### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

# What is a Confidence Interval?

Statisticians use a confidence interval to describe the amount of uncertainty associated with a sample estimate of a population parameter.

## How to Interpret Confidence Intervals

Suppose that a 90% confidence interval states that the population mean is greater than 100 and less than 200. How would you interpret this statement?

Some people think this means there is a 90% chance that the population mean falls between 100 and 200. This is incorrect. Like any population parameter, the population mean is a constant, not a random variable. It does not change. The probability that a constant falls within any given range is always 0.00 or 1.00.

The
confidence level
describes the uncertainty associated
with a *sampling method*.
Suppose we used the same sampling method to select
different samples and to compute a different interval estimate
for each sample.
Some interval estimates would include the true population
parameter and some would not. A 90% confidence level means
that we would expect 90% of the interval estimates to include
the population parameter; a 95% confidence level means that
95% of the intervals would include the parameter; and so on.

## Confidence Interval Data Requirements

To express a confidence interval, you need three pieces of information.

Given these inputs, the range of the confidence interval is
defined by the *sample statistic* __+__
*margin of error*. And the uncertainty associated with
the confidence interval is specified by the confidence level.

Often, the margin of error is not given; you must calculate it. Previously, we described how to compute the margin of error.

## How to Construct a Confidence Interval

There are four steps to constructing a confidence interval.

- Identify a sample statistic. Choose the statistic (e.g, sample mean, sample proportion) that you will use to estimate a population parameter.
- Select a confidence level. As we noted in the previous section, the confidence level describes the uncertainty of a sampling method. Often, researchers choose 90%, 95%, or 99% confidence levels; but any percentage can be used.
- Find the margin of error. If you are working on a homework
problem or a test question, the margin of error may be given.
Often, however, you will need to compute the margin of error,
based on one of the following equations.
Margin of error = Critical value * Standard deviation of statistic

Margin of error = Critical value * Standard error of statistic

For guidance, see how to compute the margin of error. - Specify the confidence interval. The uncertainty is denoted
by the confidence level. And the range of the confidence
interval is defined by the following equation.
Confidence interval = sample statistic

__+__Margin of error

The sample problem in the next section applies the above four steps to construct a 95% confidence interval for a mean score. The next few lessons discuss this topic in greater detail.

## Sample Size Calculator

As you may have guessed, the four steps required to specify a confidence interval can involve many time-consuming computations. Stat Trek's Sample Size Calculator does this work for you - quickly, easily, and error-free. In addition to constructing a confidence interval, the calculator creates a summary report that lists key findings and documents analytical techniques. Whenever you need to construct a confidence interval, consider using the Sample Size Calculator. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.

Sample Size Calculator## Test Your Understanding

**Problem 1**

Suppose we want to estimate the average weight of an adult male in Dekalb County, Georgia. We draw a random sample of 1,000 men from a population of 1,000,000 men and weigh them. We find that the average man in our sample weighs 180 pounds, and the standard deviation of the sample is 30 pounds. What is the 95% confidence interval.

(A) 180 __+__ 1.86

(B) 180 __+__ 3.0

(C) 180 __+__ 5.88

(D) 180 __+__ 30

(E) None of the above

**Solution**

The correct answer is (A). To specify the confidence interval, we work through the four steps below.

- Identify a sample statistic. Since we are trying to estimate the mean weight in the population, we choose the mean weight in our sample (180) as the sample statistic.
- Select a confidence level. In this case, the confidence level is defined for us in the problem. We are working with a 95% confidence level.
- Find the margin of error. Previously, we described
how to compute the margin of error.
The key steps are shown below.
- Find standard error. The standard error (SE) of the
mean is:
SE = s / sqrt( n )

SE = 30 / sqrt(1000) = 30/31.62 = 0.95

- Find critical value. The critical value is a factor used to
compute the margin of error. To express the critical value
as a
t score
(t*), follow these steps.
- Compute alpha (α):
α = 1 - (confidence level / 100) = 0.05

- Find the critical probability (p*):
p* = 1 - α/2 = 1 - 0.05/2 = 0.975

- Find the
degrees of freedom (df):
df = n - 1 = 1000 - 1 = 999

- The critical value is the t statistic having 999 degrees of freedom and a cumulative probability equal to 0.975. From the t Distribution Calculator, we find that the critical value is 1.96.

**Note:**We might also have expressed the critical value as a z-score. Because the sample size is large, a z-score analysis produces the same result - a critical value equal to 1.96. - Compute alpha (α):
- Compute margin of error (ME):
ME = critical value * standard error

ME = 1.96 * 0.95 = 1.86

- Find standard error. The standard error (SE) of the
mean is:
- Specify the confidence interval. The range of the confidence
interval is defined by the
*sample statistic*__+__*margin of error*. And the uncertainty is denoted by the confidence level. Therefore, this 95% confidence interval is 180__+__1.86.

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