Understanding Confidence Intervals for Proportions

Confidence intervals (CIs) are important tools that help us estimate certain values about a whole group (population) by looking at a smaller part (sample).

When we deal with proportions, confidence intervals show a range in which we think the true population proportion is likely to be. This is useful in many situations, like opinion polls, checking product quality, or studying health issues.

Step 1: Know the Proportion and Sample Size

To find a confidence interval for a population proportion, we first need to know two things:

1. Sample size ($n$): This is how many people we surveyed.
2. Number of successes ($x$): This is how many people had the opinion or preference we are interested in.

We can find the sample proportion ($\hat{p}$) using this formula:

$$\hat{p} = \frac{x}{n}$$

For example, if we asked 200 people and 60 said they preferred Brand A over Brand B, we calculate $\hat{p}$ like this:

$$\hat{p} = \frac{60}{200} = 0.3$$

Step 2: Choose the Confidence Level

Next, we need to select a confidence level ($C$), which is usually 90%, 95%, or 99%. Each of these levels has a Z-score that helps us know how far from the average we need to look to capture that percentage:

• For a 90% CI, $Z \approx 1.645$
• For a 95% CI, $Z \approx 1.96$
• For a 99% CI, $Z \approx 2.576$

These Z-scores come from a standard normal distribution.

Step 3: Calculate the Standard Error (SE)

Now, we calculate the standard error (SE) of the proportion using this formula:

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

Using our example, we find the SE:

$$SE = \sqrt{\frac{0.3(1 - 0.3)}{200}} = \sqrt{\frac{0.3 \cdot 0.7}{200}} \approx 0.0346$$

Step 4: Find the Confidence Interval

Next, we use this formula to find the confidence interval:

$$CI = \hat{p} \pm Z \cdot SE$$

For a 95% confidence level, it looks like:

$$CI = 0.3 \pm (1.96 \cdot 0.0346)$$

Calculating the margin of error:

$$ME = 1.96 \cdot 0.0346 \approx 0.0678$$

So, our 95% confidence interval is:

$$CI = (0.3 - 0.0678, 0.3 + 0.0678) \implies (0.2322, 0.3678)$$

Step 5: Understanding the Confidence Interval

We can understand this confidence interval like this: We are 95% sure that the real proportion of people who prefer Brand A is between 23.22% and 36.78%.

Conclusion

Calculating confidence intervals for proportions helps us make good guesses about a population based on sample data. By learning how to calculate sample proportions, find standard errors, and build confidence intervals, students can use these skills in many areas, including math and everyday life.