How to Estimate a Mean or Proportion from a Simple Random Sample

This lesson describes how to estimate a population mean or proportion, given survey data from a simple random sample. A good analysis should provide two outputs:

• A point estimate of the population mean or proportion.
• A quantitative measure of uncertainty associated with the point estimate (e.g., a margin of error and/or a confidence interval).

First, we describe how to conduct a good analysis step-by-step. Then, we will illustrate the analysis with a sample problem.

How to Analyze Survey Data

Any good analysis of survey sample data includes the same seven steps:

• Estimate a population parameter.
• Estimate population variance.
• Compute standard error.
• Specify a confidence level.
• Find the critical value (often a z-score or a t-score).
• Compute margin of error.
• Define confidence interval.

Let's look a little bit closer at each step - what we do in each step and why we do it. When you understand what is really going on, it will be easier for you to apply formulas correctly and to interpret analytical findings.

Note: The formulas presented below are only appropriate for simple random sampling.

Estimating a Population Mean or Proportion

The first step in the analysis is to develop a point estimate for the population mean or proportion. Use this formula to estimate the population mean:

Sample mean = x = Σx / n

where Σx is the sum of all the sample observations, and n is the number of sample observations.

A proportion is a special case of the mean. It represents the number of observations that have a particular attribute divided by the total number of observations in the group. Use this formula to estimate the population proportion:

p =	Sample observations with attribute Total sample size (n)

Because different samples can produce different point estimates, you can be fairly sure that the estimate from your sample does not equal the true value of the population parameter exactly.

Therefore, you need a way to express the uncertainty inherent in your estimate. The remaining six steps in the analysis are geared toward quantifying the uncertainty in your estimate.

Estimating Population Variance

The variance is a numerical value used to measure the variability of observations in a group. If individual observations vary greatly from the group mean, the variance is big; and vice versa.

Given a simple random sample, the best estimate of the population variance is:

s² = Σ ( x_i - x )² / ( n - 1 )

where s² is a sample estimate of population variance, x is the sample mean, x_i is the ith element from the sample, and n is the number of elements in the sample.

With a proportion, the population variance can be estimated from a sample as:

s² = [ n / (n - 1) ] * p * (1 - p)

where s² is a sample estimate of population variance, p is a sample estimate of the population proportion, and n is the number of elements in the sample.

Why do we care about population variance? The variance is needed to compute the standard error. And why do we care about the standard error? Read on.

Computing Standard Error

The standard error is possibly the most important output from our analysis. It allows us to compute the margin of error and the confidence interval.

When we estimate a mean or a proportion from a simple random sample, the standard error (SE) of the estimate is:

SE = sqrt [ (1 - n/N) * s² / n ]

where n is the sample size, N is the population size, and s is a sample estimate of the population standard deviation.

Think of the standard error as the standard deviation of a sample statistic. In survey sampling, there are usually many different subsets of the population that we might choose for analysis. Each different sample might produce a different estimate of the value of a population parameter. The standard error provides a quantitative measure of the variability of those estimates.

Specifying Confidence Level

In survey sampling, different samples can be randomly selected from the same population; and each sample can often produce a different confidence interval. Some confidence intervals include the true population parameter; others do not.

A confidence level refers to the percentage of all possible samples that produce confidence intervals that include the true population parameter. For example, suppose all possible samples were selected from the same population, and a confidence interval were computed for each sample. A 95% confidence level implies that 95% of the confidence intervals would include the true population parameter.

As part of the analysis, survey researchers choose a confidence level. Probably, the most frequently chosen confidence level is 95%.

Finding Critical Value

Often expressed as a t-score or a z-score, the critical value is a factor used to compute the margin of error. 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 you estimate a mean or proportion from a simple random sample, degrees of freedom is equal to the sample size minus one.
  • 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*).

Researchers use a t-score when sample size is small; a z-score when it is large (at least 30). You can use the Normal Distribution Calculator to find the critical z-score, and the t Distribution Calculator to find the critical t statistic.

