PSY 230 Module 3 Assignment (A-3) For this assignment you will use multiple data sets, each embedded within the assignment below. Q1) Find the z values that form the boundaries of the critical region for a two-tailed test with = .05 for each of the following sample size (n) values. a) n = 9 b) n = 16 c) n = 24 Q2) Although there is a popular belief that herbal remedies such as Ginkgo biloba and Ginseng may improve learning and memory in healthy adults, these effects are usually not supported by wellcontrolled research (Persson, Bringlov, Nilsson, & Nyberg, 2004). In a typical study, a researcher obtains a sample of n = 36 participants and has each person take the herbal supplements every day for 90 days. At the end of 90 days, each person takes a standardized memory test. For the general population, scores for the test are normally distributed with Population M = 80 and Population SD = 18. The sample of research participants had an average of M = 84. a) Identify the independent (IV) and dependent (DV) variables for this study. b) Use the information above to conduct a two-tailed hypothesis test using p = .05 as your alpha level. Be sure you complete all 5 steps of hypothesis testing and show your work/answers for each step. Q3) There is some evidence indicating that people with visible tattoos are viewed more negatively than people without visible tattoos (Resenhoeft, Villa, & Wiseman, 2008). To test this hypothesis, a researcher first obtained overall ratings of attractiveness for a woman with no tattoos shown in a color photograph. On a 7-point scale, the woman received an average rating of Population M = 4.9, and the distribution of ratings was normal with Population SD = 0.84. The researcher then modified the photo by adding a tattoo of a butterfly on the woman's left arm. The modified photo was then shown to a sample of n = 16 students at a local university and the students used the same 7 -point scale to rate the attractiveness of the woman. The average score for the photo with the tattoo was M = 4.2 a) Would the researcher conduct a 1-tailed or 2-tailed hypothesis test? Explain why. b) Use the information above to conduct a hypothesis using p = .05 as your alpha level. Be sure you complete all 5 steps of hypothesis testing and show your work/answers for each step. c) Calculate a 95% confidence interval for the above sample. Q4) Childhood participation in sports, cultural groups, and youth groups appears to be related to selfesteem for adolescents (McGee, Williams, Howden-Chapman, Martin, & Kawachi, 2006). In a representative study, a sample of n = 100 adolescents with a history of group participation is given a standardized self-esteem questionnaire. For the general population of adolescents, scores on this questionnaire form a normal distribution with Population M = 40 and Population SD = 12. The sample of group participants has an average of M = 43.84. a) Would the commander conduct a 1-tailed or 2-tailed hypothesis test? Explain why. b) Use the information above to conduct a hypothesis test using p = .01 as your alpha level. Be sure you complete all 5 steps of hypothesis testing and show your work/answers for each step. c) Calculate a 99% confidence interval for the above sample. Extra Credit (3 points): In no less than 5 sentences, explain a research idea you would test using hypothesis testing. Be sure to identify your independent (IV) and dependent (DV) variables, and report your null and alternative hypotheses. Why are you interested in this particular research question? Module 5 Assignment (A-5) For this assignment you will use multiple data sets, each embedded within the assignment below. Q1 and Q2: Steven Schmidt (1994) conducted a series of experiments examining the effects of humor on memory. In one study, participants were given a mix of humorous and non-humorous sentences and significantly more humorous sentences were recalled. However, Schmidt argued that the humorous sentences were not necessarily easier to remember, they were simply preferred when participants had the choice between the two types of sentence. To test this argument, he switched to an independent measures design in which one group got a set of exclusively humorous sentences and another group got a set of exclusively non-humorous sentences. The following data are similar to the results from the published independent measures study: Humorous Sentences 4 5 2 4 6 7 6 6 2 5 4 3 3 3 5 3 Non- Humorous Sentences 6 3 5 3 3 4 2 6 4 3 4 4 5 2 6 4 Q1) Use the dataset above to calculate each of the following statistics. Note that the Humorous Sentences Group is the sample for Population 1 and the Non-Humorous Sentences Group is the sample for Population 2. a) X 1 and X 2 b) df1 df2 dftotal c) 21 22 2pooled (estimated population variances) d) standard error of the sampling distribution e) tobtained Q2) Your statistics professor is considering changing the amount of humor used in his/her lectures, but is not sure whether this will impact students' memory for the lecture material. Use the statistics calculated in Q1 to perform a hypothesis test using an independent samples t test with an alpha level of .05 and a two-tailed design (show all 5 steps of hypothesis testing). As part of your results, make a recommendation about whether your statistics professor should use humor in his/her lectures to ensure that students