Chapter 16: Confidence Intervals for Proportions Flashcards
Define ‘Standard error’.
When we estimate the standard deviation of a sampling distribution using statistics found from the data, the estimate is called a standard error.
SE(p̂ ) = sqrt ( p̂ q̂ / n)
Define ‘Confidence interval’.
A level C confidence interval for a model parameter is an interval of values often of the form
estimate ± margin of error
found from data in such a way that C% of all random samples will yield intervals that capture the true parameter value.
Define ‘One-proportion z-interval’.
A confidence interval for the true value of a proportion based on the Nominal approximation. The confidence interval is
p̂ ± zSE(p̂ ),
where z is a critical value from the standard Normal model corresponding to the specified confidence level.
Define ‘Margin of error’.
In a confidence interval, the extent of the interval on either side of the observed statistic value is called the margin of error. A margin of error is typically the product of a critical value from the sampling distribution and a standard error from the data. A small margin of error corresponds to a confidence interval that gives relatively little information about the estimated parameter. For a proportion,
ME = z* sqrt ( p̂ q̂ / n)
Define ‘Critical value’.
The number of standard errors to move away from the mean of the sampling distribution to correspond to the specified level of confidence. The critical value, denoted z*, is usually found from a table or with technology.
(Found from a standard normal model)
How can you correctly interpret a confidence interval?
You can claim to have the specified level of confidence that the interval you have computed actually covers the true value.
What is the relationship between n (sample sixe), certainty (confidence level), and precision (margin of error)?
For the same sample size and true pop. proportion, more certainty means less precision and vice versa.
What are the assumptions and conditions for finding and interpreting confidence intervals?
- Independence Assumption and Randomization Condition
- 10% Condition
- Success/Failure Condition
How can you decide on a sample size required?
Invert the calculation for margin of error, given a guess at the proportion, a confidence level, and a desired margin of error.
Throwback to confusion from previous chapter:
Randomization condition: if dependency, often understating the true SE.
10% condition: if a sample is more than 10% of population, overestimate of the true SE (be sure to note that reported confidence level is lower than true value).
Sample size assumption: Need more data as the proportion gets closer and closer to either extreme (0 or 1).