Confidence Interval for a Proportion, PA Residency Example

Introduction

This tutorial will guide you through the process of calculating a confidence interval for a proportion, using the Pennsylvania residency example from a STAT 200 course. Understanding confidence intervals is essential in statistics for estimating the range in which a population parameter lies, based on sample data.

Step 1: Understand the Data

Before calculating a confidence interval, gather the necessary data:

• Identify the sample size (n), which is the total number of observations.
• Determine the number of successes (x), which is the count of favorable outcomes.

For example, if you are assessing the proportion of Pennsylvania residents in a sample of 100 individuals who support a certain policy, you may find that 45 of them support it. Here, n = 100 and x = 45.

Step 2: Calculate the Sample Proportion

The sample proportion (p-hat) is calculated using the formula:

[ \hat{p} = \frac{x}{n} ]

Using our example:

• ( \hat{p} = \frac{45}{100} = 0.45 )

This means that 45% of your sample supports the policy.

Step 3: Determine the Confidence Level

Choose a confidence level for your interval, commonly 90%, 95%, or 99%. A higher confidence level results in a wider interval.

• For a 95% confidence level, the critical value (z*) is approximately 1.96.

Step 4: Calculate the Standard Error

The standard error (SE) of the sample proportion can be calculated as follows:

[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} ]

Using our example:

• ( SE = \sqrt{\frac{0.45(1 - 0.45)}{100}} )
• ( SE = \sqrt{\frac{0.45 \times 0.55}{100}} \approx 0.0497 )

Step 5: Calculate the Margin of Error

The margin of error (ME) is calculated by multiplying the critical value by the standard error:

[ ME = z^* \times SE ]

For our 95% confidence level:

• ( ME = 1.96 \times 0.0497 \approx 0.0975 )

Step 6: Construct the Confidence Interval

The confidence interval is calculated using the sample proportion and the margin of error:

[ \text{Confidence Interval} = \hat{p} \pm ME ]

For our example:

• Lower limit = ( 0.45 - 0.0975 = 0.3525 )
• Upper limit = ( 0.45 + 0.0975 = 0.5475 )

Thus, the 95% confidence interval for the proportion of Pennsylvania residents who support the policy is approximately (0.3525, 0.5475).

Conclusion

In this tutorial, you learned how to calculate a confidence interval for a proportion using a practical example. Key steps included understanding your data, calculating the sample proportion, determining the confidence level, and constructing the confidence interval.

Next steps could involve applying this method to different datasets or exploring how changes in sample size or confidence level affect the width of the confidence interval.