Hypothesis Testing for Population Means and Proportions

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 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.

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

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 and σ are both unknown, andμ is the parameter of interest. The standardized variable

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

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

t-test for Small Sample Normal Distribution (Cont.) To test the hypothesis H0:μ = μ0 based 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.

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

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.

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

Large Sample z-test for a Population Proportion The null hypothesis H0: p = p0 The test statistic is Alternative Hypothesis P-value Rejection Region for Level α Test Ha: p > p0 P(Z ≥ z) z ≥ zα Ha: p < p0 P(Z ≤ z) z ≤ - zα Ha: p ≠ p0 2P(Z ≥ | z |) | z | ≥ zα/2

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.

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.