Major Points An example Sampling distribution Hypothesis testing
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1 Major Points An example Sampling distribution Hypothesis testing The null hypothesisTest statistics and their distributionsThe normal distribution and testingImportant concepts
2 Distribution of M&M’s in the population Yellow 20%Brown 30%Orange 10%Blue 10%Red 20%Green 10%Original Distribution
3 Testing one sample against population There are normally 5.65 red M&M’s in bag (a population parameter)Mean number of red M&M’s in a halloween candy bag = 4.56 (a sample statistic)Are there sufficiently more red M&M’s to conclude significant differences / the two numbers come from different populations.I have provided a simplified version of the study because it is much easier to begin by comparing a sample mean with a population mean. You will probably need to explain why I did this. You can come back and work with the two-sample example later.
4 Understanding the theoretical and statistical question Theoretical QuestionDid M&M use a different proportion of red ones for Halloween?Statistical Question:Is the difference between 5.65 and 4.25 large enough to conclude that it is a real (significant) difference?Would we expect a similar kind of difference with a repeat of this experiment?Or...Is the difference due to “sampling error?”
5 Sampling Error Often differences are due to sampling error Sampling Error does not imply doing a mistakeSampling Error simply refers to the normal variability that we would expect to find from one sample to another, or one study to another
6 How could we assess Sampling Error? Take many bags of regular M&M candy.Record the number of red M&M’s.Plot the distribution and record its mean and standard deviation.This distribution is a “Sampling Distribution” of the Mean
7 Sampling Distribution Means of various samples of Brown M&M’s7.257.006.756.506.256.005.755.505.255.004.754.504.254.003.75Sampling DistributionNumber of Red M&M’s in the populationFrequency140012001000800600400200Std. Dev = .45Mean = 5.65Distribution ranges between 3.76 and 7.25Mean is 5.65SD is .45Mean of 4.00 is not likely, mean of 5 is likelyWe can calculate these prob1abilities. Right now lets focus on the logic of the analysis.We can calculate how likely it is to get a score as low as 4.25 by chance.Convert score to z score (z distribution has a mean of 0 and standard deviation of 1)=score-mean/standard deviationLook up Table E10, smaller portion4.25=-3.1=.00065.60 = -.11= .457.5=3.5=.0002
8 What is Sampling Distribution The distribution of a statistic over repeated sampling from a specified population.Can be computed for many different statistics
9 Distribution ranges between 3.76 and 7.25 Mean is 5.65, SD is .45 Mean Number Aggressive Associates7.257.006.756.506.256.005.755.505.255.004.754.504.254.003.75Number of Red M&M’sFrequency140012001000800600400200Std. Dev = .45Mean = 5.65N =Distribution ranges between 3.76 and 7.25Mean is 5.65, SD is .45Mean of 4.00 is not likely, mean of 5 is more likely
10 How likely is it to get score as low as 4.25 by chance. Convert score to z score (z distribution has a mean of 0 and standard deviation of 1)score - mean / standard deviationLook up Table E10, smaller portionScore Z Score Probability
11 Hypothesis TestingA formal way of testing if we should accept results as being significantly different or notStart with hypothesis that halloween M&M’s are from normal distributionThe null hypothesisFind parameters of normal distributionCompare halloween candy to normal distribution
12 The Null Hypothesis (H0) Is the hypothesis postulating that there is no difference, that two things are from the same distribution.The hypothesis that Halloween candy came from a population of normal M&M’sThe hypothesis we usually want to reject.Alternative Hypothesis: Halloween and Regular M&M’s are from different distributions
13 It is easier to prove Alternative Hypothesis than Null Hypothesis Hypothesis: All crows are blackSample: 3000 crowsResults: all are blackConclusion: Are all crows black?
14 Observation: One white crow Conclusion: Statement :Every crow is black” is falseIt is easier to prove alternative hypothesis (all crows are not black) than null hypothesis (all crows are black)
15 Another example of Null & Alternative Hypothesis Hypothesis: Shopping on Amazon.com is different than on BN.comStudy: 100 usability tests comparing the two shopping process on bothResults: Both sites performed similarlyConclusion: ?
16 Observation: On Test No:101 Amazon did better Conclusion: Amazon.com and BN are different.
17 Steps in Hypothesis Testing Define the null hypothesis.Decide what you would expect to find if the null hypothesis were true.Look at what you actually found.Reject the null if what you found is not what you expected.
18 Important Concepts Concepts critical to hypothesis testing Decision Type I errorType II errorCritical valuesOne- and two-tailed tests
19 DecisionsWhen we test a hypothesis we draw a conclusion; either correct or incorrect.Type I errorReject the null hypothesis when it is actually correct.Type II errorRetain the null hypothesis when it is actually false.
22 Type I ErrorsAssume Halloween and Regular candies are same (null hypothesis is true)Assume our results show that they are not same (we reject null hypothesis)This is a Type I errorProbability set at alpha () usually at .05Therefore, probability of Type I error = .05
23 Type II ErrorsAssume Halloween and Regular Candies are different (alternative hypothesis is true)Assume that we conclude they are the same (we accept null hypothesis)This is also an errorProbability denoted beta ()We can’t set beta easily.We’ll talk about this issue later.Power = (1 - ) = probability of correctly rejecting false null hypothesis.
24 Critical ValuesThese represent the point at which we decide to reject null hypothesis.e.g. We might decide to reject null when (p|null) < .05.Our test statistic has some value with p = .05We reject when we exceed that value.That value is the critical value.
25 One- and Two-Tailed Tests Two-tailed test rejects null when obtained value too extreme in either directionDecide on this before collecting data.One-tailed test rejects null if obtained value is too low (or too high)We only set aside one direction for rejection.
26 One- & Two-Tailed Example One-tailed testReject null if number of red in Halloween candies is higherTwo-tailed testReject null if number of red in Halloween candies is different (whether higher or lower)
27 Designing an Experiment: Feature Based Product Advisors Identify some good online product advisorsFeature Based Filtering:CNET Digital Camera AdvisorBasic Sony Decision GuideDealtime Feature Based ChoiceMulti-Dimension (including features) based filtrationSony Advanced Decision GuideReview Based ChoosingEpinionsCNETComputers / Cameras / both; Sample Size; Experimental Questions