Confidence Intervals for Population Proportions

Population Proportion and Sample Proportion

In statistics, a population proportion (denoted by p) represents the probability or decimal equivalent of a certain characteristic in the entire population. The sample proportion (denoted by \hat{p}) is the proportion observed in a sample drawn from the population.

• p: True proportion in the population

• \hat{p}: Proportion in the sample

• n: Sample size

Confidence Interval (Estimate)

A confidence interval is a range (or interval) of values that is likely to contain the value of the population parameter. It is constructed around the sample estimate and is used to express the uncertainty associated with the estimate.

Degree of Confidence (Level of Confidence)

The degree of confidence (or confidence level) is the probability that the population parameter is contained in the confidence interval in repeated sampling. It is usually expressed as a decimal or percentage (e.g., 0.95 or 95%).

• Common confidence levels: 90% (C = 0.90), 95% (C = 0.95), 99% (C = 0.99)

Critical Value

The critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. For confidence intervals, this is the z-value that corresponds to the desired confidence level.

• For a confidence level C, the critical value z* is such that the area between -z* and z* under the standard normal curve is C.

Example: For a 95% confidence level, z* ≈ 1.96.

Margin of Error

The margin of error (E) is the maximum likely difference between the observed sample mean (or proportion) and the true value of the population mean (or proportion). For proportions, it is calculated as:

• z*: Critical value for the desired confidence level

• \hat{p}: Sample proportion

• n: Sample size

Standard Deviation of the Sample Proportion

The standard deviation (standard error) of the sample proportion is:

In practice, since p is unknown, we use \hat{p}:

Confidence Interval Formula for a Proportion

The confidence interval for a population proportion is given by:

or equivalently,

• Round the confidence interval limits to three or four decimal places.

Steps to Find a Confidence Interval Estimate for "p"

1. Re-write relevant information and identify the random variable and population parameter in question.

2. Draw a graph (shade tails; label completely).

3. Report/identify the critical value (CV), e.g., for 95% confidence.

4. Compute/identify the margin of error (E) using the formula above.

5. Compute/report the values and .

6. Give a complete sentence answer, e.g., "We can be 95% confident that the true proportion is between X and Y."

Worked Example: Constructing a 95% Confidence Interval for a Proportion

• Sample size (n): 750

• Number voting for Governor (x): 305

• Sample proportion (\hat{p}):

• Complement (\hat{q}):

• Confidence level (C): 95% ()

• Margin of error (E):

• Confidence interval:

Interpretation: We can be 95% confident that the percentage of voters who would vote for the former Governor is between 37.2% and 44.2%.

Reporting Results and Drawing Conclusions

• When reporting, include the sample size, confidence level, and margin of error.

• Example statement: "Based on a poll of 750 randomly selected people, we estimate that 40.7% of voters would support the Governor. The margin of error is ±3.5%."

• To determine if a specific value (e.g., 45%) is plausible, check if it falls within the confidence interval. If not, it is not supported by the data at the given confidence level.

Example conclusion: No, since the confidence interval constructed is completely below 45%, it does not appear that the Governor would receive 45% of the vote.