Normal Distribution Calculate Confidence Interval

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. For normally distributed data, we can calculate confidence intervals using the sample mean and standard deviation.

What is a Confidence Interval?

A confidence interval provides an estimated range of values which is likely to contain the population parameter. The most common confidence levels are 90%, 95%, and 99%.

The width of the confidence interval depends on:

The desired confidence level
The sample size
The sample standard deviation

Higher confidence levels result in wider intervals, while larger sample sizes produce narrower intervals.

Normal Distribution Basics

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. Key properties include:

Symmetrical around the mean
Defined by mean (μ) and standard deviation (σ)
68-95-99.7 rule for standard deviations

For large sample sizes (n > 30), the sampling distribution of the mean is approximately normal, even if the population is not normally distributed.

How to Calculate a Confidence Interval

The formula for a confidence interval for a population mean (μ) when σ is known is:

Confidence Interval = x̄ ± z*(σ/√n)

Where:

x̄ = sample mean
z = z-score from standard normal distribution
σ = population standard deviation
n = sample size

For small samples (n ≤ 30) when σ is unknown, use the t-distribution:

Confidence Interval = x̄ ± t*(s/√n)

Where:

t = t-score from t-distribution
s = sample standard deviation

Always check the normality assumption before using these formulas. For non-normal data, consider non-parametric methods.

Worked Example

Example Calculation

Suppose we want to estimate the average height of adult males in a city with 95% confidence.

Given:

Sample mean (x̄) = 175 cm
Sample standard deviation (s) = 10 cm
Sample size (n) = 50

Since n > 30, we can use the z-distribution.

For 95% confidence, z = 1.96.

Margin of error = 1.96 * (10/√50) ≈ 2.78 cm

Confidence interval = 175 ± 2.78 = (172.22, 177.78) cm

We are 95% confident that the true average height falls between 172.22 cm and 177.78 cm.

Interpreting Results

When interpreting confidence intervals:

95% confidence means that if we took 100 samples and calculated 100 confidence intervals, approximately 95 would contain the true population mean.
A 95% confidence interval does not mean there's a 95% probability that the interval contains μ.
Wider intervals indicate more uncertainty about the estimate.

Common confidence levels and their corresponding z-scores:

Confidence Level	Z-Score
90%	1.645
95%	1.960
99%	2.576

FAQ

What does a 95% confidence interval mean?

It means that if we took 100 different samples and calculated 100 confidence intervals, approximately 95 of those intervals would contain the true population mean.

How does sample size affect the confidence interval?

Larger sample sizes result in narrower confidence intervals, providing more precise estimates of the population parameter.

Can I use a confidence interval for non-normal data?

For small samples (n ≤ 30) with unknown variance, the t-distribution is appropriate. For non-normal data with larger samples, consider bootstrapping or other resampling techniques.