Statistics Chapter 10 Flashcards

Confidence Intervals

• Formula: sample estimate ± multiplier × Standard error
• For a 95% confidence interval, the multiplier is 1.96 (rounded to 2).

Factors Influencing Width of Confidence Intervals

• Sample Size: Larger samples yield narrower intervals due to decreased standard error.
• Confidence Level: Higher confidence levels increase the multiplier, producing wider intervals.
• Natural Variability: More variability results in wider intervals; standard deviation measures variability for means, while sample proportion (p̂) measures it for proportions.

Definitions

• Confidence Interval: An interval of two values derived from sample data, likely encompassing the unknown value of a population parameter.
• Confidence Level: The probability that the calculated interval includes the population parameter, commonly at 95%.

Confidence Interval for Single Proportion

• Formula: p̂ ± 2 × √(p̂(1-p)/n)

Population Proportions

• First Setting: Estimating proportions of a characteristic in a population (e.g., drivers on cell phones, smokers quitting with a patch).
• Second Setting: Estimating the long-run probability of an outcome in a repeatable situation (e.g., success of a fertility procedure, likelihood of television failure).

Z* Multipliers for Confidence Levels

• 90%: 1.65
• 95%: 1.96
• 98%: 2.33
• 99%: 2.58

Example Calculation

• Sample of 1003 Americans, 56% believe in extraterrestrial life.
• 90% CI calculation: Standard error = √(0.56 × 0.44 / 1003) = 0.0157.
• Confidence interval = 56 ± 1.645 × 0.016 = (53.46%, 58.46%).

Conditions for Using CI Formulas

• Random Sampling: Samples must be randomly selected from the population.
• Sample Size Requirement: Both np̂ and n(1-p̂) should be at least 10 (or 5).

Lenient Conditions Rule

• Data should be considered representative regarding the questions of interest.

Long Run Probability Conditions

• The sample proportion p̂ must be based on repeated independent trials with a fixed probability.
• Meet the sample size requirements: Both np̂ and n(1-p̂) should be at least 10 (or 5).

Margin of Error

• Formula for 95% CI: sample estimate ± margin of error.
• Margin of error for a sample proportion: 2 × √(p̂(1-p̂)/n), equivalent to 2 standard errors.

Standard Error for Difference Between Two Proportions

• Formula: s.e.(p̂₁ - p̂₂) = z*√(p̂₁(1-p̂₁)/n₁ + p̂₂(1-p̂₂)/n₂)