will remember the content. Q3 and Q4: Siegel (1990) found that elderly people who owned dogs were less likely to pay visits to their doctors after upsetting events than were those who did not own pets. Similarly, consider the following hypothetical data. A sample of elderly dog owners is compared to a similar group (in terms of age and health) who do not own dogs. The researcher records the number of visits to the doctor during the past year for each person. The data are as follows: Control Pet Group Owners 10 8 7 9 13 7 6 12 7 4 9 3 7 Q3) Use the dataset above to calculate each of the following statistics. Note that the Control Group is the sample for Population 1 and the Dog Owners Group is the sample for Population 2. a) X 1 and X 2 b) df1 df2 dftotal c) 21 22 2pooled (estimated population variances) d) standard error of the sampling distribution e) tobtained Q4) An assisted living facility is reviewing their policy that strictly forbids their elderly residents to have pets, and they will use your data analysis and results to make a decision. Use the statistics calculated in Q3 to determine whether the researcher's data replicates the findings of the Siegel (1990) study. Perform a hypothesis test using an independent samples t test with an alpha level of .05 and a two- tailed design (show all 5 steps of hypothesis testing). As part of your results, make a recommendation to the assisted living facility about their pet policy. Extra Credit (3 points): In no less than 5 sentences, explain a research idea you would test using hypothesis testing with an independent samples t test. Be sure to identify your independent (IV) and dependent (DV) variables, and report your null and alternative hypotheses. Why are you interested in this particular research question? The Sampling Distribution What is it? The sampling distribution is a distribution of a sample statistic. When using a procedure that repeatedly samples from a population and each time computes the same sample statistic, the resulting distribution of sample statistics is a sampling distribution of that statistic. To more clearly define the distribution, the name of the computed statistic is added as part of the title. For example, if the computed statistic was the sample mean, the sampling distribution would be titled \"the sampling distribution of the sample mean.\" Suppose a population consists of the first ten integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and a process takes a random sample without replacement of size N=3 from this population. The random sampling might generate sets that look like { 8, 3, 7}, { 2, 1, 5}, { 6, 3, 5}, { 10, 7, 5}... If the mean ( X ) of each sample is found, the means of the above samples would appear as follows: 6, 2.67, 4.67, 7.33... If this process was continued indefinitely, then the infinite number of sample means would form a sampling distribution of the mean. Graphically, this sampling distribution of the mean would appear as follows: Every statistic has a sampling distribution. For example, suppose that instead of the mean, medians (Md) were computed for each sample. That is, within each sample the scores would be rank ordered and the middle score would be selected as the median. Using the samples above, the medians would be: 7, 2, 5, 7... The infinite number of medians would be called the sampling distribution of the median and could be graphically shown as follows: It is possible to make up a new statistic and construct a sampling distribution for that new statistic. For example, by rank ordering the three scores within each sample and finding the mean of the highest and lowest scores a new statistic could be created. Let this statistic be called the mid-mean and be symbolized by M . For the above samples the values for this statistic would be: 5.5, 3, 4.5, 7.5... and the sampling distribution of the mid-mean could be graphically displayed as follows: Just as the population distributions can be described with parameters, so can the sampling distribution. The expected value and variance of any distribution can be represented by the symbols (mu) and (Sigma squared), respectively. In the case of the sampling distribution, the symbol is often written with a subscript to indicate which sampling distribution is being described. For example, the expected value of the sampling distribution of the mean is represented by the symbol X , that of the median by M d , and so on. The value of X can be thought of as the theoretical mean of the distribution of means. In a similar manner the value of M d , is the theoretical mean of a distribution of medians. The square root of the variance of a sampling distribution is given a special name, the standard error. In order to distinguish different sampling distributions, each has a name tagged on the end of \"standard error\" and a subscript on the symbol. The theoretical standard deviation of the sampling distribution of the mean is called the standard error of the mean and is symbolized by X . Similarly, the theoretical standard deviation of the sampling distribution of the median is called the standard error of the median and is symbolized by Md . In each case the standard error of the sampling distribution of a statistic describes the degree to which the computed statistics may be expected to differ from one another when calculated from a sample of