Statistics Lecture 12

zToday: Finish 8.4; begin Chapter 9 zMid-Term Next Thursday zReview Next Tuesday

Small Sample Confidence Interval for the Population Mean zIf x 1, x 2, …, x n is a random sample from a normal population with mean, and standard deviation, then a confidence interval for the population mean is: zIf you have use a distribution instead!

Example: zHeights of males are believed to be normally distributed zRandom sample of 25 adult males is taken and the sample mean & standard deviation are and 4.15 inches respectively zFind a 95% confidence interval for the mean

Small Sample Hypothesis Test for the Population Mean zHave a random sample of size n ; x 1, x 2, …, x n z zTest Statistic:

Small Sample Hypothesis Test for the Population Mean (cont.) zP-value depends on the alternative hypothesis: y zWhere T represents the t-distribution with (n-1 ) degrees of freedom

Example: zAn ice-cream company claims its product contains 500 calories per pint on average zTo test this claim, 24 of the company’s one-pint containers were randomly selected and the calories per pint measured zThe sample mean and standard deviation were found to be 507 and 21 calories zAt the 0.01 level of significance, test the company’s claim

zWhat assumptions do we make when using a t-test? zHow can we check assumptions? zCan use t procedures even when population distribution is not normal. Why?

Practical Guidelines for t-Tests zn<15: Use t procedures if the data are normal or close to normal zn<15: If the data are non-normal or outliers are present DO NOT use t procedures zn>15: t procedures can be used except in the presence of outliers or strong skewness zt>30: t procedures tend to perform well

Relationships Between Tests and CI’s zConfidence interval gives a plausible range of values for a population parameter based on the sample data zHypothesis Test assesses whether data gives evidence that a hypothesized value of the population parameter is plausible or implausible zSeem to be doing something similar

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

Example (3.96) zBased 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] zWhat is the conclusion of the t-test at the 5% level for: y

zSuppose we reject the second null hypothesis at the 5% level zAnother experimenter wishes to perform the test at the 10% level…would they reject the null hypothesis zAnother experimenter wishes to perform the test at the 1% level…would they reject the null hypothesis zWhat 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: 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