A Confidence Interval (CI) is a range of values used to estimate the true value of a population parameter, such as the mean or proportion, based on sample data. The formula for a confidence interval depends on the parameter being estimated (e.g., population mean, population proportion) and the type of data (e.g., normal distribution or binomial).

General Formula for Confidence Interval

For a population mean with a known population standard deviation ($\sigma$), the confidence interval is given by:

$CI = \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$

Where:

• $\bar{x}$ = sample mean
• $Z_{\alpha/2}$ = Z-value corresponding to the desired confidence level (e.g., for 95% confidence, $Z_{\alpha/2} \approx 1.96$)
• $\sigma$ = population standard deviation
• $n$ = sample size

For a population mean with an unknown population standard deviation, the confidence interval is:

$CI = \bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}}$

Where:

• $t_{\alpha/2}$ = t-value from the t-distribution, based on the desired confidence level and degrees of freedom ($df = n - 1$)
• $s$ = sample standard deviation
• The rest of the variables are the same as above.

Example of Confidence Interval for a Population Mean

Let’s say you are conducting a study on the average height of a population of adult women. You take a sample of 50 women, and you find the following:

• Sample mean ($\bar{x}$) = 160 cm
• Sample standard deviation ($s$) = 8 cm
• Sample size ($n$) = 50
• You want a 95% confidence interval.

1. Find the critical value: For a 95% confidence level, the $t$-value for 49 degrees of freedom (since $n-1 = 50-1 = 49$) can be found from a t-distribution table. For 95% confidence, the $t_{\alpha/2}$ value is approximately 2.0096.

2. Calculate the margin of error: $ME = t_{\alpha/2} \times \frac{s}{\sqrt{n}} = 2.0096 \times \frac{8}{\sqrt{50}} = 2.0096 \times 1.1314 \approx 2.275$

3. Calculate the confidence interval: $CI = \bar{x} \pm ME = 160 \pm 2.275$ This gives us: $CI = (157.725, 162.275)$

Thus, the 95% confidence interval for the population mean height is approximately (157.73 cm, 162.28 cm). This means we are 95% confident that the true mean height of adult women in the population lies within this range.

Interpretation:

This confidence interval implies that if we were to repeatedly sample from the population and compute a confidence interval each time, about 95% of those intervals would contain the true mean height.

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Would you like a more detailed breakdown of any of the steps or concepts used?

Here are 5 related questions:

1. How does the sample size impact the width of the confidence interval?
2. How do you choose between using a Z-distribution or a t-distribution?
3. What does it mean when we say a confidence interval has a "95% confidence"?
4. How can you calculate the confidence interval for a population proportion?
5. What happens if the confidence interval includes zero?

Tip: The larger your sample size, the narrower the confidence interval, because more data leads to a more precise estimate of the population parameter.