Estimation by Intervals

Confidence Interval

Suppose we wanted to estimate the proportion of blue candies in a VERY large bowl. We could take a sample of candies and compute the proportion of blue candies in our sample.

How much confidence do you have in the point estimate?

Would you have more confidence if your answer were

an interval?

Confidence intervalsA confidence interval (CI) for a population characteristic is an interval of plausible values for the characteristic.

It is constructed so that, with a chosen degree of confidence, the actual value of the characteristic will be between the lower and upper endpoints of the interval.

The primary goal of a confidence interval is to estimate an unknown population characteristic.

Rate your confidence0 – 100%

How confident (%) are you that you can ...

Guess Berry’s height within 10 cm?

. . . within 5 cm?

. . . within 1 cm?

What do I mean when I say to be within 10 cm?

What happened to your level of confidence as the interval

became smaller?

Confidence levelThe confidence level associated with a confidence interval estimate is the success rate of the method used to construct the interval.

Our confidence is in the method – NOT in any one particular interval!The most common confidence levels are 90%, 95%, and 99% confidence.

What are the Properties of Sampling Distributions of p?

1.

2.

pp ˆ

npp

p

)1(ˆ

As long as the sample size is less than 10% of the population

3. As long as n is large (np > 10 and n (1-p) > 10) the sampling distribution of p is approximately normal.

These are the conditions that must be true in order to calculate a large-sample

confidence interval for p

Let’s develop the equation for the large-sample confidence interval.To begin, we will use a 95% confidence level. Use the table of standard normal curve areas to determine the value of z* such that a central area of .95 falls between –z* and z*.

-1.96 1.96

Central Area = .95

Upper tail area = .025

Lower tail area = .025

95% of these values are within 1.96 of the mean.

We can generalize this to normal distributions other than the standard

normal distribution –About 95% of the values are within 1.96

standard deviations of the mean

0

For large random samples, the sampling distribution of p is approximately normal. So about 95% of the possible p will fall within

pnpp within )1(96.1

Developing a Confidence Interval Continued . . .

If p is within of p,

this means the interval

will capture p.

)1(96.1n

pp

nppp

nppp )1(96.1ˆ to )1(96.1ˆ

And this will happen for 95% of all possible samples!

Developing a Confidence Interval Continued . . .

npp )1(96.1

npp )1(96.1

Here is the mean of the sampling distribution

p

Approximate sampling distribution of p

This line represents 1.96 standard deviations below the mean.

This line represents 1.96 standard deviations above the mean.

Suppose we get this p

p

Create an interval around p

Notice that the length of each half of the interval equals

npp )1(96.1

This p fell within 1.96 standard deviations of the mean AND its confidence interval “captures” p.

Suppose we get this p and create an interval

p

This p fell within 1.96 standard deviations of the mean AND its confidence interval “captures” p.

Suppose we get this p and create an interval

pThis p doesn’t fall within 1.96 standard deviations of the mean AND its confidence interval does NOT “capture” p.

Using this method of

calculation, the confidence

interval will not capture p 5% of

the time.When n is large, a 95% confidence interval for p is

nppp )1(96.1ˆ

* Our confidence is in the process of constructing the interval, not in any one interval itself.

** Thus, we expect 95% of all 95% confidence intervals to contain the true parameter that they are estimating.

What Does “85% 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:

Try Me• For a project, a student randomly

sampled 182 other students at Shenzhen Foreign language school to determine if the majority of students were in favor of a proposal to have longer Chinese new year break. He found that 75 were in favor of the proposal. Find a 95% CI for P.

The general formula for a confidence interval for a population proportion p when

3. if the sample is selected without replacement, the sample size is small relative to the population size (at most 10% of the population)

The Large-Sample Confidence Interval for p

1. p is the sample proportion from a random sample2. the sample size n is large ( np > 10 and n(1-p) > 10), and

The general formula for a confidence interval for a population proportion p . . . is

nppzp )ˆ1(ˆvalue) critical (ˆ

point estimate

Estimate of the standard deviation of p or standard

errorThe standard error of a statistic is the estimated standard deviation of the

statistic.

The Large-Sample Confidence Interval for p

The general formula for a confidence interval for a population proportion p . . . is

nppzp )ˆ1(ˆvalue) critical (ˆ

This is called the bound on the error estimation.

The 95% confidence interval is based on the fact that, for approximately 95% of all random samples, p is

within the bound on error estimation of p.

This is also called the margin of error.

Summary on Margin of Error– The extent of the interval on either side

of is called the margin of error (ME).

• In general, confidence intervals have the form estimate ± ME.

• The more confident we want to be, the larger our ME needs to be, making the interval wider.

p̂

Margin of Error: Certainty vs. Precision

• 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.

Finding a z Critical Value

• Finding a z critical value for a 98% confidence interval.

Looking up the cumulative area or 0.9900 in the body of the table we find z = 2.33

2.33

Some Common Critical Values

Confidence level

z critical value

80% 1.2890% 1.64595% 1.9698% 2.3399% 2.5899.8% 3.0999.9% 3.29

The article “How Well Are U.S. Colleges Run?” (USA Today, February 17, 2010) describes a survey of 1031 adult Americans. The survey was carried out by the National Center for Public Policy and the sample was selected in a way that makes it reasonable to regard the sample as representative of adult Americans. Of those surveyed, 567 indicated that they believe a college education is essential for success.

The point estimate is

What is a 95% confidence interval for the population proportion of adult

Americans who believe that a college education is essential for success?

Before computing the confidence interval, we need to verify the conditions.

College Education Continued . . . What is a 95% confidence interval for the population proportion of adult

Americans who believe that a college education is essential for success?Conditions:

2) The sample size of n = 1031 is much smaller than 10% of the population size (adult Americans).3) The sample was selected in a way designed to produce a representative sample. So we can regard the sample as a random sample from the population.

1) np = 1031(.55) = 567 and n(1-p) = 1031(.45) = 364, since both of these are greater than 10, the sample size is large enough to proceed.

All our conditions are verified so it is safe to proceed with the calculation

of the confidence interval.

Calculation:

Conclusion:We are 95% confident that the population proportion of adult Americans who believe that a college education is essential for success is between 52.1% and 57.9%

nppzp )ˆ1(ˆvalue) critical (ˆ

)579,.521(.1031)45(.55.96.155.

What does this interval mean in the context of

this problem?

College Education Revisited . . . A 95% confidence interval for the population proportion of adult Americans who believe

that a college education is essential for success is:

Compute a 90% confidence interval for this proportion.

Compute a 99% confidence interval for this proportion.

)579,.521(.1031)45(.55.96.155.

)575,.524(.1031)45(.55.645.155.

)590,.510(.1031)45(.55.58.255.

What do you notice about the

relationship between the

confidence level of an interval and the

width of the interval?

Recall the “Rate your Confidence”

Activity

Quick surveyI feel I understand “Point and Interval estimates”a) Very wellb) With some review, I’ll be goodc) Not reallyd) Not at all