OPRE504
Chapter Study Guide
Chapters 9
I
Confidence Intervals for Proportions
One Proportion Z-Interval
1.
Find standard error of sampling distribution
𝑝̂𝑞̂
SE(𝑝̂ ) = √ 𝑛 , when 𝑝̂ = sample proportion of a success, 𝑞̂ = 1- 𝑝̂ , n = sample size
2.
Calculate Critical Value (Z*) for a probability associated with the stated confidence level,
using Two-way Z-table (A-32/33), where one-way probability = confidence level + (1 –
confidence level)/2
𝑝̂𝑞̂
3.
ME = z* x SE(𝑝̂ ) = z* x√ 𝑛
4.
CI = 𝑝̂ ± ME = 𝑝̂ ± z* x SE(𝑝̂ ) =[ (𝑝̂ -z* x SE(𝑝̂ ), 𝑝̂ + z* x SE(𝑝̂ ) ]
Q9.1 [Sharpe 2011, Exercise 25, p.282] A small company involved in e-commerce is interested
n statistics concerning the use of email. A poll found that 38% of a random sample of 1012
adults, who use a computer at their home, work or school, said they do not send or receive email.
a)
Find the margin of error for this poll if we want 90% confidence in our estimate of
the percent of American adults who do not use email.
b)
If we want to be 99% confident, will the margin of error be larger or smaller?
c)
Find the 99% confidence interval for the proportion of adults who use email.
Q9.2 [Sharpe 2011, Exercise 15, p.315] In 2008, a Gallup Poll asked 2336 U.S. adults aged 18
or over, how they rated economic conditions. In a poll conducted from January 27 through
February 1, 2008, 24% rated the economy as Excellent/Good. A recent media outlet claimed that
the percentage of Americans who felt the economy was in Excellent/Good shape was, in fact,
28%. Does the Gallup Poll support this claim?
Chaodong Han
OPRE504 Data Analysis and Decisions Class Handout
Page 1 of 3
a) Find a 95% confidence interval for the proportion of U.S. adults who rated economy
Excellent / Good.
b) Does your confidence interval provide evidence to support the Media’s claim?
c) What is the significance level of the test in this case?
More Exercises:
Guided example – Credit Card Promotion (pp.301-302)
Textbook Exercises:
Chapter 9: 27, 28, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43,44, 45, 46, 47,48, 49, 50, 60, 61, 62,
63, 64
II
Determine Sample Size
𝑝𝑞
1. Desired Margin of Error: ME = Z*x SE(𝑝̂ ) = Z*x √ 𝑛
2. Z* is associated with the one-way probability corresponding to the confidence level,
where one-way probability = confidence level + (1 – confidence level)/2
3. If we have no priori knowledge about the proportion, we have to start with a random
probability p=.50 and q=0.50 (the worst scenario).
4. Solve for n
Q9.3 In preparing a report on the economy, we need to estimate the percentage of business that
plan to hire additional employees in the next 60 days.
a)
How many randomly selected employers must we contact in order to create an
estimate in which we are 98% confident with a margin of error of 5%?
Chaodong Han
OPRE504 Data Analysis and Decisions Class Handout
Page 2 of 3
b)
Suppose we want to reduce margin of error to 3%, what sample size will suffice?
c)
Why might it not be worth the effort to try to get an interval with a margin of error
of only 1%?
More exercises:
Textbook example: How Much of A Difference Can It Make? (p.272)
Textbook Chapter 9: 51, 52, 53, 54, 55, 56, 57, 58, 59
Chaodong Han
OPRE504 Data Analysis and Decisions Class Handout
Page 3 of 3