Lecture 4.1-6 confidence interval for a mean

Overview of Confidence Intervals

  • Confidence Intervals (CI) provide a range of values which are likely to contain the population parameter.

  • Important for estimating the mean with a specified level of confidence (e.g. 95%).

Normal Distribution and Confidence Intervals

  • Normal Distribution:

    • Mean (µ) = 0 and Standard Deviation (σ) = 1.

    • Used for calculating probabilities and creating confidence intervals.

  • Area under the curve representation helps visualize the likelihood (e.g. P(-1.96 < x < 1.96) = 95%).

Confidence Interval for a Mean

  • Components:

    • Point Estimate: Single value estimate of the population parameter.

    • Margin of Error: ± some amount dependent on the standard error and t-score.

  • Calculation Steps:

    1. Calculate the sample mean and standard error.

    2. Determine the t-score based on desired confidence level and degrees of freedom (df = n - 1).

    3. CI formula: CI = point estimate ± margin of error.

Estimation and Requirements

  • Estimation involves:

    • Using sample data to estimate a population parameter.

  • Requirements for valid CIs:

    • Quantitative data.

    • Random sampling for inference.

    • Normal distribution of the variable or large sample size (n > 15).

Example: Hours Worked per Week

  • Source: 2014 General Social Survey.

  • Basic statistics based on n = 1501:

    • Mean (y) = 41.9 hours.

    • Standard deviation (s) = 14.4 hours.

    • 95% CI calculated: Lower limit = 41.17, Upper limit = 42.63.

    • Interpretation: We are 95% confident that the mean hours worked per week is between these limits.

T-Distribution

  • Used when sample size is small or population standard deviation is unknown.

  • Properties:

    • Symmetric and bell-shaped.

    • More spread out than the normal distribution; the shape changes with degrees of freedom.

    • Degrees of freedom affect the t-scores (e.g., df = 100 results in different t-values compared to df = 10).

Calculating Confidence Intervals

  • Key steps for CI calculation using t-distribution:

    1. Determine standard error (se = s/√n).

    2. Find the appropriate t-value based on df and confidence level.

    3. Calculate the confidence interval limits.

  • Example calculation leads to conclusions about hours worked:

    • Result: CI from 41.18 to 42.63 hours.

Width of Confidence Intervals

  • Factors Affecting CI Width:

    • Higher confidence levels yield wider intervals.

    • Larger sample sizes produce narrower intervals.

  • Examples are provided comparing 90%, 95%, and 99% CIs with varying sample sizes, displaying how they affect margin of error and error probability.

Summary Points

  • Consistently interpreting results from confidence intervals can aid understanding of data.

  • Ability to use statistical software enhances capacity to find descriptive and inferential statistics.

  • Proper interpretation bridges the gap between raw data and population-level conclusions.