### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

# 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 we wanted to know the percentage of adults that exercise daily. We could devise a sample design to ensure that our sample estimate will not differ from the true population value by more than, say, 5 percent (the margin of error) 90 percent of the time (the confidence level).

## How to Compute the Margin of Error

The margin of error can be defined by either of the following equations.

Margin of error = Critical value x Standard deviation of the statistic

Margin of error = Critical value x Standard error of the statistic

If you know the standard deviation of the statistic, use the first equation to compute the margin of error. Otherwise, use the second equation. Previously, we described how to compute the standard deviation and standard error.

## How to Find the Critical Value

The **critical value** is a factor used to compute
the margin of error. This section describes how to find the
critical value, when the
sampling distribution
of the statistic is
normal
or nearly normal.

When the sampling distribution is nearly normal, the critical value can be expressed as a t score or as a z-score. To find the critical value, follow these steps.

- Compute alpha (α): α = 1 - (confidence level / 100)
- Find the critical probability (p*): p* = 1 - α/2
- To express the critical value as a z-score, find the z-score having a cumulative probability equal to the critical probability (p*).
- To express the critical value as a t statistic, follow these steps.
- Find the degrees of freedom (DF). When estimating a mean score or a proportion from a single sample, DF is equal to the sample size minus one. For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.
- The critical t statistic (t*) is the t statistic having degrees of freedom equal to DF and a cumulative probability equal to the critical probability (p*).

## T-Score vs. Z-Score

Should you express the critical value as a t statistic or as a z-score? One way to answer this question focuses on the population standard deviation.

- If the population standard deviation is known, use the z-score.
- If the population standard deviation is unknown, use the t statistic.

Another approach focuses on sample size.

- If the sample size is large, use the z-score. (The central limit theorem provides a useful basis for determining whether a sample is "large".)
- If the sample size is small, use the t statistic.

In practice, researchers employ a mix of the above guidelines. On this site, we use z-scores when the population standard deviation is known and the sample size is large. Otherwise, we use the t statistics, unless the sample size is small and the underlying distribution is not normal.

**Warning:** If the sample size is small and the population distribution is not normal,
we cannot be confident that the sampling distribution of the statistic will be normal. In this
situation, neither the t statistic nor the z-score should be used to compute critical values.

You can use the Normal Distribution Calculator to find the critical z-score, and the t Distribution Calculator to find the critical t statistic. You can also use a graphing calculator or standard statistical tables (found in the appendix of most introductory statistics texts).

## Test Your Understanding

**Problem 1**

Nine hundred (900) high school freshmen were randomly selected for a national survey. Among survey participants, the mean grade-point average (GPA) was 2.7, and the standard deviation was 0.4. What is the margin of error, assuming a 95% confidence level?

(A) 0.013

(B) 0.025

(C) 0.500

(D) 1.960

(E) None of the above.

**Solution**

The correct answer is (B). To compute the margin of error, we need to find the critical value and the standard error of the mean. To find the critical value, we take the following steps.

- Compute alpha (α):
α = 1 - (confidence level / 100) = 1 - 0.95 = 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 = 900 -1 = 899

- Find the critical value. Since we don't know the population standard deviation, we'll express the critical value as a t statistic. For this problem, it will be the t statistic having 899 degrees of freedom and a cumulative probability equal to 0.975. Using the t Distribution Calculator, we find that the critical value is 1.96.

Next, we find the standard error of the mean, using the following equation:

SE_{x} = s / sqrt( n )
= 0.4 / sqrt( 900 ) = 0.4 / 30 = 0.013

And finally, we compute the margin of error (ME).

ME = Critical value x Standard error = 1.96 * 0.013 = 0.025

This means we can be 95% confident that the mean grade point average in the population is 2.7 plus or minus 0.025, since the margin of error is 0.025.

**Note:** The larger the sample size, the more closely the t distribution
looks like the normal distribution. For this problem, since the sample size is very large, we would have
found the same result with a z-score as we found with a t statistic. That is,
the critical value would still have been 1.96. The choice of t statistic versus z-score does not
make much practical difference when the sample size is very large.