1. Intro to Statistics for the Behavioral Sciences PSYC 1900 Lecture 5: Probability and Hypothesis Testing

2. Probability Relative Frequency Perspective Relative Frequency Perspective Probability of some event is the limit of the relative frequency of occurrence as the number of draws (i.e., samples) approaches infinity. Probability of some event is the limit of the relative frequency of occurrence as the number of draws (i.e., samples) approaches infinity. If we have 8 blue marbles and 2 red marbles, the probability of drawing a red = 2/10 = 20% on any trial (i.e., analytic perspective). If we have 8 blue marbles and 2 red marbles, the probability of drawing a red = 2/10 = 20% on any trial (i.e., analytic perspective). Across repeated trials, we would find that 20% of them produce a red marble. Across repeated trials, we would find that 20% of them produce a red marble. Note that we’re sampling with replacement. Note that we’re sampling with replacement.

3. Terminology Sampling with replacement Sampling with replacement After an event, the draw or event goes back into the pool. After an event, the draw or event goes back into the pool. Sampling in which an item drawn on trial N is replaced before the drawing of the N+1 trial. Sampling in which an item drawn on trial N is replaced before the drawing of the N+1 trial. Event Event The outcome of a trial The outcome of a trial Independent events Independent events Events where the occurrence of one has no effect on the probability of the occurrence of others Events where the occurrence of one has no effect on the probability of the occurrence of others Voting behavior of random citizens, marble draw Voting behavior of random citizens, marble draw Mutually exclusive events Mutually exclusive events Two events are mutually exclusive when the occurrence of one precludes the occurrence of the other. Two events are mutually exclusive when the occurrence of one precludes the occurrence of the other. Gender, religion, handedness Gender, religion, handedness

4. Basic Laws of Probability Probabilities range from 0 to 1, where a 1 means the event must occur. Probabilities range from 0 to 1, where a 1 means the event must occur. Additive Rule Additive Rule Gives probs of occurrence for one or more mutually exclusive envents. Gives probs of occurrence for one or more mutually exclusive envents. 30 red marbles, 15 blue, 55 green = 100 total 30 red marbles, 15 blue, 55 green = 100 total p(red)=.30, p(blue)=.15, p(green) =.55 p(red)=.30, p(blue)=.15, p(green) =.55 Probability of drawing a red or blue? Probability of drawing a red or blue? Given a set of mutually exclusive events, the probability of one event or the other equals the sum of their separate probabilities. Given a set of mutually exclusive events, the probability of one event or the other equals the sum of their separate probabilities. p(red)=.30 + p(blue)=.15=.45 p(red)=.30 + p(blue)=.15=.45

5. Basic Laws of Probability Multiplicative Law Multiplicative Law Gives the probability of the joint occurrence of independent events. Gives the probability of the joint occurrence of independent events. 30 red marbles, 15 blue, 55 green = 100 total 30 red marbles, 15 blue, 55 green = 100 total p(red)=.30, p(blue)=.15, p(green) =.55 p(red)=.30, p(blue)=.15, p(green) =.55 Probability of drawing a red on the first trial and a red on the second? Probability of drawing a red on the first trial and a red on the second? The prob of a joint occurrence of two or more independent events equals the product of their individual probabilities. The prob of a joint occurrence of two or more independent events equals the product of their individual probabilities. p(red) X p(red) =.3X.3 =.09 p(red) X p(red) =.3X.3 =.09

6. Sequence of coin flips: H,H,T,H,T,T,T,H,T,T, __ What is the probability of H on next draw? Prob=.5 Events are independent What is the probability of H and H on the next two draws? Prob=.5X.5=.25 Events are independent Conditional probability of independent events

7. Joint Probabilities The probability of the co-occurrence of two or more events The probability of the co-occurrence of two or more events Probability of sampling a red cube from a sample of red and blue marbles and cubes Probability of sampling a red cube from a sample of red and blue marbles and cubes p(red,cube) = p(red) x p(cube) p(red,cube) = p(red) x p(cube) If the events are independent If the events are independent If not independent (i.e., a correlation among events), computation of prob is more complex If not independent (i.e., a correlation among events), computation of prob is more complex

8. Conditional Probabilities The prob of one even given the occurrence of another event The prob of one even given the occurrence of another event The prob that a person will fracture a bone given that he/she has osteoporosis The prob that a person will fracture a bone given that he/she has osteoporosis p(fracture|osteoporosis) = Y p(fracture|osteoporosis) = Y If the null hypothesis is true, the probability of obtaining a difference between sample means of X size If the null hypothesis is true, the probability of obtaining a difference between sample means of X size

9. p(no fracture) =258/358=.72 p(no fracture) =258/358=.72 p(norm den, no frac)=153/358=.43 p(norm den, no frac)=153/358=.43 Why not p(norm) x p(no frac) =.49x.72=.35? Why not p(norm) x p(no frac) =.49x.72=.35? p(frac|osteo) =.42; p(frac|norm)=.14 p(frac|osteo) =.42; p(frac|norm)=.14 Other conditional prob examples? Other conditional prob examples? Bone DensityNo FractureFractureTotal Normal15324177 Row%861449% Column%5924 Cell%437 Osteoporosis10576181 Row%584251% Column%4176 Cell%2921 Total258100 Column%72%28%