Computing Margin of Error

The margin of error expresses the maximum expected difference between the true population parameter and a sample estimate of that parameter.

Here is the formula for computing margin of error (ME):

ME = SE * CV

where SE is standard error, and CV is the critical value.

Defining Confidence Interval

Statisticians use a confidence interval to express the degree of uncertainty associated with a sample statistic. A confidence interval is an interval estimate combined with a probability statement.

Here is how to compute the minimum and maximum values for a confidence interval.

Mean	Proportion
CImin = x - SE * CV CImax = x + SE * CV	CImin = p - SE * CV CImax = p + SE * CV

In the table above, x is the sample estimate of the population mean, p is the sample estimate of the population proportion, SE is the standard error, and CV is the critical value (either a z-score or a t-score). And, the confidence interval is an interval estimate that ranges between CI_min and CI_max.

Sample Problem

This section presents a sample problem that illustrates how to analyze survey data when the sampling method is simple random sampling, and the parameter of interest is a mean score. (In a subsequent lesson, we re-visit this problem and see how simple random sampling compares to other sampling methods.)

Problem 1

At the end of every school year, the state administers a reading test to a simple random sample drawn without replacement from a population of 20,000 third graders. This year, the test was administered to 36 students selected via simple random sampling. The test score from each sampled student is shown below:

50, 55, 60, 62, 62, 65, 67, 67, 70, 70, 70, 70, 72, 72, 73, 73, 75, 75,
75, 78, 78, 78, 78, 80, 80, 80, 82, 82, 85, 85, 85, 88, 88, 90, 90, 90 

Using sample data, estimate the mean reading achievement level in the population. Find the margin of error and the confidence interval. Assume a 95% confidence level.

Solution: To solve this problem, we follow the seven-step process described above.

• Estimate the population mean. Since we are trying to estimate a population mean, we choose the sample mean as the sample statistic. The sample mean is:

  x = Σ ( x_i ) / n

  x = ( 50 + 55 + 60 + ... + 90 + 90 + 90 ) / 36 = 75

  Therefore, based on data from the simple random sample, we estimate that the mean reading achievement level in the population is equal to 75.
• Estimate population variance. We need to estimate population variance now, so we can compute the standard error in the next step.

  s² = Σ ( x_i - x )² / ( n - 1 )

  s² = [ (50 - 75)² + (55 - 75)² + (60 - 75)² + ... + (90 - 75)² + (90 - 75)² ] / 35 = 98.97

• Compute standard error. The standard error measures the variability of our sample estimate of the population mean. We will use standard error to compute the margin of error and to define a confidence level.

  SE = sqrt [ (1 - n/N) * s² / n ]

  SE = sqrt [ ( 1 - 36/20,000 ) * 98.97 / 36 ]

  SE = sqrt [ ( 0.998 ) * 98.97 / 36 ] = 1.66

• Select a confidence level. In this analysis, the confidence level is defined for us in the problem. We are working with a 95% confidence level.
• Find the critical value. The critical value is a factor used to compute the margin of error. To find the critical value, we take these steps.
  • Compute alpha (α):

    α = 1 - (confidence level / 100)

    α = 1 - 95/100 = 0.05

  • Find the critical probability (p*):

    p* = 1 - α/2 = 1 - 0.05/2 = 0.975

  • Since the sample size (n = 36) is greater than 30, we can express the critical value as a z-score. The critical value is the z-score having a cumulative probability equal to 0.975. From the Normal Distribution Calculator, we find that the critical value is about 1.96.

• Compute the margin of error (ME):

  ME = critical value * standard error

  ME = 1.96 * 1.66 = 3.25

• Specify the confidence interval. The minimum and maximum values of the confidence interval are:

  CI_min = x - SE * CV = 75 - 1.66 * 1.96 = 71.75

  CI_max = x + SE * CV = 75 + 1.66 * 1.96 = 78.25

In summary, here are the results of our analysis. Based on sample data, we estimate that the population mean is 75. Given a 95% confidence level, the margin of error around that estimate is 3.25; and the 95% confidence interval is 71.75 to 78.25.