Chapter 19: Confidence Intervals
Warning: This guided worksheet should not replace your reading of the textbook but should be used to
enhance what you read in the textbook.
The goal of confidence intervals is to estimate a parameter. A parameter summarizes a
characteristic of a population. Parameters are typically, and one could argue the case always,
never known. Parameters are thought to be the true values. The best that we can achieve is an
estimate, which is a statistic. A statistic estimates a parameter. The concept of a confidence
interval is very general, but in Chapter 19 we will use the population proportion p as an example.
We could have used the population mean, µ, as the parameter to estimate.
Key vocabulary –
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Point estimate
Range estimate
Confidence level
Margin of Error
Standard Error
Critical Values (Z*)
Sample size (n)
Parameter
Statistic
Key concepts
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How does a confidence interval
work?
How do you interpret the result of
a confidence interval
Determine the sample size.
Normal Approximation – is the
sample size large enough?
Independence
Point Estimates and the First Condition
1. During an election year, a random set of voters is selected an asked if they voted for the Democratic
nominee? The voters are contacted by looking at county records of those who voted, the names are
enumerated, selected randomly by a random number generator, and then contacted by city officials.
Below is the response from 20 randomly selected voters in Multnomah County. The “D” means the
person voted for the Democratic nominee and O means they voted for another nominee.
{D, D, D, D, O, D, O, O, D, D, D, D, D, O, O, D, O, O, O, D}
A. What is the value of p̂ the proportion of voter’s samples that voted Democratic?
B. What is the general name for the value you just calculated in part 1a?
C. Is the method described to find the sample proportion have the potential to be biased as far as you
can tell?
2. What is the passing rate of MTH 243 at PCC? All students in Jessica Bernard’s MTH 243 class were
sampled. Below is the result. The letter P indicates that the person sampled Pass while N indicates that
the person did not pass.
{P, P, P, P, P, P, P, P, P, P, P, P, N, P, P, P, P, N, N, N, N, P, P, P, P, P, P, N}
A. What is the value of p̂ the proportion of voter’s samples that voted Democratic?
B. What is the general name for the value you just calculated in part 1a?
C. Is the method described to find the sample proportion have the potential to be biased as far as you
can tell?
3. So what do you think is a necessary condition about our sampling method if we are attempting to
estimate a parameter?
Required Condition Two for our Particular Calculation Method: Sample Size
In order for us to use the formula to estimate p found in this particular chapter, the
sample size must meet the criteria np  10 and n(1 – p)  10 which interprets to we need
at least ten successes and ten failures.
4. Why do you suppose we need to meet the mentioned criteria above regarding sample size?
5. What is the smallest sample we can have in order to estimate p using the methods of this chapter?
6. Consider the following scenarios determine which meet the required criteria.
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Question 1 of this worksheet.
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Question 2 of this worksheet.
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Do people like the taste of product A over product B. A blind taste test of 120 participants
showed that 80 liked product A and the rest chose product B.
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Are landline telephones as the primary telephone in the minority in Washington County
households? A random sample of 500 telephone users showed that 42% still had landline
telephones as their primary phone.
What is a Confidence Interval and How Does it Work
A point estimate attempts to estimate a parameter using a single value. For example p̂ is an
estimate of the population proportion p. The statistic p̂ is a statistic. Likewise in order to estimate
, the population mean we can use the point estimate, x . A confidence interval attempts to
estimate a parameter using a range of numbers instead of a single number; it is a range estimate. In
the center of this range estimate is our statistic, point estimate.
The graphs shown below are idealized distributions of p-hat for a given sample size that meets the
criteria np  10 and n (1 – p)  10; need at least 10 successes and 10 failures. Note that while the graphs
shown below are continuous the distribution of p-hat is discrete. We are using a continuous model in
our procedure to simplify the procedure. What we are suggesting is that the method we are about to
employ while not correct gives us good results when attempting to estimate the parameter p. The
correct method would not produce significantly better results.