similar size and selected from similar population models. The larger the standard error of a given statistic, the greater the differences between the computed statistics for the different samples. From the example population, sampling method, and statistics described earlier, we would find X = M d = M = 5.5 and X =1.46, Md = 1.96, and M =1.39. Why is the sampling distribution important? Properties of statistics Statistic have different properties as estimators of a population parameters. The sampling distribution of a statistic provides a window into some of the important properties. For example if the expected value of a statistic is equal to the expected value of the corresponding population parameter, the statistic is said to be unbiased. In the example above, all three statistics would be unbiased estimators of the population parameter X . Consistency is another valuable property to have in the estimation of a population parameter, as the statistic with the smallest standard error is preferred as an estimator of the corresponding population parameter, everything else being equal. Statisticians have proven that the standard error of the mean is smaller than the standard error of the median. Because of this property, the mean is generally preferred over the median as an estimator of X . Selection of distribution type to model scores The sampling distribution provides the theoretical foundation to select a distribution for many useful measures. For example, the central limit theorem describes why a measure, such as intelligence, that may be considered a summation of a number of independent quantities would necessarily be distributed as a normal (Gaussian) curve. Hypothesis testing The sampling distribution is integral to the hypothesis testing procedure. The sampling distribution is used in hypothesis testing to create a model of what the world would look like given the null hypothesis was true and a statistic was collected an infinite number of times. A single sample is taken, the sample statistic is calculated, and then it is compared to the model created by the sampling distribution of that statistic when the null hypothesis is true. If the sample statistic is unlikely given the model, then the model is rejected and a model with real effects is more likely. In the example process described earlier, if the sample {3, 1, 4} was taken from the population described above, the sample mean (2.67), median (3), or mid-mean (2.5) can be found and compared to the corresponding sampling distribution of that statistic. The probability of finding a sample statistic of that size or smaller could be found for each e.g. mean (p< .033), median (p<.18), and mid-mean (p<.025) and compared to the selected value of alpha (). If alpha was set to .05, then the selected sample would be unlikely given the mean and midmean, but not the median. How can sampling distributions be constructed? Mathematically Using advanced mathematics statisticians can prove that under given conditions a sampling distribution of some statistic must be a specific distribution. Hogg and Tanis (1997, p. 256) prove the following theorem: If X1, X2, ..., Xn are observations of a random sample of size n from the normal distribution N(,), n 1 X= X i n i=1 and S 2= n 1 ( X i X )2 n1 i=1 then ( n1 ) S 2 2 is 2(n-1) The given conditions describe the assumptions that must be made in order for the distribution of the given sampling distribution to be true. For example, in the above theorem, assumptions about the sampling process (random sampling) and distribution of X (a normal distribution) are necessary for the proof. Of considerable importance to statistical thinking is the sampling distribution of the mean, a theoretical distribution of sample means. A mathematical theorem, called the Central Limit Theorem, describes the relationship of the parameters of the sampling distribution of the mean to the parameters of the probability model and sample size. Monte Carlo Simulations It is not always easy or even possible to derive the exact nature of a given sampling distribution using mathematical derivations. In such cases it is often possible to use Monte Carlo simulations to generate a close approximation to the true sampling distribution of the statistic. For example, a non-random sampling method, a non-standard distribution, or may be used with the resulting distribution not converging to a known type of probability distribution. When much of the current formulation of statistics was developed, Monte Carlo techniques, while available, were very inconvenient to apply. With current computers and programming languages such as Wolfram Mathematica (Kinney, 2009), Monte Carlo simulations are likely to become much more popular in creating sampling distributions. Summary The sampling distribution, a theoretical distribution of a sample statistic, is a critical concept in statistical thinking. The sampling distribution allows the statistician to hypothesize about what the world would look like if a statistic was calculated an infinite number of times. David W. Stockburger US Air Force Academy USAF Academy, CO 80840-6200 References Hogg, R. V. and Elliot, A. T. (1997) PROBABILITY AND STATISTICAL INFERENCE, FIFTH EDITION . Upper Saddle river, NJ: Prentice Hall. Kinney, J. J. (2009) A PROBABILITY AND STATISTICS COMPANION . Hoboken, NJ: John Wiley & Sons, Inc