Presentation on theme: "Probability Population:"— Presentation transcript:
1 Probability Population: The set of all individuals of interest (e.g. all women, all college students)Sample:A subset of individuals selected from the population from whom data is collectedprobability
2 What we learned from Probability The mean of a sample can be treated as a random variable.By the central limit theorem, sample means will have a normal distribution (for n > 30) with andBecause of this, we can find the probability that a given population might randomly produce a particular range of sample means.Use table E.10
3 Inferential statistics Population:The set of all individuals of interest (e.g. all women, all college students)Sample:A subset of individuals selected from the population from whom data is collectedInferential statistics
4 Once we’ve got our sample The key question in statistical inference:Could random chance alone have produced a sample like ours?Patterns may not be indicative of some underlying factorPatterns may be natural fluctuations
5 Once we’ve got our sample Distinguishing between 2 interpretations of patterns in the data:Random Causes:Fluctuations of chanceSystematic Causes Plus Random Causes:True differences in the populationBias in the design of the studyInferential statistics separatesPatterns may not be indicative of some underlying factorPatterns may be natural fluctuations
6 Reasoning of hypothesis testing Make a statement (the null hypothesis) about some unknown population parameter.Collect some data.Assuming the null hypothesis is true, what is the probability of obtaining data such as ours? (this is the “p-value”).If this probability is small, then reject the null hypothesis.
7 Step 1: Stating hypotheses Null hypothesisH0Straw man: “Nothing interesting is happening”Alternative hypothesisHaWhat a researcher thinks is happeningMay be one- or two-sided
8 Step 1: Stating hypotheses Hypotheses are in terms of population parametersOne-sided Two-sidedH0: µ=110 H0: µ = 110H1: µ < 110 H1: µ ≠ 110
9 Step 2: Set decision criterion Decide what p-value would be “too unlikely”This threshold is called the alpha level.When a sample statistic surpasses this level, the result is said to be significant.Typical alpha levels are .05 and .01.
10 More on setting a criterion The retention region.The range of sample mean values that are “likely” if H0 is true.If your sample mean is in this region, retain the null hypothesis.The rejection region.The range of sample mean values that are “unlikely” if H0 is true.If your sample mean is in this region, reject the null hypothesisthe range of extreme sample mean values that are unlikely to be obtained by chance in cases where the “treatment” mean is the same as the population mean
11 Setting a criterion Accept H0 Reject H0 Reject H0 Null distribution ZcritZcrit
12 Step 3: Compute sample statistics A test statistic (e.g. Ztest, Ttest, or Ftest) is information we get from the sample that we use to make the decision to reject or keep the null hypothesis.A test statistic converts the original measurement (e.g. a sample mean) into units of the null distribution (e.g. a z-score), so that we can look up probabilities in a table.
13 Test Statistics Accept H0 Reject H0 Reject H0 Null distribution Ztest? ZcritZcrit
14 Accept H0Reject H0ZcritIf we want to know where our sample mean lies in the null distribution, we convert X-bar to our test statistic ZtestIf an observed sample mean were lower than z=-1.65 then it would be in a critical region where it was more extreme than than 95% of all sample means that might be drawn from that population
15 Step 4: Make a decisionIf our sample mean turns out to be extremely unlikely under the null distribution, maybe we should revise our notion of µH0We never really “accept” the null. We either reject it, or fail to reject it.
16 Steps of hypothesis testing State hypothesis (H0, HA)Select a criterion (alpha, Zcrit)Compute test statistic (Ztest) and get a p-valueMake a decision
17 Z as test statisticZ test-statistic converts a sample mean into a z-score from the null distribution.Zcrit is the criterion value of Z that defines the rejection regionZtest is the value of Z that represents the sample mean you calculated from your dataStandard error!!!!p-value is the probability of getting a Ztest as extreme as yours under the null distribution
18 Z as test statisticAll test statistics are fundamentally a comparison between what you got and what you’d expect to get from chance aloneDeviation you gotDeviation from chance aloneIf the numerator is considerably bigger than the denominator, you have evidence for a systematic factor on top of random chance
19 Example ITim believes that his “true weight” is 187 lbs with a standard deviation of 3 lbs.Tim weighs himself once a week for four weeks. The average of these four measurements isAre the data consistent with Tim’s belief?
20 Example I H0: = 187 HA: > 187 Criterion? Let’s say alpha=.05. That would be Zcrit = 1.65An X-bar of is what Ztest? What is the probability of getting a Ztest as high as ours?If H0 were true, there would be only about a 1% chance of randomly obtaining the data we have. Reject H0.
21 Example I illustrated Reject H0 Zcrit Ztest z = 190.5-187 = 2.33 3 4 0.01x = 187x= 1.5190.51.652.33ZcritZtest
22 ExerciseWe have a sample of 500 students whose average score on some standardized test is 461. We think they are a particularly gifted bunch.Assume the general student population has a distribution of scores that is approximately normal with µ = 450 and = 100.Does our sample come from a population with a mean of 450? Or are they a better test-taking species?H0: µ = 450H1: µ > 450
23 Exercise How to proceed? Let’s: Select a criterion Calculate a z-score Compare our sample z with our criterionMake a decision
24 ExerciseWe have a sample of 500 students whose average score on some standardized test is 461. We think they are a particularly gifted bunch.Assume the general student population has a distribution of scores that is approximately normal with µ = 450 and = 100.Does our sample come from a population with a mean of 450? Or are they a better test-taking species?H0: µ = 450H1: µ > 450
25 Exercise illustratedWe reject the null hypothesis because sample means of 461 or larger have a very small probability. (We expect such large means less than 1% of the time.)
26 When we reject a null hypothesis, it is because (a) if we believe the null hypothesis, there is only a small probability of getting data like ours by chance alone.(b) if we believe our data, and don’t think it came from an unlikely chance event, the null distribution is probably not true.
27 One-tailed testsIf HA states is < some value, critical region occupies left tailIf HA states is > some value, critical region occupies right tailIf observed p-value is less than , reject HoIf observed p-value is greater that or equal to , do not reject HoGraphic from
30 One- vs. two-tailed tests In theory, should use one-tailed when1. Change in opposite direction would be meaningless2. Change in opposite direction would be uninteresting3. No rival theory predicts change in opposite directionBy convention/default in the social sciences, two-tailed is standardWhy? Because it is a more stringent criterion (as we will see). A more conservative test.
31 Two-tailed hypothesis testing HA is that µ is either greater or less than µH0HA: µ ≠ µH0 is divided equally between the two tails of the critical region
32 Two-tailed hypothesis testing Means less than or greater thanReject H0Fail to reject H0Reject H0alphaZcrit100ZcritValues that differ significantly from 100
33 differ “significantly” One tailReject H0Fail to reject H0.05Values thatdiffer “significantly”from 100Zcrit100100Values that differ significantly from 100Fail to reject H0Reject H0Two tail.025Zcrit
34 ExampleWe have a sample of 36 children of geniuses. They have an average IQ of We want to know whether they are significantly different from the general population of children, who have µ=100 and σ=25.Test the hypothesis that the mean of this group is higher than that of the population.What is Ztest?What is Zcrit for alpha = .05? For alpha = .01? Do we reject the null for either case?What is the exact p-value for this test?
36 ExampleWe have a sample of 36 children of geniuses. They have an average IQ of We want to know whether they are significantly different from the general population of children, who have µ=100 and σ=25.Test the hypothesis that the mean of this group is not equal to that of the populationWhat is Ztest?What is Zcrit for alpha = .05? For alpha = .01? Do we reject the null for either case?What is the exact p-value for this test?