10. Discrete vs. Continuous Probability Distributions For discrete distributions, we can calculate probs for specific events. For discrete distributions, we can calculate probs for specific events. p(Harvard, vanilla) = p(Harvard, vanilla) =7/20=.35

11. Discrete vs. Continuous Probability Distributions For continuous distributions, case is slightly different. For continuous distributions, case is slightly different. Prob that baby will crawl at 35 weeks? Prob that baby will crawl at 35 weeks? Almost zero at 35.00001 weeks. Almost zero at 35.00001 weeks. Events at a very specific point are infrequent. Events at a very specific point are infrequent. Density gives probability for specific range Density gives probability for specific range 35 weeks means from 34.5 to 35.5 weeks. 35 weeks means from 34.5 to 35.5 weeks. Integrate to find area under curve which provides a probability as a function of proportion of interval area to entire area under curve (where total area is set to equal 1) Integrate to find area under curve which provides a probability as a function of proportion of interval area to entire area under curve (where total area is set to equal 1)

12. Sampling Distributions & Hypothesis Testing Until now, we have primarily focused on descriptive statistics. Until now, we have primarily focused on descriptive statistics. Although such statistics are quite useful for assessing the characteristics of samples, they cannot answer questions related to inference. Although such statistics are quite useful for assessing the characteristics of samples, they cannot answer questions related to inference. Is the difference between two means likely to represent chance variation? Is the difference between two means likely to represent chance variation? To answer such questions, the remainder of this course will focus on the statistical process of inference. To answer such questions, the remainder of this course will focus on the statistical process of inference.

13. Basic Form of Inference The most basic question is one in which we might compare the means of two groups. The most basic question is one in which we might compare the means of two groups. If one group has a mean of 50 and the other a mean of 42 following some manipulation, can we infer that the manipulation lowered the score? If one group has a mean of 50 and the other a mean of 42 following some manipulation, can we infer that the manipulation lowered the score?

14. Sampling Error To answer this question, we have to understand sampling error. To answer this question, we have to understand sampling error. Sampling error is the variability of a statistic from sample to sample due to chance. Sampling error is the variability of a statistic from sample to sample due to chance. If I took samples from a population, the descriptives of the samples would cluster around, but not always equal the parameters of the population. If I took samples from a population, the descriptives of the samples would cluster around, but not always equal the parameters of the population.

15. Hypothesis Testing The basic question in hypothesis testing is: The basic question in hypothesis testing is: Is the given difference large enough that it does not likely stem from sampling error? Is the given difference large enough that it does not likely stem from sampling error? Hypothesis Testing Hypothesis Testing A process by which decisions are made regarding the values of parameters. A process by which decisions are made regarding the values of parameters.

16. Sampling Distributions The distribution of a statistic over repeated sampling from a specified population. The distribution of a statistic over repeated sampling from a specified population. Both descriptive and inferential statistics (e.g., t, F, r) have sampling distributions. Both descriptive and inferential statistics (e.g., t, F, r) have sampling distributions. Tell us what values we might expect given certain conditions. Tell us what values we might expect given certain conditions. A conditional probability A conditional probability

17. Sampling Distribution of the Mean To determine if the difference between two means is likely due to sampling error, we need to know the sd of a distribution of means from the population. To determine if the difference between two means is likely due to sampling error, we need to know the sd of a distribution of means from the population. Standard Error of the Mean Standard Error of the Mean sd of a sampling distribution of means sd of a sampling distribution of means Sampling distritribution of the mean is the distribution of means collected from repeated sampling of the same population. Sampling distritribution of the mean is the distribution of means collected from repeated sampling of the same population.

18. Distribution of Sample Means

19. Hypothesis Testing Sampling distributions allow us to test hypotheses. Sampling distributions allow us to test hypotheses. Sampling distributions can be derived mathematically. Sampling distributions can be derived mathematically. If the aggression mean of kids viewing a violent video is 6.5, and the “normal” population mean for kids is 5.65, does this difference imply that the such videos increase aggressive thoughts? If the aggression mean of kids viewing a violent video is 6.5, and the “normal” population mean for kids is 5.65, does this difference imply that the such videos increase aggressive thoughts?

