### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

# Estimation in Statistics

In statistics, **estimation** refers to the process by
which one makes inferences about a population, based on information
obtained from a sample.

## Point Estimate vs. Interval Estimate

Statisticians use sample statistics to estimate population parameters. For example, sample means are used to estimate population means; sample proportions, to estimate population proportions.

An estimate of a population parameter may be expressed in two ways:

**Point estimate**. A point estimate of a population parameter is a single value of a statistic. For example, the sample mean x is a point estimate of the population mean μ. Similarly, the sample proportion*p*is a point estimate of the population proportion*P*.**Interval estimate**. An interval estimate is defined by two numbers, between which a population parameter is said to lie. For example,*a*< x <*b*is an interval estimate of the population mean μ. It indicates that the population mean is greater than*a*but less than*b*.

## Confidence Intervals

Statisticians use a **confidence interval** to express the
precision and uncertainty associated with a particular
sampling method.
A confidence interval consists of three parts.

- A confidence level.
- A statistic.
- A margin of error.

The confidence level describes the uncertainty of a sampling method.
The statistic and the margin of error define an interval estimate
that describes the precision of the method. The interval estimate
of a confidence interval is defined by the
*sample statistic* __+__ *margin of error*.

For example, suppose we compute an interval estimate of a population
parameter. We might describe this interval estimate as a 95%
confidence interval. This means that if we used the same
sampling method to select different samples and compute different
interval estimates, the true
population parameter would fall within a range defined by the
*sample statistic* __+__ *margin of error*
95% of the time.

Confidence intervals are preferred to point estimates, because confidence intervals indicate (a) the precision of the estimate and (b) the uncertainty of the estimate.

## Confidence Level

The probability part of a confidence interval is called a
**confidence level**. The confidence level describes
the likelihood that a particular sampling method will
produce a confidence interval that includes the true population
parameter.

Here is how to interpret a confidence level. Suppose we collected all possible samples from a given population, and computed confidence intervals for each sample. Some confidence intervals would include the true population parameter; others would not. A 95% confidence level means that 95% of the intervals contain the true population parameter; a 90% confidence level means that 90% of the intervals contain the population parameter; and so on.

## Margin of Error

In a confidence interval, the range of values above and below the
sample statistic is called the
**margin of error**.

For example, suppose the local newspaper conducts an election survey and reports that the independent candidate will receive 30% of the vote. The newspaper states that the survey had a 5% margin of error and a confidence level of 95%. These findings result in the following confidence interval: We are 95% confident that the independent candidate will receive between 25% and 35% of the vote.

Note: Many public opinion surveys report interval estimates, but not confidence intervals. They provide the margin of error, but not the confidence level. To clearly interpret survey results you need to know both! We are much more likely to accept survey findings if the confidence level is high (say, 95%) than if it is low (say, 50%).

## Test Your Understanding

**Problem 1**

Which of the following statements is true.

I. When the margin of error is small, the confidence level is high.

II. When the margin of error is small, the confidence level is low.

III. A confidence interval is a type of point estimate.

IV. A population mean is an example of a point estimate.

(A) I only

(B) II only

(C) III only

(D) IV only.

(E) None of the above.

**Solution**

The correct answer is (E). The confidence level is not affected
by the margin of error. When the margin of error is small, the
confidence level can low or high or anything in between. A
confidence interval is a type of interval estimate, not a type of point
estimate. A *population* mean is not an example of a point
estimate; a *sample* mean is an example of a point estimate.

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