Section 4.1: Introducing Hypothesis Tests Extrasensory Perception Is there such a thing as extrasensory perception (ESP) or a sixth sense?
One way to test for ESP is with Zener cards: Subjects draw a card at random and telepathically communicate this to someone who then guesses the symbol
There are five cards with five different symbols. If ESP does not exist, then the population proportion of all guesses p must equal 1/5 Statistics vary from sample to sample; even if the population proportion is 1/5, not every sample proportion will be exactly 1/5 How do we determine when a sample proportion is far enough above 1/5 to provide evidence of ESP?
Statistical Test: A formal procedure that uses data from a sample to judge a claim about a population. ESP Example In the ESP experiment, we want to use sample data to judge whether the population proportion of correct guesses is really higher than 1/5.
Hypothesis: A claim about a population that cannot be proven exactly. A statistical test is framed in terms of two competing hypotheses: H 0 Null Hypothesis Claim that there is no effect or difference. H a Alternative Hypothesis Claim for which we seek evidence.
H o : Null hypothesis H a : Alternative hypothesis Competing claims about a population Hypotheses are always about population parameters Sample data contradicts the null hypothesis and supports the alternative hypothesis The null is usually an equality statement and the alternative more general
Example: ESP Hypotheses For the ESP experiment: H o : p = 1/5 No "effect" or "no difference" H a : p > 1/5 Claim we seek evidence for Helpful hints: H 0 usually includes = H a usually includes >, <, or The inequality in H a depends on the question
Sleep versus Caffeine Students were given words to memorize, then randomly assigned to take either a 90 min nap, or a caffeine pill. 2½ hours later, they were tested on their recall ability. Explanatory variable: sleep or caffeine Response variable: number of words recalled Is there a difference in average word recall between sleep and caffeine?
Let µ s and µ c be the mean number of words recalled after sleeping and after caffeine. H 0 : µ s = µ c H a : µ s µ c The following hypotheses are equivalent, and either set can be used: H 0 : µ s µ c = 0 H a : µ s µ c 0
Practice: State the population parameters and hypotheses for each of the following test situations: 1. Does the proportion of people who support gun control differ between males and females? p f : proportion of females who support gun control p m : proportion of males who support gun control H 0 : p f = p m H a : p f p m
2. Is the average hours of sleep per night for college students less than 7? µ: average hours of sleep per night for college students H 0 : µ =7 H a : µ < 7
Statistical Significance When results as extreme as the observed sample statistic are unlikely to occur by random chance alone (assuming the null hypothesis is true), we say the sample results are statistically significant. If our sample is statistically significant, we have convincing evidence against H 0, in favor of H a If our sample is not statistically significant, our test is inconclusive
Extrasensory Perception p = Proportion of correct guesses H 0 : p = 1/5 H a : p > 1/5 Statistically significant result: The sample proportion of correct guesses is higher than is likely just by random chance. We have evidence that p > 1 and thus have evidence of ESP. NOT a statistically significant result: The sample proportion of correct guesses is not high enough, it could happen by random chance alone. We do not have enough evidence to conclude that p > 1/5, or that ESP exists.
Summary Statistical tests use data from a sample to assess a claim about a population Statistical tests are usually formalized with competing hypotheses: Null hypothesis (H 0 ): no effect or no difference Alternative hypothesis (H a ): what we seek evidence for If data are statistically significant, we have convincing evidence against the null hypothesis, and in favor of the alternative