Statistics6.3: Confidence Intervals for the Population Proportion, p
Objective 1: I can find a point estimate for the population proportion, p.
Recall from section 4.2, that p is the probability of a ______in a binomial experiment. q is the probability of a ______and is found by ______. This probability, p, is called the ______. In this section, we will find a CI for the population proportion, p.
To create a CI for p, we need a ______for p, just like we used ____ as a point estimate for _____. The point estimate for p, the population proportion is _____, the sample proportion, and it is pronounced ______.
The formula to find is:
where ____ is the number of ______in the sample and ____ is the ______.
The ______for the proportion of failures is _____ and is found by ______. The symbol is pronounced ______.
TIY 1: In a survey of 1006 adults from the U.S., 181 said Abraham Lincoln was the greatest president. Find a point estimate for the population proportion of adults who say Abraham Lincoln was the greatest president.
Objective 2: I can create a confidence interval for p, the population proportion.
Constructing a CI for p is similar to constructing a CI for ___. You start with a ______and calculate a ______.
To calculate the margin of error, ____, use the formula:
where _____ is the proportion of successes, _____ is the proportion of failures (______) and
____ is the ______
Steps to constructing a CI for p by hand:
1) Identify the sample statistics / 1)2) Find the point estimates _____ and ______/ 2)
3) Find the critical value, ____, that corresponds to the given confidence level. / 3) Use the standard normal table.
4) Calculate the margin of error, E. / 4)
5) Compute the left and right endpoints and for the ______. / 5) Left endpoint:
Right endpoint:
Confidence Interval:
Just like a CI for the population mean,, you can write your interval in one of three ways.
1)
2)
3)
Example 2: Construct a 95% CI for the proportion of adults in the US who say that football is their favorite sport to watch on TV. Use the information from Example 1.
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TIY 2: Use the data given in TIY 1 to construct a 90% CI to find the proportion of US adults who say that Abraham Lincoln is the greatest president.
Example 3: The graph below is a survey of 900 US adults. Construct a 99% confidence interval for the proportion of adults who think that teenagers are the more dangerous drivers.
TIY 3: Use the survey above to construct a 99% CI for the proportion of adults who think that people over age 75 are the more dangerous drivers.
Objective 3: I can find the minimum sample size needed for a given level of confidence.
One way to increase the precision of the CI without decreasing the level of confidence is to increase the ______. If we take the margin of error formula,
and solve it for ___, we can find the sample size needed for a given c-level and a given margin of error.
Given a confidence level and a margin of error, ____, the minimum sample size ___ needed to estimate ____ isThis formula assumes you have a preliminary estimate for ____ and ____.
If not, use ______and ______.
Example 4: You are running a political campaign and wish to estimate, with 95% confidence, the proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population proportion. Find the minimum sample size needed if
(1) no preliminary estimate is available and (2) a preliminary estimate gives . Compare your results.
TIY 4: You wish to estimate, with 90% confidence and within 2% of the true population, the proportion of males aged 20 to 34 who have high blood pressure. Find the minimum sample size needed if (1) no preliminary estimate is available and (2) a previous survey found that 6.4% of males in this age group had high blood pressure. Compare your results.