Statistics Chapter 10 Flashcards

Questions and Answers

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?

1.96 which is rounded to 2

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?

An interval of 2 values computed from sample data that is likely to include the unknown value of a population parameter. Signup and view all the answers

What is the confidence level for an interval?

The likelihood that the procedure used will give an interval that contains the unknown value of the population parameter, most common is 95%. Signup and view all the answers

What is the formula for a 95% CI for a single proportion?

p̂ ± 2 X √p̂(1-p)/n Signup and view all the answers

What is a population proportion?

The proportion of a population with a certain trait, opinion, or characteristic. Signup and view all the answers

What are the z* multipliers for confidence levels of 90%, 95%, 98%, and 99%?

1.65 for 90%, 1.96 for 95%, 2.33 for 98%, 2.58 for 99% Signup and view all the answers

For a sample of 1003 Americans, where 56% believe in life on other planets, calculate the 90% confidence interval.

The confidence interval is 53.46% to 58.46%. Signup and view all the answers

What is the confidence interval formula for a population proportion p̂?

p̂ ± z*√p̂(1-p̂)/n Signup and view all the answers

What are the two conditions for using the formula for a confidence interval for a proportion?

The sample is randomly selected from the population, and both np̂ and n(1-p̂) should be at least 10 (some say at least 5). Signup and view all the answers

What more lenient condition is there for calculating a confidence interval for a proportion?

If the data can be considered to be representative with regard to the question(s) of interest. Signup and view all the answers

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?

The sample proportion p̂ must reflect the frequency of the specific outcome in repeated trials, and both np̂ and n(1-p̂) should be at least 10 (some say at least 5). Signup and view all the answers

What is the formula for a 95% confidence interval?

sample estimate ± margin of error Signup and view all the answers

What is the formula for a 95% confidence level for a margin of error for a sample proportion?

margin of error = 2√p̂(1-p̂)/n Signup and view all the answers

What is the standard error formula for the difference between two sample proportions?

s.e.(p̂₁ - p̂₂) = z*√(p̂₁(1-p̂₁)/n₁ + p̂₂(1-p̂₂)/n₂) Signup and view all the answers

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.