Lecture Notes: Confidence Intervals

Summary of Confidence Intervals

• When Sigma is Known (for mean (\mu)):
  • Formula: (\bar{x} \pm Z \cdot \sigma_{\bar{x}})
• When Sigma is Unknown:
  • Use a t-distribution:
  • Formula: (\bar{x} \pm t \cdot s_{\bar{x}})
• For a Proportion (p):
  • Formula: (\hat{p} \pm Z \cdot \sigma_{\hat{p}})_

Understanding the Formula

• The basic formula for a confidence interval remains consistent with adjustments for different types of problems (means vs. proportions).
• It's important to differentiate between (\sigma_{\bar{x}}) and (\sigma_{\hat{p}}).

Key Concepts

• Identifying the Type of Problem:

  • Determine if the question concerns means or proportions.
  • Example: "67 out of 120 people said yes" requires calculating a proportion (mistaking this for means is common).

• Common Mistakes:

  • Misidentifying problems as means instead of proportions.
  • Not calculating (\hat{p}) correctly from the data.

Solving Confidence Interval Problems

• Example Problem: Determine if people support stricter gun control laws.
  • Out of 400 people surveyed, 296 said yes.
  • Calculate a 96% confidence interval.

Steps to Solution

1. Identify the Problem Type:
   • It's a proportion problem.
2. Calculate (\hat{p}):
   • (\hat{p} = \frac{296}{400} = 0.74)
   • (\hat{p}) represents the sample proportion.
3. Find (\sigma_{\hat{p}}):
   • (\sigma_{\hat{p}} = \sqrt{\frac{0.74 \times (1 - 0.74)}{400}} \approx 0.0219)
4. Check Normality Conditions:
   • Are both (np) and (n(1-p)) at least 10?
5. Draw the Distribution Picture:
   • Place (\hat{p} = 0.74) and find the Z values.
6. Calculate Confidence Interval:
   • (0.74 \pm 2.05 \times 0.0219)
   • Final interval: ( [0.695, 0.785])
   • Expressed as percentages: 69.5% to 78.5% support stricter laws.

Additional Problem

• Finding Sample Size for Desired Margin of Error
  • Given (e = 0.05) at 99% confidence, (\hat{p} = 0.5).
  • Solve using: (n = \frac{Z^2 \cdot \hat{p} \cdot (1 - \hat{p})}{e^2})
  • Determine Z using distribution picture; example: (Z \approx 2.5758).
  • Compute: (n \approx 664).

Important Tips

• Always start by identifying whether the problem is about means or proportions.
• Ensure calculations for (\hat{p}) are accurate and interpret results in context.
• Utilize normality checks and sketches to aid understanding and accuracy.
• Practice using formulas extensively to become proficient in solving these problems.