Confidence Intervals Chapter 19

Rate your confidence Name Mr. Holloways age within 10 years? within 5 years? within 1 year? Shooting a basketball at a wading pool, will make basket? Shooting the ball at a large trash can, will make basket? Shooting the ball at a carnival, will make basket?

What happens to your confidence as the interval gets smaller? The larger your confidence, the wider the interval.

Point Estimate singleUse a single statistic based on sample data to estimate a population parameter Simplest approach variationBut not always very precise due to variation in the sampling distribution

Confidence intervals Are used to estimate the unknown population parameter Formula: estimate + margin of error

Margin of error Shows how accurate we believe our estimate is more preciseThe smaller the margin of error, the more precise our estimate of the true parameter Formula:

Found from the confidence level The upper z-score with probability p lying to its right under the standard normal curve Confidence leveltail areaz* Critical value (z*).05 z*= z*= z*= % 95% 99%

For the sampling distribution of, and for large* n the sampling distribution of p is approximately normal. * np 10 and n(1-p) 10 Recall

Confidence interval for a population proportion: estimate Critical value Standard deviation of the statistic Margin of error

Standard Error The standard error of a statistic is the estimated standard deviation of the statistic, using the sample values since we dont know the true population values. For sample proportions, the standard deviation of the sampling distribution is This means that the standard error of the sample proportion is

So our confidence interval is actually: estimate Critical value Standard error Margin of error

Assumptions: Assumptions for inference with proportions: –Data values must be independent –Large enough sample for the sampling distribution to be approximately normal We cant actually check all of the assumptions, so we check related conditions

Conditions: SRS n<10% of the population At least10 Successes/Failures

Example For a project, a student randomly sampled 182 other students at a large university to determine if the majority of students were in favor of a proposal to build a parking garage. He found that 75 were in favor of the proposal. Use a 95% confidence interval to estimate the proportion of the student body in favor of the proposal. Define the Parameter of interest –Let p = the true proportion of all students at the university that favor the proposal.

Example - Conditions We are told that he took a random sample. 182 students is certainly less than 10% of all students at a large university Our sampling distribution is approximately normal because

Example - Calculations We will use a 1-proportion z-interval to approximate the true proportion of students who favor the proposal. A 95% confidence interval for p can be found using

Example – Calculations (continued) The 95% confidence interval for p is ( , )

Example – Conclusion We are 95% confident that the true proportion of students at this university that support the proposal to build a new parking garage on campus is between and

Confidence level Is the success rate of the method used to construct the interval Using this method, ____% of the time the intervals constructed will contain the true population parameter

What does it mean to be 95% confident? 95% chance that p is contained in the confidence interval The probability that the interval contains p is 95% The method used to construct the interval will produce intervals that contain p 95% of the time.

Example – Interpretation of the confidence level If we were to repeat this process many times, 95% of the confidence intervals we constructed would capture the true proportion of students at this university that support the proposal to build a new parking garage on campus. Note: You only need to interpret the confidence level if it you are specifically asked to.

Interpreting a confidence interval: We are ________% confident that the true proportion of context is between ______ and ______.

Interpreting the confidence level: If we were to repeat this process many times, ________% of the confidence intervals we constructed would capture the true proportion of context.

A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghosts.

Conditions: Have an SRS of adults 1012 adults is certainly <10% of adults We have at least 10 successes & 10 failures, so the sampling distribution is approximately normal Step 2: check conditions! p = the true proportion of all adults that believe in ghosts Step 1: define parameter!

We are 95% confident that the true proportion of adults who believe in ghosts is between.35 and.41 Step 3: do the calculations Step 4: conclusion in context

Another Gallop Poll istaken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval? To find sample size: However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!

What p-hat (p) do you use when trying to find the sample size for a given margin of error?.1(.9) =.09.2(.8) =.16.3(.7) =.21.4(.6) =.24.5(.5) =.25 By using.5 for p-hat, we are using the worst- case scenario and using the largest SD in our calculations.

Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval? Use p-hat =.5 Divide by 1.96 Square both sides Round up on sample size

How can you make the margin of error smaller? z* smaller (lower confidence level) p smaller n larger (to cut the margin of error in half, n must be 4 times as big) Really cannot change!