A confidence interval is a range of possible values for an unknown population parameter.

• Instead of a single point estimate, we get an interval
• We are “X% confident” that the interval contains the true population value
• Confidence levels EX: 90%, 95%, 99%

Example: “We are 95% confident that the true mean weight is between 7 grams and 9 grams”

For a population mean \(\mu\) with known standard deviation \(\sigma\):

\[\bar{X} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}\]

Where:

• \(\bar{X}\) = sample mean
• \(z_{\alpha/2}\) = critical value from standard normal distribution
• \(\sigma\) = population standard deviation
• \(n\) = sample size

What does 95% confidence mean?

If we repeated our sampling process many times and constructed a 95% CI each time:

\[P(\bar{X} - 1.96\frac{\sigma}{\sqrt{n}} \leq \mu \leq \bar{X} + 1.96\frac{\sigma}{\sqrt{n}}) = 0.95\]

• 95% of the intervals would contain the true parameter!
• Any single interval either contains the parameter or doesn’t.

Let’s estimate average number of hours of studying for college students.

studyHours <- rnorm(50, mean = 7.2, sd = 1.3)

n <- length(studyHours)
sampleMean <- mean(studyHours)
sampleSd <- sd(studyHours)
standard_error <- sampleSd / sqrt(n)

ciLower <- sampleMean - 1.96 * standard_error
ciUpper <- sampleMean + 1.96 * standard_error

 #or use t tests
t.test(studyHours, conf.level = 0.95)$conf.int

Three factors affect Confidence Interval width:

1. Sample size: (\(n\)): Larger \(n\) → Smaller Interval
2. Confidence level: Higher confidence → Wider Interval
   • 90% Confidence is slimmer than 95% Confidence
   • 99% Confidence is wider than 95% Confidence
3. Population variability: (\(\sigma\)): More variability → Wider Confidence Interval

So you need to decide between Precision or Confidence.

We are 95% confident that the true average studying hours for college students is between 6.91 and 7.58 hours.