SECTION 7.2

Estimating a Population Proportion

Where Have We Been?

In Chapters 2 and 3 we used “descriptive statistics”.

We summarized data using tools such as graphs, mean, and standard deviation.

Where Are We Going?

In Chapter 6 we began using “inferential statistics”.

We will be using sample data to make inferences about population parameters.

Goals – Ideas to Understand

1. How to find the best point estimate of the population proportion.

2. How to construct and interpret confidence intervals.

3. How to find the sample size necessary to estimate a population proportion.

Topic 1 – Point Estimate of a Population

Definition Point Estimate: A single value (or point)

used to approximate a population parameter.

Important Notes If we want to estimate a population

proportion with a single value, the best estimate is the sample proportion . Unbiased (Section 6.7) Most consistent

Topic 1- Example

In a Pew Research Center poll, respondents were asked “From what you’ve read and

heard, is there solid evidence that the average temperature on earth has been increasing

over the past few decades, or not? 70% of 1501 randomly selected adults in the

U.S answered yes.

Find the best point estimate of the proportion of all adults in the U.S who

believe in Global Warming. The best estimate is 70%.

Topic 1 - Flaws

70% was our best point estimate of the population proportion p, but we have no idea of just how GOOD our best estimate

is.

THINK: I could give you my best estimate for how awesome cats are.

However, given my experience is based on only two cats, how good of a estimate

is that for every cat?

Introducing …

TOPIC 2 – CONFIDENCE INTERVALS

Topic 2 – Confidence Intervals

Definition Confidence Interval: A range (or interval) of

values used to estimate the true value of a population parameter. Abbreviated CI.

Important Notes A confidence level is expressed as the

probability or area 1 – , where is the complement of the confidence level. Most common choices are 90%, 95%, or 99%.

This means (α = 10%), (α = 5%), (α = 1%)

Topic 2 – Interpreting Confidence Intervals

Correct “We are 95% confident that the interval from

0.677 to 0.723 actually does contain the true value of the population proportion p.”

This means that if we were to select many different samples of size n and construct the corresponding confidence intervals, 95% of them would actually contain the value of the population proportion p.

Refers to the process.

Topic 2 – Interpreting Confidence Intervals

Incorrect “There is a 95% chance that the true value

of p will fall between 0.677 and 0.723.” “95% of sample proportions fall between

0.677 and 0.723.” “Mr. Llorens is more of a slick daddy than

Ms. Pobuda.”

A confidence interval either contains p or it does not. That is why it is

incorrect to say there is a 95% chance that p will fall between 0.677 and

0.723.

Topic 2 – Critical Values

A standard z score can be used to distinguish between sample statistics that are likely to occur and those that are unlikely to occur. Such a z score is called a critical value.

Definition Critical Value: The number on the

borderline separating sample statistics that are likely to occur from those that are unlikely to occur.

Topic 2 – Critical Clarification

The number z/2 is a critical value that is a z score with the property that it separates an area of /2 in the right tail of the standard normal distribution.

Topic 2 – Critical Example

Finding zα/2 (z*) for a 95% Confidence Level

1. Find /2

2. Subtract the result from 1. 3. Find the corresponding z score.

Topic 2 – Critical Practice

Confidence Level

Critical Value, z/2

90% 0.10 1.64595% 0.05 1.9699% 0.01 2.575

Your turn. Complete the table.

Topic 2 – Margin of Error

Definition Margin of Error: When data from a simple

random sample are used to estimate a population proportion p, the margin of error, denoted by E, is the maximum likely difference (with probability 1 – , such as 0.95) between the observed proportion and the true value of the population proportion p.

Important Notes Mr. Llorens has a larger margin of error than

Ms. Pobuda.

Topic 2 – Margin of Error Formula

Important NotesMargin of Error for Proportions

2

ˆ ˆpqE z

n/2

margin of error

z * critical value

ˆ proportion of successes

ˆ proportion of failures

sample size

E

z

p

q

n

Topic 2 – Margin of Error Example

Assume that a sample is used to estimate a population proportion p.

Find the margin of error, E, that corresponds to the given statistics and confidence level. Round the margin of

error to three decimal places. a) 90% confidence, sample size is 500, of

20% are successes. b) 99% confidence, sample size is 300, of

25% are successes.

Topic 2 – Expressing A Margin of Error

3 different ways to write a confidence interval

ˆ ˆ

ˆ

ˆ ˆ,

p E p p E

p E

p E p E

Now express your answers from before as an interval.

Topic 3 – Point Estimate and E

Finding the Point Estimate and E from a Confidence Interval

point estimate of :

upper confidence limit lower confidence limitˆ

2

p

p

Margin of Error:

upper confidence limit lower confidence limit

2E

Breathing Point

Learned about point estimate.

Defined confidence intervals.

Discovered how to find critical values and the margin of

error. Now the MAIN

EVENT:CONSTRUCTING

CONFIDENCE INTERVALS

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Topic 2 – Putting it All Together

How To Guide: Constructing a Confidence Interval

1. Verify the sample is a simple, random sample and that the conditions for a binomial distribution are met.

2. Find the critical value zα/2

3. Evaluate the margin of error. 4. Find the values of the confidence interval limits. 5. Round the resulting confidence interval limits to

three significant figures.

Topic 2 – Example

Constructing a Confidence Interval: Poll Results

A poll of 1501 randomly selected U.S adults showed that 70% of the respondents believe in global

warming. 1. Find the margin of error that corresponds to a

95% confidence interval. 2. Find the 95th confidence interval estimate of the

population proportion p. 3. Based on the results, can we safely conclude that

the majority of adults believe in global warming?

Topic 3 – Determining Sample Size

Objective Determine how large the sample should

be in order to estimate the population proportion p.

Requirements The sample must be a simple random

sample of independent subjects. When an estimate of is known:

When an estimate of is unknown:

2

/2

2

ˆ ˆz pqn

E

2

/2

2

0.25zn

E

Topic 3 – Determining Sample Size

Important Note

Round-off Rule for Determining Sample Size

If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.

Topic 3 – Determining Sample Size Example

How Many Adults Use the Internet? An executive for E-Bay wants to determine

the current percentage of U.S adults who now use the Internet. How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points? Use this result from a Pew Research Center poll:

In 2006, 73% of U.S adults used the Internet. Assume that we have no prior information

suggesting a possible value of the proportion.

Topic 3 – Point Estimate and E

Finding the Point Estimate and E from a Confidence Interval

point estimate of :

upper confidence limit lower confidence limitˆ

2

p

p

Margin of Error:

upper confidence limit lower confidence limit

2E

Topic 3 – Point Estimate and E Example

The article “High-Dose Nicotine Patch Therapy,” by Dale Hurt includes the statement: “Of the 71 subjects, 70%

were abstinent from smoking at 8 weeks (95% confidence interval, 58% to 81%).”

Use that statement to find the point estimate and margin of error E.

Practice

Pg. 340-341 #5-8 (Finding Critical Values) #9-10 (Expressing/Interpreting CI) #17-20 (Finding Margin of Error) #21-24 (Constructing Confidence

Intervals) #25-28 (Determining Sample Size) #31, 32 (Application Problems)