1. Confidence Interval Estimation for a Population Proportion
Lecture 43
Section 9.4
Fri, Apr 9, 2004

2. Point Estimates The statistic p^ gives us a point estimate.
A point estimate is a single number that estimates the parameter.
The problem with point estimates is that we have no idea how close we can expect it to be to the parameter.

3. Interval Estimates
A more informative estimate is an interval estimate.
An interval estimate is an interval of numbers that has a certain probability of containing the parameter.
The probability associated with the interval is called the confidence level.

4. Interval Estimates
If the confidence level is 95%, then the interval is called a 95% confidence interval.

5. Approximate 95% Confidence Intervals
How do we find a 95% confidence interval for p?
Begin with the sample size and the sampling distribution of p^.
We know that the sampling distribution is normal with mean p and standard deviation
p^ = (p(1 – p)/n).

6. Approximate 95% Confidence Intervals
Therefore, approximately 95% of the time, p^ is within 2 standard deviations of p.
That is, for a single random p^, there is a 95% probability that it is within 2 standard deviations of p.
Therefore, there is a 95% chance that p is within 2 standard deviations of p^ (for a single random p^).

7. Approximate 95% Confidence Intervals
Thus, the confidence interval is
p^  2p^ .
The trouble is, to know p^, we must know p.
The best we can do is to use p^ in place of p to estimate p^.

8. Approximate 95% Confidence Intervals
That is,
p^  (p^(1 – p^)/n).
This is called the standard error of p^ and is denoted SE(p^).
Now the 95% confidence interval is
p^  2SE(p^).

9. Example See Example 9.6, p. 539. The answer is (0.178, 0.206).
That means that we are 95% confident, or sure, that p is somewhere between and

10. Let’s Do It!
Let’s do it! 9.5, p. 540 – When Do You Turn Off Your Cell Phone?

11. Confidence Intervals
We are using the number 2 as a rough approximation for a 95% confidence interval.
We can get a more precise answer if we use the normal tables.
A 95% confidence interval cuts off the upper 2.5% and the lower 2.5%.
What value of z does that?

12. Standard Confidence Levels
The standard confidence levels are 90%, 95%, 99%, and 99.9%.
Confidence Level z
90%
1.645
95%
1.960
99%
2.576
99.9%
3.291

13. The Confidence Interval
The confidence interval is given by the formula
p^  zSE(p^).
where z is given by the previous chart or is found in the normal table.

14. Probability of Error
We use the symbol  to represent the probability that the confidence interval is in error.
That is, p is not in the confidence interval.
In a 95% confidence interval,  = 0.05.

15. Probability of Error Thus, the area in each tail is /2.
The value of z can be found by using the invNorm function on the TI-83.
For example,
invNorm(0.05) = –1.645.
invNorm(0.025) = –1.960.
invNorm(0.005) = –2.576.
invNorm(0.0005) = –3.291.

16. Think About It Think about it, p. 542. Which is better?
A wider confidence interval.
A narrower confidence interval.
A low level of confidence.
A high level of confidence.

17. Think About It Which is better?
A smaller sample.
A larger sample.
Is it possible to increase the level of confidence and make the confidence narrower at the same time?

18. Confidence Intervals on the TI-83
The TI-83 will compute a confidence interval for a population proportion.
Press STAT.
Select TESTS.
Select 1-PropZInt.

19. Confidence Intervals on the TI-83
A display appears requesting information.
Enter x, the numerator of the sample proportion.
Enter n, the sample size.
Enter the confidence level, as a decimal.
Select Calculate and press ENTER.

20. Confidence Intervals on the TI-83
A display appears with several items.
The title “1-PropZInt.”
The confidence interval, in interval notation.
The sample proportion p^.
The sample size.
How would you find the margin of error?

21. Assignment
To be posted later today.