Confidence Intervals

Solution & Explanation

What is a Confidence Interval?

A confidence interval (CI) is a range of values, derived from a data sample, that is likely to contain the value of an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval. For example, a 95% confidence interval suggests that if you were to take 100 different samples and compute a confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the population parameter.

Importance of Understanding Confidence Intervals

• Quantifies Uncertainty: Confidence intervals provide a range of plausible values for the population parameter, offering a more informative picture than a single point estimate.
• Decision-Making: They help in making informed decisions by providing a range within which the true parameter value is likely to lie.
• Statistical Significance: Confidence intervals can be used to determine the statistical significance of an estimate. If a confidence interval for a mean difference does not include zero, it indicates a statistically significant difference.

Steps to Calculate a Confidence Interval

1. Identify the Statistic of Interest: Determine the sample statistic (e.g., sample mean or proportion) you want to estimate the confidence interval for.

2. Select Confidence Level: Decide on the confidence level (e.g., 90%, 95%, 99%). A higher confidence level means a wider interval.

3. Calculate the Sample Statistic: Compute the sample mean or proportion from your data.

4. Determine the Standard Error (SE):

   • For a mean: $SE = \frac{\sigma}{\sqrt{n}}$ where $\sigma$ is the standard deviation of the sample and $n$ is the sample size.
   • For a proportion: $SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ where $\hat{p}$ is the sample proportion.

5. Find the Critical Value: Use the z-score or t-score corresponding to your confidence level. This value can be found in statistical tables or calculated using statistical software.

6. Calculate the Margin of Error (ME): $ME = \text{Critical Value} \times SE$

7. Construct the Confidence Interval:

   • For a mean: $\text{CI} = \bar{x} \pm ME$ where $\bar{x}$ is the sample mean.
   • For a proportion: $\text{CI} = \hat{p} \pm ME$

Example

Suppose you have a sample of 100 adults, with a sample mean height of 175 cm and a standard deviation of 10 cm. You want to compute a 95% confidence interval for the mean height.

1. Sample Statistic: Sample mean $\bar{x} = 175$
2. Confidence Level: 95%
3. Standard Error: $SE = \frac{10}{\sqrt{100}} = 1$
4. Critical Value: For 95% confidence, the z-score is 1.96.
5. Margin of Error: $ME = 1.96 \times 1 = 1.96$
6. Confidence Interval: $CI = 175 \pm 1.96 = (173.04, 176.96)$

This means we are 95% confident that the true population mean height is between 173.04 cm and 176.96 cm.