Section 7.4

1. Section 7.4 How Do We Choose the Sample Size for a Study?

2. How are the Sample Sizes Determined in Polls? • It depends on how much precision is needed as measured by the margin of error • The smaller the margin of error, the larger the sample size must be

3. Choosing the Sample Size for Estimating a Population Proportion? • First, we must decide on the desired margin of error • Second, we must choose the confidence level for achieving that margin of error • In practice, 95% confidence intervals are most common

4. Example: What Sample Size Do You Need For An Exit Poll? • A television network plans to predict the outcome of an election between two candidates – Levin and Sanchez • They will do this with an exit poll that randomly samples votes on election day

5. Example: What Sample Size Do You Need For An Exit Poll? • The final poll a week before election day estimated Levin to be well ahead, 58% to 42% • So the outcome is not expected to be close • The researchers decide to use a sample size for which the margin of error is 0.04

6. Example: What Sample Size Do You Need For An Exit Poll? • What is the sample size for which a 95% confidence interval for the population proportion has margin of error equal to 0.04?

7. Example: What Sample Size Do You Need For An Exit Poll? • The 95% confidence interval for a population proportion p is: • If the sample size is such that 1.96(se) = 0.04, then the margin of error will be 0.04

8. Example: What Sample Size Do You Need For An Exit Poll? • Find the value of the sample size n for which 0.04 = 1.96(se):

9. Example: What Sample Size Do You Need For An Exit Poll? • A random sample of size n = 585 should give a margin of error of about 0.04 for a 95% confidence interval for the population proportion

10. How Can We Select a Sample Size Without Guessing a Value for the Sample Proportion • In the formula for determining n, setting = 0.50 gives the largest value for n out of all the possible values to substitute for • Doing this is the “safe” approach that guarantees we’ll have enough data

11. Sample Size for Estimating a Population Parameter • The random sample size n for which a confidence interval for a population proportion p has margin of error m (such as m = 0.04) is

12. Sample Size for Estimating a Population Parameter • The z-score is based on the confidence level, such as z = 1.96 for 95% confidence • You either guess the value you’d get for the sample proportion based on other information or take the safe approach of setting = 0.50

13. Sample Size for Estimating a Population Mean • The random sample size n for which a 95% confidence interval for a population mean has margin of error approximately equal to m is • To use this formula, you guess the value you’ll get for the sample standard deviation, s

14. Sample Size for Estimating a Population Mean • In practice, since you don’t yet have the data, you don’t know the value of the sample standard deviation, s • You must substitute an educated guess for s • You can use the sample standard deviation from a similar study

15. Example: Finding n to Estimate Mean Education in South Africa • A social scientist plans a study of adult South Africans to investigate educational attainment in the black community • How large a sample size is needed so that a 95% confidence interval for the mean number of years of education has margin of error equal to 1 year?

16. Example: Finding n to Estimate Mean Education in South Africa • No prior information about the standard deviation of educational attainment is available • We might guess that the sample education values fall within a range of about 18 years

17. Example: Finding n to Estimate Mean Education in South Africa • If the data distribution is bell-shaped, the range from – 3 to + 3 will contain nearly all the distribution • The distance – 3 to + 3 equals 6s • Solving 18 = 6s for s yields s = 3 • So ‘3’ is a crude estimate of s

18. Example: Finding n to Estimate Mean Education in South Africa • The desired margin of error is m = 1 year • The required sample size is:

19. What Factors Affect the Choice of the Sample Size? • The first is the desired precision, as measured by the margin of error, m • The second is the confidence level

20. What Other Factors Affect the Choice of the Sample Size? • A third factor is the variability in the data • If subjects have little variation (that is, s is small), we need fewer data than if they have substantial variation • A fourth factor is financial • Cost is often a major constraint

21. What if You Have to Use a Small n? • The t- methods for a mean are valid for any n • However, you need to be extra cautious to look for extreme outliers or great departures from the normal population assumption

22. What if You Have to Use a Small n? • In the case of the confidence interval for a population proportion, the method works poorly for small samples

23. Constructing a Small-Sample Confidence Interval for a Proportion • Suppose a random sample does not have at least 15 successes and 15 failures • The confidence interval formula: • Is still valid if we use it after adding ‘2’ to the original number of successes and ‘2’ to the original number of failures • This results in adding ‘4’ to the sample size n