1. Confidence Intervals with Proportions

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

3. 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

4. Confidence intervals
Are used to estimate the unknown population parameter
Formula:
statistic + margin of error
Point estimate
The variability of the estimate in repeated simple random samples

5. 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:

6. 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?

7. % What happens to your confidence as the interval gets smaller?
Your confidence level decreases with smaller intervals % % % %

8. 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

9. Interpreting Confidence Level and Confidence Intervals
Interpreting Confidence Levels and Confidence Intervals
Interpreting Confidence Level and Confidence Intervals
Confidence level: To say that we are 95% confident is shorthand for “95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.”
Confidence interval: To interpret a C% confidence interval for an unknown parameter, say, “We are C% confident that the interval from _____ to _____ captures the actual value of the [population parameter in context].”

10. Interpreting Confidence Levels and Confidence Intervals
The confidence level tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times.
The confidence level does not tell us the chance that a particular confidence interval captures the population parameter.
Instead, the confidence interval gives us a set of plausible values for the parameter.

11. 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%

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

13. Before calculating a confidence interval for µ or p there are three important conditions that you should check.
1) Random: The data should come from a well-designed random sample or randomized experiment.
2) Normal: The sampling distribution of the statistic is approximately Normal.
For proportions: We can use the Normal approximation to the sampling distribution as long as np ≥ 10 and n(1 – p) ≥ 10.
3) Independent: Individual observations are independent. When sampling without replacement, the sample size n should be no more than 10% of the population size N (the 10% condition) to use our formula for the standard deviation of the statistic.

14. 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

15. 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
Independent because Population is at least 10n
3.) Calculations
4.) Conclusion (in context of problem)

16. Conclusion Statement: (memorize!!)
Interpreting Confidence Interval: “We are ________% confident that the interval from ______to _____ captures the true proportion of (in context).”

17. 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 . . .

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

20. 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.

21. 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.

22. Step 2: check assumptions!
One sample confidence interval for proportions
Step 1: Identify!
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 2: check assumptions!
Step 3: make calculations
Step 4: conclusion in context
We are 95% confident that the interval between 35% and 41% captures the true proportion of adults who believe in ghosts.

23. 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.

24. Step 2: check assumptions!
One sample confidence interval for proportions
Step 1: Identify!
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 2: check assumptions!
Step 3: make calculations
Step 4: conclusion in context
We are 90% confident that the interval between 12.2% and 19.8% contains the true proportion of egg cartons with at least one broken egg.

25. 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?
Margin of error
To find sample size:
However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!

26. 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.

27. 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