# Confidence Intervals with Proportions

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Confidence Intervals with Proportions
Chapter 9

Suppose we wanted to estimate the proportion of the earth that is covered by water?
Create a jar with different types of coins . . .

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

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

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

Rate your confidence 0 - 100 Guess my age within 10 years?
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?
Your confidence level decreases with smaller intervals % % % %

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

Critical value (z*) z*=1.645 z*=1.96 z*=2.576 .05 .025 .005
Found from the confidence level The upper z-score with probability p lying to its left under the standard normal curve Confidence level tail area z* .05 z*=1.645 .025 .005 z*=1.96 z*=2.576 90% 95% 99%

Confidence interval for a population proportion:
But do we know the population proportion? Statistic + Critical value × Standard deviation of the statistic Margin of error

What are the steps for performing a confidence interval?
1.) Identify the interval by name or formula (CI for one-sample proportion) 2.) Assumptions SRS of context Approximate Normal distribution because np > 10 & n(1-p) > 10 Population is at least 10n 3.) Calculations 4.) Conclusion (in context of problem)

Conclusion Statement: (memorize!!)
We are ________% confident that the true proportion context is between ______ and ______.

Suppose we wanted to estimate the proportion of the earth that is covered by water? Let’s “throw” the earth around and record the number of times that we point to water or land. Repeat the sampling for 50 trials. Calculate a 90% confidence interval for the amount of water on earth. Create a jar with different types of coins . . .

Calculate a 95% confidence interval for the true proportion of water on the earth.
What do you notice?

As the confidence level increases, do the intervals generally get wider or more narrow? Explain.
As the sample size increases, do the intervals generally get wider or more narrow? Explain. More narrow When 100 confidence intervals are generated, why are they all different? Sampling variability If the confidence level selected is 90%, about how many of 100 intervals will cover the true percentage of orange balls? Will exactly this number of intervals cover the true percentage each time 100 intervals are created? Explain.

Where are the last two assumptions from?
SRS of context Approximate Normal distribution because np > 10 & n(1-p) > 10 Population is at least 10n

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 ghost.

Step 1: check assumptions!
Have an SRS of adults np =1012(.38) = & n(1-p) = 1012(.62) = Since both are greater than 10, the distribution can be approximated by a normal curve Population of adults is at least 10,120. Step 1: check assumptions! Step 2: make calculations Step 3: conclusion in context We are 95% confident that the true proportion of adults who believe in ghosts is between 35% and 41%.

The manager of the dairy section of a large supermarket took a random sample of 250 egg cartons and found that 40 cartons had at least one broken egg. Find a 90% confidence interval for the true proportion of egg cartons with at least one broken egg.

Step 1: check assumptions!
Have an SRS of egg cartons np =250(.16) = 40 & n(1-p) = 250(.84) = 210 Since both are greater than 10, the distribution can be approximated by a normal curve Population of cartons is at least 2500. Step 1: check assumptions! Step 2: make calculations Step 3: conclusion in context We are 90% confident that the true proportion of egg cartons with at least one broken egg is between 12.2% and 19.8%.

Another Gallop Poll is. taken
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? To find sample size: However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!

Remember that, in a binomial distribution, the histogram with the largest standard deviation was the one for probability of success of 0.5. 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