the value of p would be .5

since it is equally likely to have either a

head or tail.

The mean value of the Bernoulli distri-

bution is p. The standard deviation

is !p^112 p^2. In the fl ipping coin

example, the mean value is equal

to 0.5 and the standard deviation is

!0.5^112 0.5^25 0.5.

a. You toss a die, recording a 1 for the

values 1 through 3, and a 0 for values

4 through 6. What is the mean value?

What is the standard deviation?

b. You toss a die, recording a 1 for a

value of 1 or 2, and a 0 for the values

3, 4, 5, and 6. What is the mean value?

What is the standard deviation?

c. You toss a die, recording a 1 for a value

of 1, and a 0 for all other values. What

is the mean value? What is the stan-

dard deviation?

5. If you fl ip 10 coins, what is the probabil-
   ity of getting exactly 5 heads? To answer
   this question, you have to refer to the
   Binomial distribution, which is the dis-
   tribution of repeated trials of a Bernoulli
   random variable. The probability func-
   tion for the Binomial distribution is
   Binomial Distribution

P^1 Y 5 y^25 a

n

y

bpy^112 p^2 n^2 y y 5 0,1,2,c, n

where a

n

y

b 5

n!

y!^1 n 2 y^2!

Exercises