Sampling Distribution Models and Confidence Intervals for Proportions

Chapter 13: Sampling Distribution Models and Confidence Intervals for Proportions

Introduction to Sampling Distributions and Confidence Intervals

In inferential statistics, we use sample data to estimate population parameters. Rather than relying on a single value, we provide a range of plausible values—known as a confidence interval—to express our uncertainty about the true parameter value.

Population Parameters and Sample Statistics

• Parameter: A numerical summary of a population (e.g., population proportion p).

• Statistic: A numerical summary calculated from a sample (e.g., sample proportion \hat{p}).

• Statistical Inference: The process of drawing conclusions about a population based on sample data.

Confidence Intervals for Proportions

A confidence interval (CI) for a population proportion provides a range of plausible values for the parameter, based on the sample statistic and the margin of error.

• General form: Statistic ± Margin of Error

• Margin of Error: Reflects sampling variability and the confidence that the interval captures the parameter.

For a proportion, the CI is:

$\hat{p} \pm z^* \cdot SE(\hat{p})$

where $SE(\hat{p})$ is the standard error of the sample proportion.

Standard Error for a Proportion

The standard error (SE) estimates the standard deviation of the sampling distribution of the sample proportion. Since the population proportion p is usually unknown, we use the sample proportion \hat{p}:

$SE(\hat{p}) = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

Critical Values and Confidence Levels

The critical value $z^*$ depends on the desired confidence level (C). It represents the number of standard errors to extend from the sample statistic to capture the central C% of the normal distribution.

Confidence Level (C)	Critical Value ($z^*$)
90%	1.645
95%	1.960 (≈2)
99%	2.576
99.9%	3.291

To find $z^*$, use a normal probability table, calculator, or statistical software.

Assumptions and Conditions for Constructing a Confidence Interval

Before constructing a confidence interval for a proportion, check these conditions:

• Independence Assumption: Data should be collected using a simple random sample (SRS) or randomized experiment.

• 10% Condition: Sample size $n$ should be less than 10% of the population size $N$ ($n < 0.1N$).

• Success/Failure Condition: There must be at least 10 expected successes and 10 expected failures in the sample: $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$.

One-Proportion Z-Interval

If the above conditions are satisfied, the confidence interval for a population proportion $p$ is:

$\hat{p} \pm z^* \cdot SE(\hat{p})$

where $SE(\hat{p}) = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$ and $z^*$ is the critical value for the desired confidence level.

Naming the Confidence Interval

• One-proportion z-interval: "One" for one population, "proportion" for the categorical variable, "z" for the normal approximation, and "interval" for the range of plausible values.

Interpreting Confidence Intervals

Correct interpretation involves three steps:

1. State the level of confidence (e.g., 95%).

2. State the parameter of interest (e.g., the proportion of all townspeople who agree with school redistricting).

3. State the confidence interval (e.g., between 0.632 and 0.788).

Example: "I am 95% confident that the interval from 0.632 to 0.788 captures the true proportion of all townspeople who agree with school redistricting."

Common Misinterpretations

• It is incorrect to say there is a 95% probability that the true proportion is in the interval. The parameter is fixed; the interval is random.

• It is incorrect to say that other samples will have proportions in the interval. Each sample yields its own interval.

• The confidence interval is about the parameter, not the sample statistic.

Worked Example

Suppose a random sample of 131 townspeople yields 93 who agree with school redistricting. Calculate the 95% confidence interval for the true proportion:

• Sample proportion: $\hat{p} = \dfrac{93}{131} \approx 0.71$

• Standard error: $SE(\hat{p}) = \sqrt{\dfrac{0.71 \times 0.29}{131}} \approx 0.078$

• Critical value for 95%: $z^* = 1.96$

• Confidence interval: $0.71 \pm 1.96 \times 0.078 = (0.632, 0.788)$

Interpretation: "I am 95% confident that between 63.2% and 78.8% of all townspeople agree with school redistricting."

Summary Table: Steps for Constructing a Confidence Interval for a Proportion

Step	Description
1	Check assumptions and conditions (independence, 10%, success/failure)
2	Calculate sample proportion $\hat{p}$
3	Compute standard error $SE(\hat{p})$
4	Find critical value $z^*$ for desired confidence level
5	Construct interval: $\hat{p} \pm z^* \cdot SE(\hat{p})$
6	Interpret the interval in context

What You Cannot Say About the Interval

• Do not claim the interval contains the sample statistic with a certain probability.

• Do not claim the parameter varies; it is fixed, but unknown.

• Do not claim the interval applies to other samples.