Confidence Interval Lecture Notes

Introduction to Confidence Intervals

• Definition: Questions regarding the range within which the true mean (population parameter) is expected to lie with a certain level of confidence.
• Example Question: What is the 98% confidence interval for the average age of college students?
  • Given: ( \bar{X} = 21.9 ), ( \Sigma = 6.1 ), ( N = 120 )

Checking Conditions for Normal Distribution

• Use a normal distribution if:
  • Sigma ( \Sigma ) is known
  • Sample size ( N ) is over 30

Formula for Confidence Interval

• Basic Formula: [ \bar{X} \pm Z \times \left( \frac{\Sigma}{\sqrt{N}} \right) ]
  • ( \bar{X} ): Sample mean
  • ( Z ): Z-score (depends on confidence level)
  • ( \frac{\Sigma}{\sqrt{N}} ): Standard error of the mean

Calculating the Standard Error

• ( \text{Standard error} = \frac{6.1}{\sqrt{120}} = 0.5569 ) (rounded)

Drawing the Picture

• Purpose: Helps visualize the confidence interval
• Components:
  • Mean ( \bar{X} = 21.9 )
  • Two tails (each side of the mean)
  • Values in the tails represent the error margin

Determining the Z-Score

• 98% Confidence Level:
  • Area in the middle: 0.98
  • Area in each tail: 0.01
  • ( Z \approx 2.3264 )
• Using software or Z-tables to find Z-scores

Calculating the Confidence Interval

• For 98% Confidence:
  • ( 21.9 \pm 2.3264 \times 0.5569 )
  • Confidence Interval: [20.6044, 23.1956]

Understanding the Results

• Confidence Interval Interpretation:
  • We are 98% confident that the true mean is between these two values
  • Could be written as: ( 21.9 \pm 1.2956 ) (where 1.2956 is the margin of error)

Margin of Error

• Definition: ( Z \times \frac{\Sigma}{\sqrt{N}} )
• Represents the possible error range of the estimate

Point Estimate

• Definition: ( \bar{X} ) is the best estimate of the population mean ( \mu )

Re-evaluating with Different Confidence Level

• 95% Confidence Interval:
  • ( Z \approx 1.96 )
  • New Confidence Interval: [20.8085, 22.9915]

Writing the Confidence Interval

• Multiple ways to express:
  • "We are 98% confident that..."
  • Simply stating the range with confidence level
  • Using plus/minus format for clarity and brevity

Conclusion

• Always work with two tails for confidence intervals
• Different ways exist to express the intervals, all acceptable
• Key to remember the meaning behind the numbers and formulas used