Hypothesis Testing for Population Means and Proportions

1. Hypothesis Testing for Population Means and Proportions

2. Topics • Hypothesis testing for population means: • z test for the simple case (in last lecture) • z test for large samples • t test for small samples for normal distributions • Hypothesis testing for population proportions: • z test for large samples

3. z-test for Large Sample Tests • We have previously assumed that the population standard deviationσis known in the simple case. • In general, we do not know the population standard deviation, so we estimate its value with the standard deviation s from an SRS of the population. • When the sample size is large, the z tests are easily modified to yield valid test procedures without requiring either a normal population or known σ. • The rule of thumb n > 40 will again be used to characterize a large sample size.

4. z-test for Large Sample Tests (Cont.) • Test statistic: • Rejection regions and P-values: • The same as in the simple case • Determination of βand the necessary sample size: • Step I: Specifying a plausible value of σ • Step II: Use the simple case formulas, plug in theσestimation for step I.

5. t-test for Small Sample Normal Distribution • z-tests are justified for large sample tests by the fact that: A large n implies that the sample standard deviation s will be close toσfor most samples. • For small samples, s and σare not that close any more. So z-tests are not valid any more. • Let X1,…., Xn be a simple random sample from N(μ, σ). μ andσ are both unknown, andμ is the parameter of interest. • The standardized variable

6. The t Distribution • Facts about the t distribution: • Different distribution for different sample sizes • Density curve for any t distribution is symmetric about 0 and bell-shaped • Spread of the t distribution decreases as the degrees of freedom of the distribution increase • Similar to the standard normal density curve, but t distribution has fatter tails • Asymptotically, t distribution is indistinguishable from standard normal distribution

7. α = .05 Table A.5 Critical Values for t Distributions

8. t-test for Small Sample Normal Distribution (Cont.) • To test the hypothesis H0:μ = μ0based on an SRS of size n, compute t test statistic • When H0 is true, the test statistic T has a t distribution with n -1 df. • The rejection regions and P-values for the t tests can be obtained similarly as for the previous cases.

10. Recap: Population Proportion • Let p be the proportion of “successes” in a population. A random sample of size n is selected, and X is the number of “successes” in the sample. • Suppose n is small relative to the population size, then X can be regarded as a binomial random variable with

11. Recap: Population Proportion (Cont.) • We use the sample proportion as an estimator of the population proportion. • We have • Hence is an unbiased estimator of the population proportion.

12. Recap: Population Proportion (Cont.) • When n is large, is approximately normal. Thus is approximately standard normal. • We can use this z statistic to carry out hypotheses for H0:p = p0 against one of the following alternative hypotheses: • Ha: p > p0 • Ha: p < p0 • Ha: p ≠ p0

13. Large Sample z-test for a Population Proportion • The null hypothesis H0: p = p0 • The test statistic is

14. Determination of β • To calculate the probability of a Type II error, suppose that H0 is not true and that p = pinstead. Then Z still has approximately a normal distribution but , • The probability of a Type II error can be computed by using the given mean and variance to standardize and then referring to the standard normal cdf.

16. Determination of the Sample Size • If it is desired that the level αtest also have β(p) = β for a specified value of β, this equation can be solved for the necessary n as in population mean tests.