Download Introduction to the Practice of Statistics

Introduction to the Practice of Statistics
Sixth Edition
Moore, McCabe
Section 5.1 Homework Answers
5.10 What is wrong? Explain what is wrong in each of the following scenarios.
a) In the binomial setting X is a proportion.
Given what is written in this chapter, X is not a proportion but a count. To be more precise, when you
hear binomial setting, automatically the random variable counts successes. As for the actual symbol
used to represent the count, you as the user need to define it. Meaning there is absoutely nothing
sacred about the letter X representing the count; just like in algebra we can change our variable to any
other symbol besides x.
b) The variance for a binomial count is
p (1 − p )
.
n
The expression shown is the standard deviation for proportions,
σ p̂ .
c) The Normal approximation to the binomial distribution is always accurate when n is greater Aan 1000.
The criteria for using a normal approximation for a binomial distribution is np ≥ 10 and n(1 – p) ≥ 10.
A sample of 1000 should for most cases be enough, however consider the case where p = 0.007.
1000(0.007) = 7 which is not greater than 10, thus, in this scenario a normal approximation is not
appropriate.
5.14 Visits to Web sites. What kinds of Web sites do males aged 18 to 34 visit most often? Pornographic
sites take first place, but about 50% of male Internet users in this age group visit an auction site such as eBay
at least once a month.2 Interview a random sample of 15 male Internet users aged 18 to 34.
(a) What is the distribution of the number who have visited an online auction site in the past month?
The distribution consists of the measurements being made and the probability of the measurements.
Don’t assume because this section is binomial, that the answer is a binomial distribution; verify
instead.
1. Are we counting successes? Yes: the number who have visited an online auction site in the past month?
The question suggests we are counting the males that have visited the online auction site.
2. Fixed sample count? Yes, the situation says we are sampling 15 people.
3. Independence? Technically no, since everytime I sample a male that changes the probability of
the gettting a male who has visited an auction site for the next male sample. But, we are only
sampling 15 males out of a population of millions. Thus, we have for practical purposes
independence since my probability is changing but by very little.
4. Does p stay fixed during sampling? This question you have to ask yourself if some dramatic
event has occurred during the sampling that would cause a change to p? For this question
assume not.
Thus, the distribution is binomial.
(b) What is the probability that at least 8 of the 15 have visited an auction site in the past month?
Let X count the number of males that have visited an on-line auction site.
P(X = 8) = 0.1964
Using Excel
=Binomdist(8,15, 0.5, false)
Using TI83/84
binomialpdf(15,0.5,8)
5.16 Visits to Web sites. Suppose that 50% of male Internet users aged 18 to 34 have visited an auction site
at least once in the past month.
(a) If you interview 15 at random, what is the mean of the count X who have visited an auction site?
What is the mean of the proportion p in your sample who have visited an auction site?
µX = 15(0.5) = 7.5
µ p̂ = 0.5
(b) Repeat the calculations in (a) for samples of size 150 and 1500. What happens to the mean count of
successes as the sample size increases? What happens to the mean proportion of successes?
µX = 150(0.5) = 75
µX = 1500(0.5) = 750
µ p̂ = 0.5
µ p̂ = 0.5
The mean of the count increases as n increases since it is a count, and the sample is increasing.
However, the proportion mean will not change since it is a proportion.
5.30 Scuba-diving trips. The mailing list of an agency that markets scuba-diving trips to the Florida Keys
contains 60% males and 40% females. The agency calls 30 people chosen at random from its list.
(a) What is the probability that 20 of the 30 are men? (Use the binomial probability formula.)
What I want you to recognize is that the situation is binomial.
Use the calculator to get the probability.
Define your variable. Let X count the number of men contacted
out of a sample of 30.
P(X = 20) = 0.1152
Excel: =binomdist(20,30,0.6,false)
TI83/84: binomialpdf(30,0.6,20)
(b) What is the probability that the first woman is reached on the
fourth call? (That is, the first 4 calls give MMME)
P(M And M And M And E) = 0.6(0.6)(0.6)(0.4) =0.0864
This situation is not binomial. Why? This question asks to get a success on the fourth attempt.
Binomial does not do this. The question suggests the sample size is four. A binomial situation can only
calculate the probability of a woman being called on four attempts, not on which particular attempt
the woman is reached as is asked on this question. Again , notice that I assumed I had independence to
answer the question. This is reasonable since my sample size is small relative to the population on the
Florida Keys.