1. Estimating a Population Proportion
Goal: Given a sample proportion, estimate the value of the population proportion p.
Example: In a sample of 750 people, 27% said they feel that health care is the most important issue facing our state. What proportion of the population feels that health care is the most important issue?

2. Assumptions The sample is a simple random sample.
The conditions for the binomial distribution apply: There are a fixed number of trials, the trials are independent, there are two categories of outcomes, and the probabilities remain constant for each trial.
The normal distribution can be used to approximate the distribution of sample proportions, since and
Since p and q are not known, we use the sample proportion to estimate their values.

3. New Notation p = population proportion
sample proportion (of x successes in a sample of size n)

4. The sample proportion is the best point estimate (single value approximation) of the population proportion p.
Problem: Using to approximate p doesn’t convey how accurate we expect our estimate to be. To do that, we need confidence intervals

5. Confidence Intervals (CI)
A confidence interval is a range (or interval) of values used to estimate the true value of the population parameter.
A confidence level is the probability that our confidence interval contains the true value of p.
The confidence level is expressed as a probability 1- α

6. Common Values 90% confidence level (α = 0.10)

7. An example of a Confidence Interval
Based on our survey earlier,
The 95% confidence interval estimate of the population proportion p is: < p < 0.305
This means that there is 95% chance that this interval contains the actual population proportion p.
In other words, 95% of the time that we do a sample, the confidence interval will contain the true population proportion.

8. Critical Values
A critical value is a z-score that separates outcomes that are likely to occur from those that are unlikely to occur.
An example: For 95% confidence interval:

9. For the 95% confidence interval, α = .05
Notice that falls above the critical value, and falls below the opposite (negative) critical value. Each of these areas is α/2.
Notation
The critical value zα/2 is the positive z-value that separates the top area of α/2. -zα/2 is the boundary of the bottom area of α/2.

10. Another Example
So if our confidence level was 99%, the critical value zα/2 would be the score that separates the top 0.5% of data, and –zα/2 would separate the bottom 0.5% of data.
Leaving 99% of the data between –zα/2 and zα/2

11. Finding Critical Values
Example:
For the 95% confidence interval, the area above zα/2 is .025, so the area below is = .975
So P(z < zα/2) = From the table, we find zα/2 =1.96

12. Common Critical Values
(listed at bottom of z-score table)

13. Creating a Confidence Interval

14. Margin of Error
The margin of error E is the maximum likely (with probability 1-α) difference between the observed proportion and the population proportion p.

15. Summary of Procedure for finding a Confidence Interval
Verify the required assumptions are satisfied
Find the critical value that corresponds to the desired confidence level
Evaluate the margin of error E
Find the values and Write the confidence interval
Round values to three decimal places

16. Example
In a sample of 750 people, 27% said they feel that health care is the most important issue facing our state.
95% confidence level, so

17. Example continued so our confidence interval is

18. Example continued 0.238 < p < 0.302
Based on our survey results, we are 95% confident that the true percentage of Washingtonians who feel that health care is the most important issue facing the state is between 23.8% and 30.2%.

19. From a Confidence Interval
If you know a confidence interval, the middle value is the point estimate (in this case, ). You can find it by calculating
If you know a confidence interval, the Margin of Error is half the width of the confidence interval. You can find it by calculating

20. Determining Sample Size from desired Margin of Error
when is known
when is not known

21. Homework
6.2: 5, 13, 17, 21, 27, (29)