1. 8.2

2. Activity Everybody grab a die You’re going to roll the die 40 times
Counting what proportion of the time you get a 1 or a 6
So if you get 5 ones and 7 sixes, then your proportion is 12/40, or 0.3

3. Activity This proportion is your point estimate ( 𝑝 )
Let’s check our conditions for normality
Since this is a proportion
np≥ ∗ 𝑝 ≈13.3
N(1-p)≥ ∗(1− 𝑝 )≈26.7
We often will need to check independence, but since out population of dice rolls is infinitely large, this is not an issue in this case

4. Activity Because it is normal, we can calculate the standard deviation
In this case, let’s pretend that we don’t know the exact value for p in the population
In this case, we could calculate it, but in most cases we won’t know it
𝑝(1−𝑝) 𝑛
𝑝 (1− 𝑝 ) 𝑛 ≈.0745

5. Activity
Now we’re ready to construct a confidence interval for the proportion of times we get a 1 or a 6
Remember the general form for a confidence interval:
Let’s start with a 95% confidence interval
The “critical value” means how many standard deviations away from the mean we need to go to capture that much area
So for 95% confidence interval, it is 2 standard deviations

6. Activity So we plug in 𝑝 for our statistic (point estimate)
We plug in 2 for our critical value
And the standard error for the standard deviation
Everybody’s answer will be a little bit different
.333±
.333± This is one way to write my interval Or
Or (.184, .482)
So I am 95% confident that the true proportion is between and .482

7. Activity The true value in the population is 1/3 or .333333
Did anybody get an interval that did not include this value?
Meaning that is OUTSIDE of your interval
On average, we should get one person (ish)

8. Activity
So we are 95% confident that the true proportion is between and .482
This is a pretty wide range
As discussed last class, we have 2 options to make this range narrower:
Increase sample size (roll more than 40 times)
Decrease the confidence level
Let’s do option #2

9. Activity Now we want an 80% confidence interval
The point estimate and the standard error do not change
But the critical value does
We want to know how many standard deviations I need to go away from the mean (of the sampling distribution) to capture 80% of the data
Any thoughts on how we can do that?

10. Activity
We can use Table A or InvNorm

11. Activity So our critical value now is 1.28 .333± 1.28 .0745 .333±.095
.333±
.333±.095
( .238, .428)
Narrower than before
We are 80% confident that the true value for p is between .238 and .428
Did anyone get an interval that did not contain .333?
Now try a 99% confidence interval

12. Activity
.333± (.0745)
.333±.192
Yours will be different
Did anyone get an interval that did not contain .333?

13. Formal Definition

14. Using our Calculator We can also ask our calculator to do these for us
HOWEVER, you need to be ABLE to do them by hand
Because you might be asked a questions like “what is the critical value for this interval?” or “what is your standard error?”
STAT—TESTS-- “1-PropZInt…”
(NOT “1-PropZTest”)
Let’s replicate our 99% interval
X=# of successes
N=sample size
C-Level= confidence level
So we will use .99 as our confidence level

15. An Example
The EPA recommends taking remedial action in any home that has radon levels above 4 picocuries per liter (pCi/L). A random sample of 200 houses in Pennsylvania revealed that 127 of them had radon levels above 4 pCi/L. Construct a 95% confidence interval for the proportion of houses that have radon levels above 4 pCi/L

16. An Example
The EPA recommends taking remedial action in any home that has radon levels above 4 picocuries per liter (pCi/L). A random sample of 200 houses in Pennsylvania revealed that 127 of them had radon levels above 4 pCi/L. Construct a 95% confidence interval for the proportion of houses that have radon levels above 4 pCi/L, then interpret this interval in context
(.56828, )
“We are 95% confident that the true proportion of houses in Pennsylvania with high radon levels is between and ” OR
“We are 95% confident that the interval to captures the true proportion of houses in Pennsylvania with high radon levels”

17. Determining the Correct Sample Size
We may sometimes want a particular margin of error and a particular confidence level
And choose our sample size in order to get this margin of error
Because we have not taken the sample yet, we don’t even know 𝑝 yet
But we do know the critical value

18. Determining the Correct Sample Size
In order to be conservative, we will assume that 𝑝 =0.50
This will always give us a big enough sample, regardless of what p-hat actually is
Because 𝑝 ∗ 1− 𝑝 is maximized at 0.5

19. Example
Let’s think back to our Pennsylvania example. Let’s say we wanted to do the same study in Colorado. Assume that the proportion of houses with radon in Pennsylvania tells us nothing about the proprtion of houses in Colorado. We want a 95% confidence interval with a margin of error of 5% (.05) or less
ME=(Crit value)(St error)
.05= 1.96 ( 𝑛 )
Solve for n
N=384.16
So we need a sample of at least 385 people

20. You try
You want to estimate the true proportion of students at a certain school that have a tattoo with 95% confidence and a margin of error of at most How many people should be surveyed?

21. You try
You want to estimate the true proportion of students at a certain school that have a tattoo with 95% confidence and a margin of error of at most How many people should be surveyed?
At least 97 people