Statistics: Confidence Intervals and Margin of Error

Confidence Level and Interval

  • Confidence Level: Specifies the degree of certainty (e.g., 95% confidence). This means we are 95% confident that the true population parameter lies within the interval we construct.
  • Example: Shannon's coach wants 95% confidence that Emily averages 10 assists per set.
  • Confidence Interval: Given an average claim (10 assists/set), the confidence interval might range between 9 and 11 assists.
  • Unknown Standard Deviation: When population standard deviation (c3) is unknown, we use the sample standard deviation instead.

Margin of Error

  • Margin of Error Formula: E=timessnE = t imes \frac{s}{\sqrt{n}}
    • Where:
    • E is the margin of error
    • t is the t-value from the t-distribution based on confidence level and degrees of freedom
    • s is the sample standard deviation
    • n is the sample size

Degrees of Freedom

  • Definition: The number of observations (n) that are free to vary. Calculated as:
    Degrees of Freedom=n−1\text{Degrees of Freedom} = n - 1
  • Example: For 6 tests, if 5 scores are free, the degrees of freedom is 5. It relates to how the remaining score is determined based on desired average.

Example Calculation: Confidence Interval for Sleeping Bags

  • Scenario: Prices of sleeping bags keep you warm between 20 and 45 degrees Fahrenheit.

  • Given Data:

    • Sample standard deviation (s) = $28.97
    • Mean price = $83.75
    • Sample size (n) = 20
    • Confidence level = 90%

  • Steps:

    1. Find degrees of freedom:
    • Degrees of Freedom=n−1=20−1=19\text{Degrees of Freedom} = n - 1 = 20 - 1 = 19
    1. Find the t-value for 90% confidence and df = 19 (from t-table):
    • t≈1.729t \approx 1.729
    1. Calculate margin of error:
    • E=1.729×28.9720≈11.2E = 1.729 \times \frac{28.97}{\sqrt{20}} \approx 11.2
    1. Construct confidence interval:
    • 83.75 \pm 11.2
    • ext{Interval} = (72.55, 94.95)

  • Interpretation: With 90% confidence, the mean price of all sleeping bags is between $72.55 and $94.95.

Higher Confidence Levels

  • Implication: Increasing confidence level (e.g., 99%) results in a wider confidence interval, as a greater spread reduces the risk of misrepresenting the true population value.
  • Example for 99% Confidence:
    • New alpha level = 0.01
    • Find t-value for df = 19 and 99% confidence:
    • t≈2.861t \approx 2.861
    • New margin of error then calculated leading to a wider interval.

Sample Size Estimation

  • Formula: n=(z×σE)2n = \left( \frac{z \times \sigma}{E} \right)^2
    • Where:
    • z is the z-value corresponding to the desired confidence level
    • c3 is population standard deviation (not known), use sample standard deviation if unavailable
    • E is the margin of error
  • Rule of Thumb: Always aim for a sample size of at least 30 for sufficient data representation.