Calculate the confidence interval for the population proportion using a population proportion confid...

To calculate the confidence interval for a population proportion, we will follow a series of steps that involve statistical concepts and formulas. The confidence interval gives us a range within which we expect the true population proportion to lie, with a certain level of confidence. Let's assume we are working with a 95% confidence level, which is commonly used in statistical analysis.

Here are the steps to calculate the confidence interval for a population proportion:

Step 1: Define the Sample Proportion

First, we need to define the sample proportion ($\hat{p}$), which is calculated by dividing the number of successes in our sample by the total number of observations in the sample.

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

where:
- $x$ is the number of successes in the sample.
- $n$ is the total number of observations in the sample.

Step 2: Determine the Confidence Level

The confidence level is the probability that the calculated confidence interval will contain the true population proportion. Common confidence levels are 90%, 95%, and 99%. For this example, we will use a 95% confidence level.

Step 3: Find the Z-Score for the Confidence Level

The Z-score corresponds to the chosen confidence level and can be found using a standard normal distribution table or a Z-score calculator. For a 95% confidence level, the Z-score is approximately 1.96.

Step 4: Calculate the Standard Error

The standard error (SE) of the sample proportion is calculated using the formula:

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

Step 5: Calculate the Margin of Error

The margin of error (ME) is the amount that is added to and subtracted from the sample proportion to create the confidence interval. It is calculated using the formula:

$$ME=Z\times SE$$

where:
- $Z$ is the Z-score from Step 3.
- $SE$ is the standard error from Step 4.

Step 6: Calculate the Confidence Interval

Finally, the confidence interval (CI) is calculated by adding and subtracting the margin of error from the sample proportion:

$$CI=\hat{p}\pm ME$$

This gives us the lower and upper bounds of the confidence interval.

Example Calculation

Let's say we have a sample of 200 people, and 50 of them have a certain characteristic (success). We want to calculate the 95% confidence interval for the population proportion.

1. Calculate the sample proportion ($\hat{p}$):

$$\hat{p}=\frac{50}{200}=0.25$$

2. The confidence level is 95%, so the Z-score is approximately 1.96.

3. Calculate the standard error (SE):

$$SE=\sqrt{\frac{0.25(1-0.25)}{200}}=\sqrt{\frac{0.25\times 0.75}{200}}=\sqrt{\frac{0.1875}{200}}\approx 0.0306$$

4. Calculate the margin of error (ME):

$$ME=1.96\times 0.0306\approx 0.0600$$

5. Calculate the confidence interval (CI):

$$CI=0.25\pm 0.0600$$
$$CI=[0.25-0.0600,0.25+0.0600]$$
$$CI=[0.19,0.31]$$

Therefore, the 95% confidence interval for the population proportion is between 0.19 and 0.31. This means we can be 95% confident that the true population proportion lies within this range.