Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 1 z-scores & t-scores (unit 2) Review –Normal Curve, Deviation Scores, Standard Deviation z-scores for scores (x) –Standard Scores –Describing distance in standard deviation units z-scores for sample means (x bar ) t-scores for sample means –when σ x is unknown, and you estimate based on ŝ x Purpose –Means to determine how extreme x or xbar is –Foundation of hypothesis testing

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 2 Review Normal Curve –Most distributions fit bell-curve pattern –More extreme scores less frequent –x’s (scores) deviate around μ (population average) Deviation Scores –distance of score from mean; foundation for standard deviation Bob has an IQ of 120, when the average IQ is 100 –score – mean OR x – x bar OR x - μ –large deviation scores (pos or neg)  extreme (unlikely) score Standard Deviations –typical deviation found within a distribution –typical distance a given score falls from the mean The standard deviation for IQ is 15, and the mean is 100

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 3 IQ Female Height4’4”4’8”5’0”5’4”5’8”6’0”6’4” Anxiety Stand.Normal Curve Normal Curve (with raw scores and standard scores) Few Extreme Scores μ 1  2 σ

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 4 Relation between SDs and Percent of Scores

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 5 Check-up: Reading the Normal Curve 1.What’s the μ for female height? 2.What’s the σ for IQ? 3.What’s the σ for Anxiety? 7.What’s the standard deviation for f. height? 8.Which variable has the largest stand. dev.? 9.Which scores, on each variable, fall at –3 Stand. Dev.? 10.What percent of scores fall between 0 and –1 S.D.? 4.The most typical 68% of females are between ___ and ___ inches tall. 5.99% of people will fall within what range of anxiety scores? 6.95% of people are between what two IQ scores?

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 6 Check-up: Basic Concepts 1.Contrast Deviation score & Standard Deviation Score 2.What does it mean about a score if it falls near the end of the distribution? 3.If you describe the location of a score in the tail of the distribution with a standard score, what sort of value would the standard score have? 4.Which standard score would be more surprising, -1, +4, 0, or –3?

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 7 Standard Scores: z-scores z-scores –standard scores –location of score on standard normal curve (where μ=0 and σ=1) –distance between score and mean in std. dev. units –indicates “how extreme ” Population Formula Sample Formula

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 8 Understanding the z-score formula Deviation Score “Difference Observed” Standard Deviation “Difference Expected”

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 9 History Test #1

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 10 History Test #2

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 11 Questions about previous slide Estimating with inter-occular method –What % of people score at 75 or below on test #2? –What % of people score at 75 or better on test #2? –What percent of people score 80 or below on test #1? –What percent of people score 80 or above on test #1? Using z-score tables (found in back of text book) –What % of people score at 77 or below on test #2? –What % of people score at 77 or above on test #2? –What percent of people score 82 or below on test #1? –What percent of people score 82 or above on test #1?

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 12 Possible z-score conversions

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 13 Sampling Distributions “Distributions” – what we’ve been using so far for z-scores –Frequency distributions of scores – x’s –Z tells distance x falls from μ –e.g., Your SAT score How does your score compare to the pop. mean? “Sampling Distributions” – new type of z-scores –Distribution of sample means – x bars –Z tells distance x bar falls from μ –e.g. The average score SAT of 4 psyc majors How does the sample mean compare to the pop. mean? Which has less variability? –If you’re estimating travel time to Charleston, do you ask 1 person or 5 people? –As variability decreases, prediction accuracy_______.

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 14 Why Use Sampling Distributions? Pulling out scores: x’s x= 550, 450, 600, 525, 675, etc. use σ x Pulling out sample means : M= 530, 480, 540, 510, 490 SAT Scores μ = 500 n=1 n=4 Sample means have less variability!!!! Sample means better predictors of μ!!!  Use Standard Error of the mean: σ x bar More Accurate! Samples

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 15 Bottom-Line We use samples in research because they better represent the populations than individual scores do. Standard Error of the Mean –Definition: Typical deviation of sample means around the population mean Measure of variability in a sampling distribution –Symbol: –Formula:

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 16 Comparing frequency and sampling distributions Frequency DistributionSampling Distribution Have scores (x ’s) sample means (x bars ) Compare Amt. of Variab.  Meas. of Variab. standard deviation σ x standard error (of the mean) σ xbar Formula

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 17 Frequency Distribution z-score

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 18 Sampling Distribution z-score

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p Graphing Frequency Distribution Std. Scores (z) Raw Scr. (x) What’s always in the center? What measure of variability? This distance equals what?

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p Graphing Sampling Distribution Std. Scores (z) Raw Scores (x) What’s always in the center? What measure of variability? This distance equals what?