When the the conditions from the previous lesson are met, a C% confidence interval for the unknown proportion p is:
p̂ ± z* √((p̂(1-p̂))/n)
Where z* (z-star) is the critical value for the standard normal curve with C% of its area between -z* and z*.
Confidence Intervals––PANIC style:
Parameter Assumptions/Conditions Name the Interval Interval Conclude in Context
Panic Explained P: We are interested in estimating the true proportion of [variable] with C% confidence. A: Check for the Random, 10%, and Large Counts conditions N: [1 or 2]-Sample [t or z]-[interval or test] for population [proportion or mean] I: Calculate the interval C: We are C% confident that the interval from [###] to [###] captures the true proportion of [variable].
To calculate a confidence interval for an unknown population proportion on a calculator, press STAT, then choose TESTS and 1-PropZInt.
X = successes P = sample proportion N = Sample Size
You must show all steps (in the PANIC method) to get credit however even if you do it entirely through your calculator.
Sample Size for Desired Margin of Error
To determine the sample size n that will yield a C% confidence interval for a population proportion p with a maximum margin of error ME, solve the following inequality for n:
z*√((p̂(1-p̂))/n) ≤ ME
Where p̂ is a guessed value for the sample proportion. The margin of error will always be less than or equal to ME if you use p̂ = 0.5.
Inference about a population proportion usually arises when we study categorical variables. To estimate a population mean, we have to record values of a quantitative variable for a sample of individuals.
To calculate a 95% confidence interval for U, we use our formula:
Statistic ± (critical value)*(standard deviation of statistic)
The critical value, z* = 1.96 (used for 95% confidence), tells us how many standardized units we need to go out to catch the middle 95% of the sampling distribution. Our interval is:
x̄ ± (z*) * (σ/√(n)).
We call such an interval a one-sample z interval for a population mean. This method isn’t useful in practice, however. In most real-world settings, if we don’t know the population mean U, then we don’t know the population standard deviationσeither. For this reason, we will rarely use z procedures to calculate a confidence interval for a population mean μ. The only time z procedures are valid is when the population standard deviation is known.