How to Calculate Confidence Interval for Normal Distribution in R

What is a Confidence Interval?

A confidence interval (CI) is a range of values that is likely to contain the true population parameter with a certain level of confidence. For example, a 95% confidence interval suggests that if the same study were repeated many times, 95% of the intervals would contain the true parameter.

Confidence intervals provide more information than point estimates by showing the precision of the estimate. They are widely used in scientific research, quality control, and decision-making processes.

Normal Distribution

Normal distribution, also known as Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve. Many natural phenomena follow a normal distribution, including heights, weights, test scores, and measurement errors.

Key properties of normal distribution:

• Symmetrical around the mean
• Defined by mean (μ) and standard deviation (σ)
• 68-95-99.7 rule applies (approximately 68% within 1σ, 95% within 2σ, 99.7% within 3σ)

When data follows a normal distribution, we can use z-scores to calculate confidence intervals.

Calculating Confidence Interval in R

R provides several functions to calculate confidence intervals for normally distributed data. The most common approach uses the t.test() function for small samples or the prop.test() function for proportions.

Step-by-Step Process

1. Collect your sample data
2. Calculate the sample mean and standard deviation
3. Determine the confidence level (typically 90%, 95%, or 99%)
4. Find the appropriate critical value from the t-distribution table
5. Calculate the margin of error
6. Determine the confidence interval

Using R Code

Here's an example of how to calculate a 95% confidence interval in R:

# Sample data
data <- c(72, 75, 78, 80, 76, 79, 82, 77, 81, 74)

# Calculate confidence interval
ci <- t.test(data, conf.level = 0.95)

# Print results
print(ci)

Worked Example

Let's calculate a 95% confidence interval for the following sample of test scores:

Scores: 85, 88, 92, 78, 89, 91, 84, 87, 90, 82

Step 1: Calculate Sample Statistics

Mean (x̄) = (85 + 88 + 92 + 78 + 89 + 91 + 84 + 87 + 90 + 82)/10 = 86.6

Standard Deviation (s) ≈ 4.5

Step 2: Determine Critical Value

For a 95% confidence level, the critical z-value is approximately 1.96.

Step 3: Calculate Margin of Error

Margin of Error = z*(s/√n) = 1.96*(4.5/√10) ≈ 2.8

Step 4: Determine Confidence Interval

CI = 86.6 ± 2.8 → (83.8, 89.4)

This means we are 95% confident that the true population mean test score falls between 83.8 and 89.4.

Interpreting Results

When interpreting confidence intervals for normal distribution:

• Wider intervals indicate less precision
• Narrower intervals indicate more precise estimates
• If the interval contains the hypothesized value, it suggests the hypothesis is plausible
• If the interval doesn't contain zero, it suggests a statistically significant effect