Chapter 19: Confidence Intervals for Proportions

Question 1

Define ‘Standard error’.

Answer 1

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)

Question 2

Define ‘Confidence interval’.

Answer 2

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.

Question 3

Define ‘One-proportion z-interval’.

Answer 3

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.

Question 4

Define ‘Margin of error’.

Answer 4

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)

Question 5

Define ‘Critical value’.

Answer 5

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)

Question 6

How can you correctly interpret a confidence interval?

Answer 6

You can claim to have the specified level of confidence that the interval you have computed actually covers the true value.

Question 7

What is the relationship between n (sample sixe), certainty (confidence level), and precision (margin of error)?

Answer 7

For the same sample size and true pop. proportion, more certainty means less precision and vice versa.

Question 8

What are the assumptions and conditions for finding and interpreting confidence intervals?

Answer 8

• Independence Assumption and Randomization Condition
• 10% Condition
• Success/Failure Condition

Question 9

How can you decide on a sample size required?

Answer 9

Invert the calculation for margin of error, given a guess at the proportion, a confidence level, and a desired margin of error.

Question 10

Throwback to confusion from previous chapter:

Answer 10

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).