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Questions and Answers
What is the formula for a confidence interval?
What is the formula for a confidence interval?
sample estimate ± multiplier X Standard error
For a 95% confidence interval, what do we use for a multiplier?
For a 95% confidence interval, what do we use for a multiplier?
1.96 which is rounded to 2
What three factors determine the width of the confidence interval?
What three factors determine the width of the confidence interval?
Sample size, confidence level, the natural variability among individual units.
What is the definition of a confidence interval?
What is the definition of a confidence interval?
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What is the confidence level for an interval?
What is the confidence level for an interval?
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What is the formula for a 95% CI for a single proportion?
What is the formula for a 95% CI for a single proportion?
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What is a population proportion?
What is a population proportion?
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What are the z* multipliers for confidence levels of 90%, 95%, 98%, and 99%?
What are the z* multipliers for confidence levels of 90%, 95%, 98%, and 99%?
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For a sample of 1003 Americans, where 56% believe in life on other planets, calculate the 90% confidence interval.
For a sample of 1003 Americans, where 56% believe in life on other planets, calculate the 90% confidence interval.
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What is the confidence interval formula for a population proportion p̂?
What is the confidence interval formula for a population proportion p̂?
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What are the two conditions for using the formula for a confidence interval for a proportion?
What are the two conditions for using the formula for a confidence interval for a proportion?
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What more lenient condition is there for calculating a confidence interval for a proportion?
What more lenient condition is there for calculating a confidence interval for a proportion?
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What are the two conditions for using the formula for a confidence interval for a proportion when the parameter is a long-run probability p?
What are the two conditions for using the formula for a confidence interval for a proportion when the parameter is a long-run probability p?
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What is the formula for a 95% confidence interval?
What is the formula for a 95% confidence interval?
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What is the formula for a 95% confidence level for a margin of error for a sample proportion?
What is the formula for a 95% confidence level for a margin of error for a sample proportion?
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What is the standard error formula for the difference between two sample proportions?
What is the standard error formula for the difference between two sample proportions?
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Study Notes
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)
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Description
Test your knowledge with these flashcards covering key concepts from Chapter 10 of Statistics. Learn about confidence intervals, the multipliers used, and the factors that influence their width. Perfect for quick revision and self-assessment.