Confidence Intervals for Proportions

p-hat

the sample proportion and an unbiased estimator of the population proportion,p, when it is unknown (varies between samples)

With one sample..

you CANNOT conclude the population proportion but you CAN come up with an INTERVAL that may contain the true proportion and how CONFIDENT you are that it falls within the interval

Remember for finding Confidence Intervals for Proportions

• The sampling model for p-hat is approximately Normal assuming np is greater than or equal to 10 and nq is greater than or equal to 10

• The mean of the sampling model is p

• The standard deviation of the sampling model is square root of pq/n assuming the population size is at least 10 times larger than the sample size (10% condition)

Standard Error Formula

SE= square root of (p-hat)(q-hat)/n (same as standard deviation but with p-hat instead of p)

According to the 68-95-99.7 (Empirical Rule)…

95% of all possible samples of size # will produce a statistic p-hat that is within 2 standard errors of the mean of our sampling model (distance between actual p value and statistic p-hat will usually (95% of the time) be less than or equal to the standard error

“#% Confidence”

Formally means that “#% of samples this size will produce confidence intervals that capture the true proportion”

To be more confident..

widen the interval (increase the ME)

critical value

number of standard errors needed so that the Margin of error size corresponds to new confidence level; found from computer, calculator, or Normal Probability table (z-table)

Assumption and Condition needed to check to be able to MAKE a confidence interval

• Independence Assumption

• Randomization Condition

• 10% Condition

• Sample Size Assumption

• Success/Failure Condition

PANIC ( how to find confidence interval)

• P(parameter of interest): “I want to find and interval that’s likely within #% to obtain true proportion

• A(Assumptions and Conditions): Check to see if any are violated

• N(name the interval): Identify type of interval

• I (Interval Calculation): find SE, ME, and Confidence Interval

• C(Conclusion): interpret the confidence interval in context

A Confidence interval procedure works if..

it’s claimed parameter capturate (confidence level) is the fraction of the resulting intervals that contain the parameter of interest when it’s applied over many random samples

What can go wrong?

• Don’t misstate what the interval means

• Don’t suggest that the parameter varies

• Don’t claim that other samples will agree

• Don’t be certain about your parameter

• Don’t forget that the confidence interval is about the parameter

• Don’t claim to know too much

• Not all intervals interpreted would have the true parameter

• Treat the whole interval equally

• Beware of too large of a margin to be useful

• Look out for violations of the assumptions