Confidence Intervals Notes

Angus Reid Strategies Survey on Evolution and Creationism

  • 59% of Canadians believe in evolution.
  • 42% agree that dinosaurs and humans co-existed, a creationist belief.
  • Survey conducted on 1,088 Canadians with a margin of error of +3.0%, 19/20 confidence.

Confidence Intervals

  • Definition: An interval of values within which the true population mean likely falls.
  • Confidence intervals account for sampling errors, providing upper and lower limits around the sample mean.

Constructing Confidence Intervals

  • Sample mean is a starting point for estimating population mean.
  • Example: Population mean of male Dawson students' heights between 5ft. and 6ft.
  • Confidence levels (common choices):
    • 95% (z = 1.96)
    • 99% (z = 2.58)

Calculating Confidence Intervals when Population Standard Deviation is Known

  • Formula: CI = ext{Sample Mean} \, ext{±} \, z imes ext{Standard Error}
  • Standard Error (SE) when sample size n is known: SE = rac{ ext{Standard Deviation}}{ ext{sqrt}(n)}

Example Calculation

  • Sample: 225 students, mean = 606, population SD = 100
  • SE = rac{100}{ ext{sqrt}(225)} = 6.67
  • 95% CI = 606 ± (1.96 imes 6.67) = 606 ± 13.13
  • CI = [592.87, 619.13]

Standard Error of the Mean

  • Definition: Average distance of sample means from the population mean.
  • Calculating SE: SE = rac{ ext{Standard Deviation}}{ ext{sqrt}(n)}

Confidence Intervals When Population Standard Deviation is Unknown

  • Use t distribution instead of z-scores.
  • Formula changes to: CI = ext{Sample Mean} \, ext{±} \, t imes S_x
  • Estimate standard error with sample standard deviation: S_x = rac{s}{ ext{sqrt}(n-1)}

Example with t-distribution

  • Mean = 100, SD = 12, Sample size = 16
  • Standard Error = rac{12}{ ext{sqrt}(15)}; Find appropriate t value (based on degrees of freedom).

Confidence Intervals for Proportions

  • Formula: CI = P \, ext{±} \, Z imes s_p
  • Where s_p = ext{sqrt} rac{P(1-P)}{n}
  • Calculate confidence interval for a sample proportion (e.g. 40% of students working 20 hours/week).

Final Note

  • Confidence intervals provide a range of values for estimating population parameters.
  • Use appropriate methods depending on what is known (standard deviation vs sample statistics).