Confidence Intervals

Overview of the Confidence Intervals for the Mean

  • The concept of confidence intervals (CI) relates to estimating population parameters using sample statistics.

Calculating the Margin of Error (E)

  • The margin of error is derived from:

    • The critical z score, denoted as $z_c$.

    • The standard deviation of the sampling distribution, often represented as $ rac{ ext{sigma}}{ ext{sqrt}(n)}$ where ( n ) is the sample size.

  • The relationship between the confidence level (denoted as c) and the critical z score is fundamental.

Confidence Level and Critical Z Score

  • Given a confidence level (e.g., 90% or 99%), the critical z scores correspond to the tails of the normal distribution.

  • For instance:

    • For a 90% confidence level, the area in the tails is 10%, with 5% in each tail, leading to critical z scores of +/- $z_c$ that contain the central 90%.

    • Connects to the z-table by looking up the area that corresponds to (1 - (0.10/2)).

Building the Confidence Interval

  • Once you determine $z_c$:

    • The left edge of the confidence interval is calculated as:

    • Left Endpoint = Sample Statistic - Margin of Error, ( E )

    • The right edge is calculated as:

    • Right Endpoint = Sample Statistic + Margin of Error, ( E )

Example of Constructing a Confidence Interval

Scenario

  • An admissions director samples 20 students to estimate their average age.

  • Sample mean ($\bar{x}$) = 22.9 years, ( ext{sigma} = 1.5 ), and confidence level = 90%.

Steps to Construct the Interval

  1. Determine Critical Z Score:

    • For 90%, $z_c$ values are 1.64 or 1.65.

  2. Calculate Margin of Error (E):

    • Using: E = z_c \cdot \frac{\text{sigma}}{\sqrt{n}}

    • Thus: E = 1.645 \cdot \frac{1.5}{\sqrt{20}} \approx 0.55

  3. Construct CI:

    • Left Endpoint = 22.9 - 0.55 = 22.35

    • Right Endpoint = 22.9 + 0.55 = 23.45

    • CI: (22.35, 23.45)

    • Interpretation: There is a 90% confidence that the mean age falls within this interval.

Adjusting Sample Size to Meet Desired Margin of Error

  • Higher confidence levels correspond to broader intervals (less precision).

  • To maintain precision while increasing the confidence level, increase sample size (n).

  • Formula for minimum sample size, given margin of error (E) and confidence level, derived from:

    • n = \left(\frac{z_c \cdot \text{sigma}}{E}\right)^2

  • Increasing n reduces the width of confidence intervals while maintaining higher confidence levels.

Switching from Z to T-Distributions

  • In practice, we may not know ( ext{sigma} ), in which case we estimate it with the sample standard deviation (s).

  • The t-distribution is used instead of z-distribution when sample size is small or when ( ext{sigma} ) is unknown:

    • T distributions have thicker tails than normal distributions and are indexed by degrees of freedom (df = n - 1).

Key Concepts

  • Critical Z Score (z_c): Represents the number of standard deviations from the mean corresponding to the specified confidence level.

  • Margin of Error (E): The maximum expected difference between the true population parameter and a sample estimate.

  • Confidence Interval: The range within which the true population parameter is expected to fall with a certain level of confidence.

Summary of Steps to Construct a Confidence Interval

  1. Determine the critical z score based on the desired confidence level.

  2. Calculate the margin of error using the standard deviation and the critical z score.

  3. Calculate the confidence interval using the sample statistic and margin of error.