PSYCH 301 1st Edition Lecture 9Outline of Last Lecture I. QuestionsII. Statistical InferenceIII. Three distributionsIV. Sampling DistributionV. QuestionsOutline of Current Lecture VI. Z TestVII. NHSTVIII. Variability IX. QuestionsCurrent LectureI. Z Test (for samples) taking you mean and comparing it to other meansa. It tells you how likely your sample mean is to happen compared to a population meanb.II. NHST∶ Does being told that a person in a picture has positive personality qualities increase their attractiveness rating?a. STEP 1: Restate the Question as a Research Hypothesis and a Null Hypothesis about the populationsThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.i. HA: Students who are told that the person has positive personality qualities will give higher attractiveness scores then students who were told nothing about the person’s personality qualitiesii. Hₒ: Students who are told that the person has positive personality qualities will NOT give higher attractiveness scores then students who were told nothing about the person’s personality qualitiesb. STEP 2: Determine the Characteristics of the Comparison Distributioni. N=64,μ=200,σ=48ii. What is μm? 1. μm= 200 alsoiii. What is σm?1. σm= σ/√na. 48/√64b. 48/8= 6c. STEP 3: Determine the Cutoff sample score in the comparison distribution at which the null hypothesis should be rejectedi. What is the cutoff z statistic for α = .05 for a one-tailed hypothesis?ii. Z=1.64 (the cut off is either at 1.65 or 1.96 for a two-tailed hypothesis; these are the two numbers we need to remember!)d. STEP 4: Determine your sample’s score on the comparison distributioni. = 220 ii. μ = 200 iii. σm = 6ii. → (220 – 200) / 6 = (20 / 6) = 3.33iii. 3.33 is our sample mean expressed as a Z-statistice. STEP 5: decide whether to reject the null hypothesisi. 3.33 > 1.64 (cutoff score)ii. YES, reject the Null hypothesis because it is well above the cutoff regionf. STEP 6: Interpret your resultsi. Students who are told that the person had positive personality qualities gave higher attractiveness scores (mean = 220) then students who were told nothing about the person’s personality qualities (mean = 200)g. There are 4 pieces of information that will verify your resultsi. ( Z = 3.33, p <.05 )III. Variability (2 ways)1) Standard error of the mean (SEM or just SE)a. σm = √ (σ²/n) = σ/√nb. SEM reflects how large your sample size is because the mean doesn’t always tell enough.c. The mean doesn’t tell how much the scored varied from one score to the next. The SEM tells the degree of variance.d. To show the SEM on a bar chart you add on “standard error bars”e. SEM is both added and subtracted from the mean and that number is marked on the graph and the lines are connected. f.Category 1Category 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5g. In the above graph, category one had less variability with an SEM=0.2 (mean +1 SEM) and (mean -1 SEM) and plot it. Category 2 has more variability, shown by the longer length of the standard error cars. Category 2 has an SEM=0.5 (added and subtracted from the mean)This tellsthat weused theZ-testThis tells thevalue of ourZ-statisticThis tells howp is comparedto α( < , > or = )This tells what αlevel was used(one/two-tailed)2) Confidence Intervals (CI): a range of scores with a good estimate of the population mean (μ) at the center. a. A larger confidence interval indicates a cutoff score that is further away from the population mean. b. CI’s only exist for two-tailed hypothesesc. 95% confidence level for z statistic (from previous example)i. -1.96 < μ < 1.96ii. Mean + (cutoff x SEM) to Mean – (cutoff x SEM)iii. μ – (1.96 x SEM) to μ + (1.96 x SEM)iv. 220 – (1.96 x 6) to 220 + (1.96 x 6)v. 208.24 to 231.76vi. Our 95% CI is from 208.24 to 231.76, the other 5% is outsideof the cutoff regiond. 99% confidence level for z statistici. -2.58< μ <2.58ii. Mean + (cutoff x SEM) to Mean – (cutoff x SEM)iii. μ – (2.58 x SEM) to μ + (2.58 x SEM)iv. 220 – (2.58 x 6) to 220 + (2.58 x 6)v. 204.52 to 235.48vi. Our 99% CI is from 204.52 to 235.48, the other 1% is outsideof the cutoff regionIV. Questionsa. Why does the sampling distribution get skinnier as n increases?i. As n increases, it become less likely to have a sample mean farther awayfrom the population mean.b. How do the formulas for z score and z statistic differ?i. They differ based on what your comparative distribution is.ii. For a z score, you are comparing 1 score to a number of scoresiii. For a z statistic, you are comparing the mean of a sample to a sampling distribution made up of meansc. What does a confidence interval express?i. A range of scores telling us the variability of scores in the data