Confidence Intervals for Population Mean

Formula and Components

• Basic formula: $\overline{x}\pm t^{\ast }\times \frac{s}{\sqrt{n}}$
• Where:
  • $\overline{x}$ = sample mean
  • $s$ = sample standard deviation
  • $n$ = sample size
  • $t^{\ast }$ = critical value from t-distribution

Required Conditions

For a valid confidence interval:

1. Random sample from the population
2. Either:
   • Sample size $n\ge 30$, or
   • Population is normally distributed

Normal vs. t-Distribution

• When $\sigma$ is known (rare): Use normal distribution with $\overline{x}\pm z^{\ast }\times \frac{\sigma }{\sqrt{n}}$
• When $\sigma$ is unknown (common): Use t-distribution with $\overline{x}\pm t^{\ast }\times \frac{s}{\sqrt{n}}$

Finding Critical Values

• For 95% CI: $t_{0.975,df=n-1}$ (in R: qt(0.975, df=n-1))
• For 90% CI: $t_{0.95,df=n-1}$ (in R: qt(0.95, df=n-1))
• For 99% CI: $t_{0.995,df=n-1}$ (in R: qt(0.995, df=n-1))

Margin of Error

• Margin of Error (ME): $t^{\ast }\times \frac{s}{\sqrt{n}}$
• Represents the precision of your estimate

Interpretation

"We are [confidence level]% confident that the true population mean μ is between [lower bound] and [upper bound]."

Key Relationships

• Higher confidence level → wider interval
• Larger sample size → narrower interval

t-Distribution Properties

• Symmetric around zero
• Bell-shaped but with heavier tails than Normal
• Approaches Normal as sample size increases
• Uses degrees of freedom (df = n-1)