The Chapter 19 Formula to Estimate p
*
The formula we will use p̂  z
ˆ  p)
ˆ
ˆˆ
p(1
* pq
,or as written in your text p̂  z
,to estimate p
n
n
assumes that np  10 and n(1 – p)  10, meaning we need at least 10 successes and 10 failures
using binomial distribution language. Study this formula carefully!!
7. What is the Standard Error and how is it different than the Standard Deviation?
8. What is the margin of error?
9. What is the confidence level?
9. What is z* and is it different than z?
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What is the z* value for a 95% confidence interval?
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What is the z* value for a 99% confidence interval?
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What is the z* value for a 85% confidence interval?
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What is the z* value for a 86% confidence interval?
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What is the z* value for a 70% confidence interval?
10. Let us take the formula out for a drive around the block with some examples. For all the problems
below write down if we meet the necessary conditions: sample size large enough, does the method
seem to have bias, do we have independence between observations. Lastly, interpret the result.
a) What is the proportion of students that would pass MTH 60 using only ALEKS (instructional
software) to instruct students? Five classes are randomly selected to participate. Out of 157
students in the five courses 110 passed the course. Use a confidence interval to estimate the
proportion of students in the general population that would pass MTH 60 only using ALEKS.
b) Two weeks before a major election for mayor of a city, the local political radio host at a popular
radio station conducts a poll to see how many people will vote for candidate A for mayor who
the host of this radio show supports. The radio host asks his listeners to go to the radio stations
web site where they can select the candidate they will vote for. Of the 320 people who
participated in poll, 275 denoted that they would vote for Candidate A.
c) The Gallup organization randomly selects people to call via phone in the state of Washington to
find out how many support making same sex marriage legal in the state; the method used to
select phone numbers using a random number generator. Out of the 1000 people contacted by
phone, 563 support same sex marriage.
Some Basic Questions
11. We take a random sample using appropriate procedures. We intended to use a 90% confidence
interval. Once the data is collected your boss says change the confidence level to 95%. How does the
change from 90% to 95% affect the margin of error? Does it get bigger, smaller or stay the same?
12. Suppose we want to estimate the number of citizens in Oregon that would support a sales tax if
income tax and property taxes were lowered. Which would yield a lower margin of error a sample size
of 1000 or a sample size of 4000?
13. True or False - The margin of error will take care of (account for) samples that are biased.
14. True or False – In general, for a fixed sample size, as the confidence level increases the margin of
error decreases.
15. The graph shows the relationship between
different values of p and how it affects the
standard deviation of p-hat.
a. What value of p produces the largest standard
deviation? At what value of p does is the margin
of error the largest for a fixed confidence level?
b. What happens to the standard deviation as p gets closer to 1 or 0? What happens to the margin of
error as p gets close to 1 or 0 for a fixed confidence level?
16. The graph shows the relationship between the
sample size and the margin of error. For this particular
scenario the population proportion p is fixed at 0.5, and
the confidence level is fixed at 95%.
a. What happens to the margin of error as the sample
size increases?
b. In your opinion, at approximately what sample size
does it seem that the ability to lessen the margin of
error significantly reaches is optimal point?
Simulation
It is difficult to appreciate what the confidence interval can say and what it does not say unless you run a
simulation. For this activity you will need a partner. You and your partner will switch roles. One of the
partners will know the true population parameter, the other will be blind to this knowledge. Now the
partner that knows the true value is never, and I mean NEVER to reveal that value to the other partner.
Not inside the classroom, later outside the classroom or next to the person in their death bed. NEVER!!!
And here is the reason why this is to be so. I want the person that does not know the value to get the
gut feeling, what it really feels like not knowing and having to trust only the result of the confidence
interval.
The scenario – You are developing a product that you plan to sell. You want to know what proportion
of the intended target audience for your product will buy your product. You gather a group of paid
volunteers. The group is to fill out detailed questionnaires in order to see if they fit the target audience
for the product. Of the volunteers that fit the required profile, a randomly selected group is selected to
try the product. A sample of 490 is
Interpretation of a Confidence Interval
17. What proportion of teens in Florida are