Module 16: Introduction to Confidence Intervals

Connection to Theoretical Sampling Distribution and Normal Model

• Theoretical sampling distribution models help in inference procedures.
• Discussions begin with simulations to highlight the sampling process.
  • Simulations illustrate the randomness and probability model nature of sampling processes.

Mathematical Model for Sampling Distribution

• Center of Sample Proportions:
  • Mean of sample proportions is equal to the population proportion ( p ).
• Spread of Sample Proportions:
  • Standard deviation (standard error) is used for spread.
  • Formula for standard error is provided but not detailed here.
• Shape Condition:
  • Normal model fits if expected successes and failures are at least 10.

Using Normal Model for Sampling Distribution

• Requires satisfaction of specific conditions.
• Empirical Rule Application:
  • Applies when a normal model can be used.
  • Rule provides percentages for values within standard deviations from the mean:
    • 68% within 1 standard deviation.
    • 95% within 2 standard deviations.
    • 99.7% within 3 standard deviations.

Confidence Intervals

• Relate to standard deviations and standard errors:
  • 68% within 1 standard error of population proportion.
  • 95% within 2 standard errors of population proportion.
  • 99.7% within 3 standard errors of population proportion.
• 95% Confidence Interval:
  • Contains population proportion 95% of the time (wrong 5% of the time).
  • Margin of error equals two standard errors.
  • Formula: ( \hat{p} \pm 2 \times \text{standard error} ).

Conditions for Normal Model

• Ensure the normal model is a good fit:
  • Check if expected successes and failures meet the necessary conditions (at least 10).

Example: Overweight Men

• Data from CDC shows 68% of US men overweight.
• Sample of 40 men found 75% overweight.
• Checking Normality Conditions:
  • Expected successes and failures: ( n \times p = 40 \times 0.68 \approx 13 ) (conditions met).
• Calculating Standard Error:
  • Formula: ( \sqrt{0.68 \times 0.32 / 40} \approx 0.074 ).
• 95% Confidence Interval Calculation:
  • Sample proportion ( \hat{p} = 0.75 ).
  • Interval: 60.2% to 89.8%.
• Interpretation:
  • 95% confidence that 60.2% to 89.8% of US men are overweight this year.