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Today • Today: Chapter 10 • Sections from Chapter 10: 10.1-10.4 • Recommended Questions: 10.1, 10.2, 10-8, 10-10, 10.17, 10.19 Small Sample Hypothesis Test for the Population Mean • Have a random sample of size n ; x1, x2, …, xn • H 0 : 0 • Test Statistic: t X S/ n Small Sample Hypothesis Test for the Population Mean (cont.) • P-value depends on the alternative hypothesis: – : : p - value P(T t ) H 1 0 – : : p - value P(T t ) H 1 0 H– 1 : 0 : p - value 2P(T | t |) • Where T represents the t-distribution with (n-1 ) degrees of freedom Example: • An ice-cream company claims its product contains 500 calories per pint on average • To test this claim, 24 of the company’s one-pint containers were randomly selected and the calories per pint measured • The sample mean and standard deviation were found to be 507 and 21 calories • At the 0.01 level of significance, test the company’s claim • What assumptions do we make when using a t-test? • Can use t procedures even when population distribution is not normal. Why? Relationships Between Tests and CI’s • Confidence interval gives a plausible range of values for a population parameter based on the sample data • Hypothesis Test assesses whether data gives evidence that a hypothesized value of the population parameter is plausible or implausible • Seem to be doing something similar • For testing: H 0 : 0 vs. H1 : 0 • If the test rejects the null hypothesis, then • If the null hypothesis is not rejected, Example (3.96) • Based on a random sample of size 18 from a normal population, an investigator computes a 95% confidence interval for the mean and gets [27.1, 39.3] • What is the conclusion of the t-test at the 5% level for: – H0 : 29 vs. H1 : 29 – H 0 : 26.8 vs. H1 : 26.8 • Suppose we reject the second null hypothesis at the 5% level • Another experimenter wishes to perform the test at the 10% level…would they reject the null hypothesis • Another experimenter wishes to perform the test at the 1% level…would they reject the null hypothesis • What does changing the significance level do to the range of values for which we would reject the null hypothesis Large Sample Inferences for Proportions Example: • Consider 2 court cases: – Company hires 40 women in last 100 hires – Company hires 400 women in last 1000 hires • Is there evidence of discrimination? • Can view hiring process as a Bernoulli distribution: • Want to test: Situation: • Want to estimate the population proportion (probability of a “success”), p • Select a random sample of size n • Record number of successes, X • Estimate of the sample proportion is: • If n is large, what is distribution of • Can use this distribution to test hypotheses about proportions p̂ Large Sample Hypothesis Test for the Population Proportion • Have a random sample of size n • H0 : p p0 • pˆ X n • Test Statistic: Z pˆ p0 p0 q0 / n • P-value depends on the alternative hypothesis: – H1 : p p0 : p - value P(Z z) – H1 : p p0 : p - value P(Z z) – H1 : p p0 : p - value 2P(Z | z |) • Where Z represents the standard normal distribution • What assumptions must we make when doing large sample hypotheses tests about proportions? • Example revisited: Example • For both court cases, find a 95% confidence interval for the probability that the company hires a woman