# Confidence Intervals for Proportions - My Math Mantra

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No Slide TitleProportions

Standard Error Both of the CLThm sampling distributions

we’ve looked at so far are Normal. For proportions (a.k.a. a “%”)

For means (quantitative)

Slide 1- Copyright © 2009 Pearson Education, Inc.

Standard Error (cont.) When we don’t know p or σ, we’re stuck, right?

Instead, we will use the sample’s statistics to estimate these population parameters.

Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

SD -> SE:

Slide 1- Copyright © 2009 Pearson Education, Inc.

Consider the 95% level: (+/-) 2 SD: There’s a 95% chance that p is no more than

2 SEs away from . In other words, if we reach out 2 SEs in either

direction of , we can be 95% confident that this interval contains the true proportion p.

This is called a “95% confidence interval”.

A Confidence Interval

A Confidence Interval (cont.)

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Does “95% Confidence” Really Mean? Each confidence interval uses a sample

statistic to estimate a population parameter.

But, since samples vary, the statistics we use, and thus the confidence intervals we construct, vary as well.

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Does “95% Confidence” Really Mean? The figure to the

right shows that some of our confidence intervals (from 20 random samples) capture the true proportion (the green horizontal line), while others do not:

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Does “95% Confidence” Really Mean? Our confidence is in the process of constructing

the interval, not in any one interval itself.

Thus, we expect 95%, or 19 of 20 samples, to contain the true parameter that they are estimating.

However, make it known that your sample stands a 5% chance that it’s Confidence Interval may NOT encompass the true population’s parameter!

Slide 1- Copyright © 2009 Pearson Education, Inc.

Margin of Error: Certainty vs. Precision In general, confidence intervals have the form

estimate ± ME.

The more confident we want to be, the larger our ME needs to be (makes the interval wider).

Slide 1- Copyright © 2009 Pearson Education, Inc.

Margin of Error: Certainty vs. Precision (cont.)

Slide 1- Copyright © 2009 Pearson Education, Inc.

Margin of Error: Certainty vs. Precision (cont.) To be more confident, we wind up being less precise.

We need more values in our confidence interval to be more certain.

Because of this, every confidence interval is a balance between certainty and precision.

The tension between certainty and precision is always there. Fortunately, in most cases we can be both

sufficiently certain and sufficiently precise to make useful statements.

The most commonly chosen confidence levels are 90%, 95%, and 99% (but any percentage can be used).

Slide 1- Copyright © 2009 Pearson Education, Inc.

Critical Values The ‘2’ in (our 95% confidence

interval) came from the 68-95-99.7% Rule.

Using a table or technology, we find that a more exact value for our 95% confidence interval is 1.96 instead of 2. Call 1.96 the critical value and denote it z*.

Slide 1- Copyright © 2009 Pearson Education, Inc.

Critical Values (cont.) Example: For a 90% confidence interval, the

critical value is 1.645:

Assumptions and Conditions All statistical models depend upon assumptions.

Different models depend upon different assumptions.

If those assumptions are not true, the model might be inappropriate and our conclusions based on it may be wrong.

You can often decide whether an assumption is plausible by checking a related condition....

Slide 1- Copyright © 2009 Pearson Education, Inc.

Assumptions and Conditions (cont.) Here are the assumptions and the corresponding

conditions you must check before creating a confidence interval for a proportion:

1. Independence Assumption: Think about whether the Independence is plausible. It’s not one you can check by looking at the data.

1a. Randomization Condition: Proper randomization can help ensure independence.

1b. 10% Condition: Sample size no more than 10% of the population.

Slide 1- Copyright © 2009 Pearson Education, Inc.

Assumptions and Conditions (cont.)

2. Sample Size Assumption: The sample needs to be large enough for us to be able to use the CLT. Success/Failure Condition: Expect at least 10

“successes” and at least 10 “failures.”

Now we are confident that the confidence interval is

where

The critical value, z*, depends on the particular confidence level, C, that you specify.

Slide 1- Copyright © 2009 Pearson Education, Inc.

One-Proportion z-Interval

According to a September 2004 Gallup poll, about 73% of 18- to 29-year-olds said that they were registered to vote. A statistics professor asked her students whether or not they were registered to vote. In a sample of 50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote.

1. Find a 95% confidence interval for the true proportion of the professor’s students who were registered to vote. (Make sure to check any necessary conditions and to state a conclusion in the context of the problem.)

(i) Conditions: a.We have a random sample of less than 10% of the professor’s students b. 35 expected successes (registered) and 15 expected failures (not registered) (ii) Standard Error: p^ = 35/50 = 0.70 , now we have

(iii) Margin of Error: 95% corresponds to a z-score of 1.96 (see table A-2) pˆ ± z∗SE ( pˆ ) = 0.70 ±1.96(0.065) = 0.70 ± 0.127 = 0.573 to 0.827

Slide 1- Copyright © 2009 Pearson Education, Inc.

