In-Depth Notes on Confidence Intervals

Confidence Intervals

• Chapter Overview: This chapter focuses on confidence intervals, including their construction, interpretation, and application in hypothesis testing.

Objectives

• Construct and interpret confidence intervals for:

  • Population proportions

  • Population means

• Determine necessary sample sizes for estimating population parameters within specified margins of error.

• Use confidence intervals to test hypotheses regarding means.

Key Terms

• Point Estimate: A single value estimate of a population parameter that estimates the population value.

• Interval Estimate: Preferred over point estimates as it includes a margin of error.

• Margin of Error (E): Indicates the extent of the interval around the point estimate.

Population Proportion

• Point Estimate Formula: p = \frac{x}{n} where:

  • p = sample proportion

  • x = number with a specified characteristic in the sample

  • n = sample size

Example: DACA Poll

• A Quinnipiac University Poll sampled 1412 voters:

  • In favor: 1158

  • Point estimate of proportion in favor:
    p = \frac{1158}{1412} = 0.82

• If margin of error is 0.03, the interval estimate is: p \pm E

  • Lower bound: p - E

  • Upper bound: p + E

Constructing Confidence Intervals for Proportions

• Critical Value: Use critical values from Table V based on desired confidence level (e.g., 0.90, 0.95, 0.99).

• Confidence Interval Formula:

  • Lower bound:
    p - \frac{z/2 \, \sqrt{p(1-p)}}{\sqrt{n}}

  • Upper bound:
    p + \frac{z/2 \, \sqrt{p(1-p)}}{\sqrt{n}}

• Conditions for normal approximation:

  • np \geq 10

  • n(1 - p) \geq 10

Example: Confidence Interval for Voters

• For 1412 voters, with confidence level at 99% (Critical value = 2.575):

  • Compute bounds using p = 0.82 :

  • Lower bound:
    0.82 - 2.575 \left(\frac{\sqrt{0.82(1-0.82)}}{\sqrt{1412}}\right)

  • Upper bound:
    0.82 + 2.575 \left(\frac{\sqrt{0.82(1-0.82)}}{\sqrt{1412}}\right)

Understanding Confidence Levels

• Confidence level indicates the proportion of intervals that include the parameter across many samples. For example, with a 95% confidence level:

  • Expected number of intervals containing the parameter is approximately 95 out of 100.

• Important Note: A specific interval either does or does not contain the true population parameter, which makes probability statements about individual intervals inappropriate.

Sample Size Determination

• Formula for Sample Size to estimate population proportion with margin of error E:
  n = \left(\frac{z/2 \cdot E}{p(1-p)}\right)^2

• Adjust according to previous estimates or set p = 0.5 for maximum variability.

Example: Sociologist Study

• Desired Margin of Error: 3% with 90% confidence level, using prior estimate p = 0.824 ;

  • Resulting sample size calculation leads to a requirement of:

  • $ n = 437 $

Point and Interval Estimates for Means

• Point Estimate for Mean:
  \bar{x} estimates population mean \mu .

• Interval Estimate for Mean: \bar{x} \pm E

  • Where E is calculated using:
    E = t/2 \cdot \frac{s}{\sqrt{n}}

• Critical value of t is determined based on confidence level and degrees of freedom.

Example: Penny Weights

• Samples collected from 17 pennies:

  • Mean weight \bar{x} = 2.464 grams

  • Calculate 99% confidence interval around this value using corresponding t-critical value.

  • Result: We are 99% confident that mean weight falls within certain bounds.

Summary of Conditions for CI Construction

• Data should come from a random sample.

• Sample size should be sufficiently small relative to the population (n ≤ 0.05N).

• Data should be normally distributed, or sample size should be at least 30.

Overall Confidence Interval Use

• Confidence intervals provide a range of plausible values for population parameters.

• They are useful in both estimating parameters and testing statistical hypotheses.