20. Logic of Hypothesis Testing Set up relevant null hypothesis [H 0 ] Set up relevant null hypothesis [H 0 ] Sample (i.e., kids who watch violent videos) represents same population. Sample (i.e., kids who watch violent videos) represents same population. Mean should equal population mean of 5.65 Mean should equal population mean of 5.65 Calculate mean of sample Calculate mean of sample Mean = 6.5 Mean = 6.5 Obtain sampling distribution and standard error Obtain sampling distribution and standard error Determine probability of obtaining a mean at least as large as the actual sample mean Determine probability of obtaining a mean at least as large as the actual sample mean On that basis, decide whether to accept or reject the null hypothesis On that basis, decide whether to accept or reject the null hypothesis

21. The Null Hypothesis At its heart, the null states that parameters are the same. At its heart, the null states that parameters are the same. For example, 2 means are equal For example, 2 means are equal The difference between the means is zero The difference between the means is zero Any differences reflect sampling error Any differences reflect sampling error Why use the null? Why use the null? Excellent starting place Excellent starting place What would the alternative be? What would the alternative be? We’d have to specify sampling distributions for exact alternative parameter values? We’d have to specify sampling distributions for exact alternative parameter values?

22. Test Statistics and Sampling Distributions The same logic applies to test statistics as well as means. The same logic applies to test statistics as well as means. t’s, F’s, r’s t’s, F’s, r’s A sampling distribution can be calculated for each statistic and used to evaluate the corresponding null. A sampling distribution can be calculated for each statistic and used to evaluate the corresponding null. For t, a sampling distribution when H 0 is true would consist of t values from an infinite number of paired samples. For t, a sampling distribution when H 0 is true would consist of t values from an infinite number of paired samples. Compare current t to sampling distribution to determine viability of null. Compare current t to sampling distribution to determine viability of null.

23. Using Normal Distribution to Test Hypotheses The normal distribution can be used to test hypotheses involving individual scores or sample means. The normal distribution can be used to test hypotheses involving individual scores or sample means. Assumes scores or sampling distributions of the mean are normally distributed Assumes scores or sampling distributions of the mean are normally distributed Going back to our example: Going back to our example: Mean of kids watching violent videos = 6.5 Mean of kids watching violent videos = 6.5 Population parameters Population parameters Mean = 5.65, sd =.45 Mean = 5.65, sd =.45

24. Using Normal Distribution to Test Hypotheses Convert 6.5 to a z score Convert 6.5 to a z score applet applet p(6.5|N(5.65,0.45))=.06 p(6.5|N(5.65,0.45))=.06

25. Terminology Significance Level Significance Level Probability with which we are willing to reject null when it is in fact correct Probability with which we are willing to reject null when it is in fact correct Also called alpha level Also called alpha level Rejection Region Rejection Region Set of outcomes that will lead to rejection of null Set of outcomes that will lead to rejection of null Alternative Hypothesis Alternative Hypothesis Hypothesis that is adopted when null is rejected Hypothesis that is adopted when null is rejected Usually the research hypothesis: Usually the research hypothesis:

26. Type I and Type II Errors As we’ve seen, determining whether a difference is “real” or due to sampling error requires a choice of a critical value or significance level. As we’ve seen, determining whether a difference is “real” or due to sampling error requires a choice of a critical value or significance level. Because we are making a choice, there is always the chance that the choice will be incorrect. Because we are making a choice, there is always the chance that the choice will be incorrect.

27. Type I and Type II Errors If we use a significance level of.05 If we use a significance level of.05 5% of the time we will reject the null hypothesis when it is true 5% of the time we will reject the null hypothesis when it is true Type I Error Type I Error p(Type I) = alpha p(Type I) = alpha If we feel this amount of error is too large, what can we do to minimize Type I errors? If we feel this amount of error is too large, what can we do to minimize Type I errors?

28. Type I and Type II Errors Use a more stringent alpha level to reduce Type I errors Use a more stringent alpha level to reduce Type I errors Alpha =.01; only 1% error in rejecting null Alpha =.01; only 1% error in rejecting null This strategy has a trade-off This strategy has a trade-off Failing to reject the null when it is false is a Type II error Failing to reject the null when it is false is a Type II error p(Type II) = beta p(Type II) = beta

29. DecisionTrue StateOf World Null TrueNull False Reject NullType 1 ErrorCorrect Decision Fail to Reject NullCorrect DecisionType II Error

30. One-Tailed vs. Two-Tailed Tests Two-tailed (nondirectional) tests are most common Two-tailed (nondirectional) tests are most common Look for extremes in both tails (i.e., positive or negative deviations from the mean) Look for extremes in both tails (i.e., positive or negative deviations from the mean) Alpha =.05 has.025 null rejection area in each tail of sampling distribution Alpha =.05 has.025 null rejection area in each tail of sampling distribution Used because one might never truly be sure what outcome to expect Used because one might never truly be sure what outcome to expect

31. One-Tailed vs. Two-Tailed Tests One-tailed (directional tests) are less commonly used One-tailed (directional tests) are less commonly used Look for extreme parameter values in only 1 tail Look for extreme parameter values in only 1 tail Researcher predicts direction of difference Researcher predicts direction of difference Alpha=.05 places total.05 null rejection area in a single tail Alpha=.05 places total.05 null rejection area in a single tail What is the benefit in terms of power? What is the benefit in terms of power? Smaller differences will be viewed as significant due to increased null rejection area Smaller differences will be viewed as significant due to increased null rejection area