PSYC 3101 1st EditionExam # 2 Study Guide Chapters: 7 - 9Chapter 7 Inferential Stats and Sampling Distributions- Researchers select a sample or portion of data from a much larger population - They then measure a sample stat they are interested in, such as the mean or variance in a sample- They do not measure a sample mean- Instead they select a sample mean to learn more about the mean in a populationEx: o Ask a question about a population: How good is your score on an exam compared to the mean performance in your class?o Population: identify all individuals of interest who were identified by the question: All students who took the exam in your class. o Observations made in the sample are generalized to the population: you compare your grade to the mean grade in the sample to determine how well you did in your class (the population) o Sample: Select a portion of individuals from the population and measure a sample statistic identified by the question: Select a few students in your class and record their grades on the exam.Sampling distribution: a distribution of all sample means or sample variances that could be obtained in samples of a given size from the same population.- To avoid bias: researchers use a random procedure- For samples to be selected at random, all individuals in a population must have an equal chance of being selected. The Sample Mean ( μ ): is calculated by summing all scores in the population, then dividing by the population size. - Sample mean is an unbiased estimator meaning the sample mean we obtain in a randomly selected sample equals the value of the population mean on average. - Central Limit Theorem explains that regardless of the distribution of scores in a population, the sampling distribution of sample means selected from that population will be approximately normally distributed. - Variance can be any positive number and gives us an idea of how far the value of a sample mean can deviate from the value of the population mean.Standard error of the mean- Sampling error is the extent to which sample means selected from the same population differ from one another. o σM = σ /sqrt(n)- Factors that decrease standard error:o As the population standard deviation decreases, standard error decreases.o As the sample size increases, standard error decreases.Convert any sampling distribution with any mean and standard error to a standard normal distribution by applying the z-transformation.z=(M- μ)/ σMChapter 8Hypothesis testing: method in which we select samples to learn more about characteristics in a given population. Four Steps to Hypothesis Testing: Memorize this!!!!1) State the null and alt hypotheses 2) Set criteria for rejecting the null (direction, alpha level, critical value)3) Collect data and compute test statistic4) Make decision about the null and draw a conclusion (5 o’clock news summary)Ex: Happiness scores on Becky’s happiness test (BHT) are normally distributed with a mean of 68 and std dev of 4.5. I think that people who are married are happier than average, so I found 24 married people and gave them the BHT. Their mean was 71. Test to see if married people have higher happiness scores, using alpha=.05 and the four steps.μ=68, σ =4.5, n=24, M=711) H0= μ=68, H1= μ>682) Alpha=.05, one-tailed, Zcv=1.6453) Zobs=(M- μ)/ σM = (71-68)/(4.5/sqrt(24))=3.274) Reject the null; married people are happier than average.Types of error:Retain the Null Reject the NullTrue Correct 1-alphaType I ErroralphaFalse Type II ErrorbetaCorrect1-betapowerCohen’s d- A decision to reject the null hypotheses means that an effect is significant. For a one-sample test, an effect is the difference between a sample mean and the population mean stated in the null hypothesis.o (M- μ)/ σDescription of Effect Effect Size (d)Small d<0.2Medium 0.2<d<0.8Large d<0.8Power- As the effect size increases, power increases- Increasing sample size increases powerChapter 9Going from z to t- It is rare that we know the variance of a population, which makes it impossible to compute the z-statistic.- To compute z-statistic formula, we need to know population standard deviation, which requires that we know the variance. If we don’t know population variance, then we can’t compute the value for z.- Therefore there is a formula for error called estimated standard error (sM ):o SD/Sqrt(n)- An alternative test statistic for one-independent sample when the population variance is unknown uses the t-statistic formula:o tobt =(M- μ)/ sMDegrees of freedom- t-distribution associated with degrees of freedom (df), which equals n-1. - As the sample size increases, the degrees of freedom of the sample variance increases.One-independent sample t-test- This test is used to test hypotheses concerning a single group mean selected from a population with an unknown variance. o Ex: Scientists study relatives of patients with OCD. As part of their study, they record the social functioning of relatives using a 36-item health survey where scores range from 0 (worst possible health) to 100 (best possible health). The mean score of the population was 77.43. The researchers then select a sample of 61 relatives and recorded a mean of 61.91+/-23.29. Test whether the mean score in this sample significantly differs from that in the general population at a .05 level of significance.  Step 1: State the hypotheses. H0: μ=77.43; H1: μ≠77.43 Step 2: Set criteria for rejecting the null (direction, alpha level, critical value)Level of significance= .05, two-tailed test with n-1 df. n=61 so df=60. Critical value=+/-2.000 Step 3: Collect data and compute test statisticSD/Sqrt(n)= 23.29/sqrt(61)=2.98tobt =(M- μ)/ sM = (61.91-77.43)/2.98=5.21 Step 4: Make decision about the null and draw a conclusion (5 o’clock news summary)Obtained value falls below the critical value Social functioning scores among relatives of patients with OCD were significantlylower than scores in the general population. - Estimated effect size most often used with t-tests is the estimated Cohen’s do d=(M- μ)/SDo From the example above: d=(61.91-77.43)/23.29=-0.67- Another measure of effect size is to estimate the proportion of variance that can be accounted for by some treatment.o Proportion of variance= variability explained/total variability o Two measures for proportion of variance: eta-squared and omega-squared Eta-squared=(t^2)/(t^2+df)From previous example:Eta-squared=(-5.21)^2/(((-5.21)^2)+60) =.31We