Two-sided p-values (1.4) and Theory-based approaches (1.5) Stat 217 – Day 10 Two-sided p-values (1.4) and Theory-based approaches (1.5)

Previously Test of Significance Define parameter of interest (e.g., p = long-run probability someone picks the left front tire) State competing hypotheses about the parameter (e.g., H0: p = .25, Ha: p > .25) Assume the null hypothesis is true and generate a “null distribution” of the statistic (e.g., distribution of sample proportions for 35 people picking a tire) If the observed statistic is in the tail of the null distribution, have evidence against the null hypothesis, in favor of the alternative

Evidence against the null hypothesis p-value is probability of getting a result at least as extreme as the observed statistic when the null hypothesis is true (e.g., about 15% of samples from a process with p = .25 have a sample proportion of .343 or larger) p-values less than .05 are usually considered strong evidence against the null Standardized score measures how far the statistic is from the hypothesized value of the parameter (center of null distribution) in terms of standard deviation units (e.g., .343 is 1.28 standard deviations above .25) Standardized statistics above 2 are usually considered strong evidence against the null

Previously But what if our sample comes from a population rather than a process? Needs to be a random sample for us to believe it is representative of the population Modelled the chance variation in our statistic from random sample to random sample

Proportion of e-words Samples of n = 10 from process with p = .47 Samples of n = 10 from population with p = .47 Coin Tossing Gettysburg Address

Key Result When the population size is large compared to the sample size (e.g., more than 20 times larger), then sampling from a population looks pretty much the same as sampling from a process So we will use the same analysis tools One Proportion Inference applet

Example 2

Summary Can calculate “two-sided” p-values for “not equal to” alternative hypotheses by considering evidence at least as extreme in either direction When validity conditions are met, can use the normal distribution (Central Limit Theorem) to obtain a “theory-based” p-value Interpret the p-value exactly the same way

To do for Friday Finish Investigation 2 Pre-lab for Lab 2 to be posted soon Finish reading Chs. 1 and 2