Example: One-Proportion z-Interval

According to a September 2004 Gallup poll, about 73% of 18- to 29-year-olds said that they were registered to vote. A statistics professor asked her students whether or not they were registered to vote. In a sample of 50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote.

2. Explain what 95% confidence means in this context.

If many random samples were taken, 95% of the confidence intervals produced would contain the actual percentage of the professor’s students who are registered to vote.

3. What is the probability that the true proportion of the professor’s students who were registered to vote is in your confidence interval?

There is no probability involved—once the interval is constructed, the true proportion of the professor’s students who were registered to vote is in the interval or it is not.

Slide 1- Copyright © 2009 Pearson Education, Inc.

Example: One-Proportion z-Interval

According to a September 2004 Gallup poll, about 73% of 18- to 29-year-olds said that they were registered to vote. A statistics professor asked her students whether or not they were registered to vote. In a sample of 50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote.

4. Does the 73% figure from Gallup seem reasonable for the professor’s class? Explain.

The 73% figure from Gallup seems reasonable since 73% lies in our confidence interval.

5. If the professor only knew the information from the September 2004 Gallup poll and wanted to estimate the percentage of her students who were registered to vote to within ±4% with 95% confidence, how many students should she sample?

Note: Since there are only 700 students in the professor’s class, she cannot sample this many students without violating the 10% condition!

Slide 1- Copyright © 2009 Pearson Education, Inc.

Example: One-Proportion z-Interval

Classwork: One-Proportion z-Interval

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Can Go Wrong? Don’t Misstate What the Interval Means: Don’t suggest that the parameter varies. Don’t claim that other samples will agree with yours. Don’t be certain about the parameter. Don’t forget: It’s the parameter (not the statistic). Don’t claim to know too much. Do take responsibility (for the uncertainty). Do treat the whole interval equally.

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Can Go Wrong? (cont.) Margin of Error Too Large to Be Useful: How precise do we need to be? One way to make the margin of error smaller is to

reduce your level of confidence. You need to think about your margin of error when

you design your study. Need less variability? Use a larger sample…

Violations of Assumptions:

Watch out for biased sampling—keep in mind what you learned in Chapter 12.

Think about independence.

Slide 1- Copyright © 2009 Pearson Education, Inc.

What have we learned? Finally we have learned to use a sample to say

something about the world at large.

This process (statistical inference) is based on our understanding of sampling models, and will be our focus for the rest of the book.

In this chapter we learned how to construct a confidence interval for a population proportion (a.k.a. a percentage %).

Standard Error Both of the CLThm sampling distributions

we’ve looked at so far are Normal. For proportions (a.k.a. a “%”)

For means (quantitative)

Slide 1- Copyright © 2009 Pearson Education, Inc.

Standard Error (cont.) When we don’t know p or σ, we’re stuck, right?

Instead, we will use the sample’s statistics to estimate these population parameters.

Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

SD -> SE:

Slide 1- Copyright © 2009 Pearson Education, Inc.

Consider the 95% level: (+/-) 2 SD: There’s a 95% chance that p is no more than

2 SEs away from . In other words, if we reach out 2 SEs in either

direction of , we can be 95% confident that this interval contains the true proportion p.

This is called a “95% confidence interval”.

A Confidence Interval

A Confidence Interval (cont.)

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Does “95% Confidence” Really Mean? Each confidence interval uses a sample

statistic to estimate a population parameter.

But, since samples vary, the statistics we use, and thus the confidence intervals we construct, vary as well.

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Does “95% Confidence” Really Mean? The figure to the

right shows that some of our confidence intervals (from 20 random samples) capture the true proportion (the green horizontal line), while others do not:

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Does “95% Confidence” Really Mean? Our confidence is in the process of constructing

the interval, not in any one interval itself.

Thus, we expect 95%, or 19 of 20 samples, to contain the true parameter that they are estimating.

However, make it known that your sample stands a 5% chance that it’s Confidence Interval may NOT encompass the true population’s parameter!

Slide 1- Copyright © 2009 Pearson Education, Inc.

Margin of Error: Certainty vs. Precision In general, confidence intervals have the form

estimate ± ME.

The more confident we want to be, the larger our ME needs to be (makes the interval wider).

Slide 1- Copyright © 2009 Pearson Education, Inc.

Margin of Error: Certainty vs. Precision (cont.)

Slide 1- Copyright © 2009 Pearson Education, Inc.

Margin of Error: Certainty vs. Precision (cont.) To be more confident, we wind up being less precise.

We need more values in our confidence interval to be more certain.

Because of this, every confidence interval is a balance between certainty and precision.

The tension between certainty and precision is always there. Fortunately, in most cases we can be both

sufficiently certain and sufficiently precise to make useful statements.

The most commonly chosen confidence levels are 90%, 95%, and 99% (but any percentage can be used).

Slide 1- Copyright © 2009 Pearson Education, Inc.

Critical Values The ‘2’ in (our 95% confidence

interval) came from the 68-95-99.7% Rule.

