1. 4.4.2 Normal Approximations to Binomial Distributions
Confidence intervals and sample size for proportions

2. Confidence Intervals, a review
Margin of error
These results are accurate to within +/- 3.7%,
19 times out of 20.
Confidence level
95% probability that x is somewhere in the range (x – 3.7, x+3.7)
Confidence interval

3. Confidence Intervals, a review
A (1 – ) or (1 – ) x 100% confidence interval for , given population standard deviation , sample size n, and sample mean , represents the range of values

4. Confidence Levels for Proportions
Often want to know proportion of population that have particular opinion or characteristic
Proportion is p, the probability of success in binomial distribution
is the proportion in the sample

5. Common confidence levels and their associated z-scores
Tail size,
z-score,
90%
0.05
1.645
95%
0.025
1.960
99%
0.005
2.576

6. Comments
The sample proportion is an estimate of the population proportion
(like sample mean is an estimate of the population mean)
For many polls, population proportion not known
In fact, purpose of poll is to estimate it!
Can estimate using sample proportion

7. Example
Voter turnout in municipal elections is often very low. In a recent election, the mayor got 53% of the voters, but only about 1500 voters turned out.
Construct a 90% confidence interval for the proportion of people who support the mayor.
Comment on any assumptions you have to make for your calculation.

8. Example 1a, sol’n Use election results to estimate proportion:
These estimated values give a 90% confidence interval of
The mayor can be 90% confident of having the support of 51-55% of the population

9. Example 1b, sol’n
Have to assume that people who voted are representative of whole population
Assumption might not be valid
people who take trouble to vote likely to be the ones most interested in municipal affairs

10. Sample Sizes & Margin of Error
Margin of Error = half the confidence interval width
the maximum difference between the observed proportion in the sample and the true value of the proportion in the population
w = 2E

11. Sample Size, n

12. Example 2
A recent survey indicated that 82% of secondary-school students graduate within five years of entering grade 9. This result is considered accurate within plus or minus 3%, 19 times out of 20. Estimate the sample size in this survey.

13. Example 2 You would need to sample about 630 students.
For a 95% confidence level, z0.975 = 1.960
w = 2E
= 2(3%)
= 6%
Use survey results to estimate p:
You would need to sample about 630 students.