z-scores & t-scores (unit 2)

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Dr. Sinn, PSYC 301 Unit 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

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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 - PowerPoint PPT Presentation

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Page 1: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 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 (xbar)

• t-scores for sample means– when σx is unknown, and you estimate based on ŝx

• Purpose– Means to determine how extreme x or xbar is– Foundation of hypothesis testing

Page 2: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 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 – xbar 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

Page 3: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 3

IQ 55 70 85 100 115 130 145

Female Height 4’4” 4’8” 5’0” 5’4” 5’8” 6’0” 6’4”

Anxiety 20 30 40 50 60 70 80

Stand.Normal Curve -3 -2 -1 0 +1 +2 +3

Normal Curve (with raw scores and standard scores)

Few Extreme Scores

Few Extreme Scores

μ

12 σ

Page 4: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 4

Relation between SDs and Percent of Scores

Page 5: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 5

Check-up: Reading the Normal Curve1. 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?

Page 6: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 6

Check-up: Basic Concepts1. 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?

Page 7: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 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”

x

z

S

Mxz

Population Formula

Sample Formula

Page 8: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 8

Understanding the z-score formula

x

z

•Deviation Score

“Difference Observed”

•Standard Deviation

“Difference Expected”

Page 9: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 9

History Test #1

Page 10: z-scores & t-scores (unit 2)

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

History Test #2

Page 11: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 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?

Page 12: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 12

Possible z-score conversions

Page 13: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 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 – xbars

– Z tells distance xbar 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_______.

Page 14: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 14

Why Use Sampling Distributions?

Pulling out scores: x’s

x= 550, 450, 600, 525, 675, etc.

use σx

Pulling out sample means :

M= 530, 480, 540, 510, 490SAT Scores

μ = 500

n=1

n=4

•Sample means have less variability!!!!•Sample means better predictors of μ!!! Use Standard Error of the mean: σxbar

More Accurate!

Samples

Page 15: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 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:

x

nx

x

Page 16: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 16

Comparing frequency and sampling distributions

Frequency Distribution Sampling Distribution

Have scores (x ’s) sample means (xbars)

Compare

Amt. of Variab.

Meas. of

Variab.standard deviation

σx

standard error (of the mean)

σxbar

Formula

x

zx

xz

xx

Page 17: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 17

Frequency Distribution z-score

Page 18: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 18

Sampling Distribution z-score

Page 19: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 19

-3 -2 -1 0 +1 +2 +3

500

Graphing Frequency Distribution

Std. Scores (z)

Raw Scr. (x)

What’s always in the center?

What measure of variability?

This distance equals what?

Page 20: z-scores & t-scores (unit 2)

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 20

-3 -2 -1 0 +1 +2 +3

500

Graphing Sampling Distribution

Std. Scores (z)

Raw Scores (x)

What’s always in the center?

What measure of variability?

This distance equals what?