Using a table or technology, we find that a more exact value for our 95% confidence interval is 1.96 instead of 2. Call 1.96 the critical value and denote it z*.

Slide 1- Copyright © 2009 Pearson Education, Inc.

Critical Values (cont.) Example: For a 90% confidence interval, the

critical value is 1.645:

Assumptions and Conditions All statistical models depend upon assumptions.

Different models depend upon different assumptions.

If those assumptions are not true, the model might be inappropriate and our conclusions based on it may be wrong.

You can often decide whether an assumption is plausible by checking a related condition....

Slide 1- Copyright © 2009 Pearson Education, Inc.

Assumptions and Conditions (cont.) Here are the assumptions and the corresponding

conditions you must check before creating a confidence interval for a proportion:

1. Independence Assumption: Think about whether the Independence is plausible. It’s not one you can check by looking at the data.

1a. Randomization Condition: Proper randomization can help ensure independence.

1b. 10% Condition: Sample size no more than 10% of the population.

Slide 1- Copyright © 2009 Pearson Education, Inc.

Assumptions and Conditions (cont.)

2. Sample Size Assumption: The sample needs to be large enough for us to be able to use the CLT. Success/Failure Condition: Expect at least 10

“successes” and at least 10 “failures.”

Now we are confident that the confidence interval is

where

The critical value, z*, depends on the particular confidence level, C, that you specify.

Slide 1- Copyright © 2009 Pearson Education, Inc.

One-Proportion z-Interval

According to a September 2004 Gallup poll, about 73% of 18- to 29-year-olds said that they were registered to vote. A statistics professor asked her students whether or not they were registered to vote. In a sample of 50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote.

1. Find a 95% confidence interval for the true proportion of the professor’s students who were registered to vote. (Make sure to check any necessary conditions and to state a conclusion in the context of the problem.)

(i) Conditions: a.We have a random sample of less than 10% of the professor’s students b. 35 expected successes (registered) and 15 expected failures (not registered) (ii) Standard Error: p^ = 35/50 = 0.70 , now we have

(iii) Margin of Error: 95% corresponds to a z-score of 1.96 (see table A-2) pˆ ± z∗SE ( pˆ ) = 0.70 ±1.96(0.065) = 0.70 ± 0.127 = 0.573 to 0.827

Slide 1- Copyright © 2009 Pearson Education, Inc.

Example: One-Proportion z-Interval

According to a September 2004 Gallup poll, about 73% of 18- to 29-year-olds said that they were registered to vote. A statistics professor asked her students whether or not they were registered to vote. In a sample of 50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote.

2. Explain what 95% confidence means in this context.

If many random samples were taken, 95% of the confidence intervals produced would contain the actual percentage of the professor’s students who are registered to vote.

3. What is the probability that the true proportion of the professor’s students who were registered to vote is in your confidence interval?

There is no probability involved—once the interval is constructed, the true proportion of the professor’s students who were registered to vote is in the interval or it is not.

Slide 1- Copyright © 2009 Pearson Education, Inc.

Example: One-Proportion z-Interval

According to a September 2004 Gallup poll, about 73% of 18- to 29-year-olds said that they were registered to vote. A statistics professor asked her students whether or not they were registered to vote. In a sample of 50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote.

4. Does the 73% figure from Gallup seem reasonable for the professor’s class? Explain.

The 73% figure from Gallup seems reasonable since 73% lies in our confidence interval.

5. If the professor only knew the information from the September 2004 Gallup poll and wanted to estimate the percentage of her students who were registered to vote to within ±4% with 95% confidence, how many students should she sample?

Note: Since there are only 700 students in the professor’s class, she cannot sample this many students without violating the 10% condition!

Slide 1- Copyright © 2009 Pearson Education, Inc.

Example: One-Proportion z-Interval

Classwork: One-Proportion z-Interval

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Can Go Wrong? Don’t Misstate What the Interval Means: Don’t suggest that the parameter varies. Don’t claim that other samples will agree with yours. Don’t be certain about the parameter. Don’t forget: It’s the parameter (not the statistic). Don’t claim to know too much. Do take responsibility (for the uncertainty). Do treat the whole interval equally.

Slide 1- Copyright © 2009 Pearson Education, Inc.

What Can Go Wrong? (cont.) Margin of Error Too Large to Be Useful: How precise do we need to be? One way to make the margin of error smaller is to

reduce your level of confidence. You need to think about your margin of error when

you design your study. Need less variability? Use a larger sample…

Violations of Assumptions:

Watch out for biased sampling—keep in mind what you learned in Chapter 12.

Think about independence.

Slide 1- Copyright © 2009 Pearson Education, Inc.

What have we learned? Finally we have learned to use a sample to say

something about the world at large.

This process (statistical inference) is based on our understanding of sampling models, and will be our focus for the rest of the book.

In this chapter we learned how to construct a confidence interval for a population proportion (a.k.a. a percentage %).