Statistics Lecture 14

zToday: Chapter 8.5 zAssign #5: 8.62, 8.70, 8.78, and yIn recent years, there has been growing concern about the health effects of video display terminals (VDTs). It is known that the miscarriage rate of a pregnancy under normal conditions is about 20%. In a survey of 15 part-time female employees who used VDTs at least 20 hours per week, there were 5 reported miscarriages. At the 1% significance level, is there evidence that exposure to VDTs is increasing the miscarriage rate? zDue next Thursday (in 9 days)

zFor testing: zIf the test reject the null hypothesis, then zIf the null hypothesis is not rejected,

Large Sample Inferences for Proportions Example: zConsider 2 court cases: yCompany hires 40 women in last 100 hires yCompany hires 400 women in last 1000 hires zIs there evidence of discrimination?

zCan view hiring process as a Bernoulli distribution: zWant to test:

Situation: zWant to estimate the population proportion (probability of a “success”), p zSelect a random sample of size n zRecord number of successes, X zEstimate of the sample proportion is:

zIf n is large, what is distribution of zCan use this distribution to test hypotheses about proportions

Large Sample Hypothesis Test for the Population Proportion zHave a random sample of size n z zTest Statistic:

zP-value depends on the alternative hypothesis: y zWhere Z represents the standard normal distribution

zWhat assumptions must we make when doing large sample hypotheses tests about proportions? zExample revisited:

Large Sample Confidence Intervals for the Population Proportion zLarge sample confidence interval for a population proportion:

Example zFor both court cases, find a 95% confidence interval for the probability that the company hires a woman

Motivation for Large Sample Inference for Proportions zCan express sample proportion a sample mean zSampling distribution of sample proportion is approximately normal (why?) zWhat assumptions are we making?

Sample Size for a Desired Margin of Error zPrior to sampling, one should have an idea of the required precision for the experiment zThe margin of error for the confidence interval is zThe required sample size is zSince we do not have p or q, we substitute

Small Sample Hypotheses for Proportions zDo not always have enough resources to take a large sample zWhat is the sampling distributions for the number of successes in n Bernoulli trials

Situation: zWant to make inferences for the population proportion (probability of a “success”), p zSelect a random sample of size n zRecord number of successes, X zDistribution of X is:

z zIf H 0 is true, what is distribution of X zP-value = z zIf H 0 is true, what is distribution of X zP-value =

Example: zGroup of 10 subjects with certain disease are given a new treatment z8 subjects showed improvement zTest the claim: majority of disease sufferers using this treatment show improvement with a 5% significance level