Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 1. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 2 Overview...

Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 1


Overview of t-scores

• Very similar to z-scores– Provides way of judging how extreme a sample mean is – A bunch of t-scores form a t-distribution

• Done when σ is unknown

• Used for hypothesis testing:– Ex: You wonder if college students really get 8 hours of sleep

• Ho: μ = 8 (College students do get eight hours of sleep)• Ha: μ 8 (College students don’t get eight hours of sleep)

• t-distribution provides foundation for t-test– can do by hand with table– can do on SPSS

• Key difference: t-test done when σ is unknown


Review: Different Measures of Stand. Dev.

nn

xx

x

2

2

Have all the scores in a population

E.g., SAT scores (ETS has every single score).

Have only scores in a sample, want to estimate variability in population

E.g., hours of sleep students in this class slept last night

(Need to adjust because you’ve only got sample data.)

1ˆ

22

nnx

xsx

* Calculate differently based on available information


Different Measures of Sampling Error

• If σx is known, do z-test

• Use σx to get measure of sampling error in distribution.

• If σx is not known, do t-test

• Use ŝx to get measure of sampling error in distribution.

x

xx

xz

son

...

x

xx

s

xt

son

ss

ˆ

...

ˆˆ


t-distributions vs. z-distributions


Comparing Frequency & Sampling Distributions (T1)

Frequency D-z Sampling D – z Sampling D - t

Havex ’s xbars xbars

Compare

Amt. of Variab. +

Meas. ofVariab.

Formula

xz

x

xz

xx x

x xs

xs

xt

ˆ

x


Practice Problem: Calculating t-test

• Do college students sleep 8 hours per night?Do college students sleep 8 hours per night?

• Follow hypothesis testing steps:1. State type of comparison

2. State null (H0) and alternative (HA)

3. Set standards:

a. State type of test (& critical values if doing by hand )• E.g., t-critical (get from table in back of book)

b. Significance level you require (eg. α = .05)

c. 1 vs. 2 tailed test (we’ll always do 2-tail tests- more conservative)

4. Calculate statistic (e.g. get t-obtained)

5. State decision and explain in English.


Finding t-critical


Homework Problem

• College graduates score 35, 45, 30, 50, 60, 55, 60, 45, 40 on a critical thinking test.

• If normal people score 45 on the test, do college graduates score significantly better?

• Do hypothesis testing steps


HW: Standard Deviation Calculation


HW: T-Calculation

• SD = 10.6066

• SE = 3.536

• t = (46.67-45) / 3.536 = .4781


HW: Hypothesis testing steps

1. Compare xbar and μ

2. Ho: μ = 45 Ha: μ 45

3. α = .05, df = n-1 = 8, two-tailed test. tcritical = 2.306

4. tobt = .471

5. Retain Ho. The hypothesis was not supported. College graduates did not score significantly better (M=46.67) on critical thinking (μ =45), t(8) = .471, n.s.


T-test Example: Speed

• The government claims cars traveling in front of your house average 55 mph. You think this is a load of…. That is, you think cars travel faster than this.

• You steal a police radar gun and clock nine cars, obtaining the following speeds:

• 45, 60, 65, 55, 65, 60, 50, 70, 60

• What’s μ?


SPSS StepsGo to Compare Means

Pick variable

Set to μ

Enter the speeds of cars you clocked.


Output part #1

One-Sample Statistics

9 58.89 7.817 2.606SPEEDN Mean

Std.Deviation

Std. ErrorMean

Average speed of these cars (sample mean).

Standard deviation of these speeds.

xsx

Standard error of the mean – the typical difference we’d expected sampling error to cause.

xs

Number of cars you measured (sample size).


Output part #2

One-Sample Test

1.492 8 .174 3.89 -2.12 9.90SPEEDt df

Sig.(2-tailed)

MeanDifference Lower Upper

95% ConfidenceInterval of the

Difference

Test Value = 55tobtained

•pobt: Proportion of time you’d see a difference of this size

simply because of sampling error •This value must fall below .05 to say we have a significant difference.

x

xs

xt

ˆ

By hand, it’s

Note: There’s no t-critical when done with SPSS


Hypothesis Testing Steps

1. Compare xbar and μ

2. Ho: μ = 55 Ha: μ 55

3. α = .05, df = n-1 = 8, two-tailed test.

4. tobt = 1.492, pobt = .174

5. Retain Ho. Average car speed (M=58.89) does not differ significantly from 55 mph speed limit, t(8) = 1.492, n.s.


Same test, different outcome

• What if we had measured slightly different speeds?

• 50,60,65,55,65,60,55,75,65

• What happens to μ? xbar?


9 61.11 7.407 2.469SPEEDN Mean Std. Deviation

Std. ErrorMean

One-Sample Test

2.475 8 .038 6.11 .42 11.80SPEEDt df Sig. (2-tailed)

MeanDifference Lower Upper

95% ConfidenceInterval of the

Difference

Test Value = 55

• In this case, we’d reject the Ho.

• Speeds appear to exceed 55 mph, t(8) = 2.475, p.05


Learning Check

1. As tobt increases, we become more likely to ___ Ho.

2. If the sample size increases tobt will _____ and tcritical will ______

3. If the difference between xbar and μ increases

a. sampling error will ______

b. tcritical will _______

c. tobtained will _______

d. ŝxbar will _______

e. you become _____ likely to reject the Ho


Learning Check

1. A researcher compares the number of workdays missed for employees who are depressed versus the company-wide average of 6 days per year.

a. Rejecting the Ho would mean what about depressed employees?

b. Would you be more likely to reject Ho with a sample mean of 8 or 10?

c. Would you be more likely to reject Ho with a ŝx of 1.5 or 3?


Decision Errors

• Educated guesses can be wrong.

• Def: Drawing a false conclusion from an hypothesis test– Never know for sure if a difference is due just to sampling error

or if it truly reflects a treatment effect.

• Two Types– Type I: Falsely rejecting null

• Seeing something that’s not there. False positive.

– Type II: Falsely retaining null

• Missing something that is there. False negative.


Decision Errors – Example #1

“Is that a burglar or am I hearing things?”• You hear a noise in your house and wonder if it means

there’s a burglar in the house. The problem is that it could just be regular background noise (___________) or it really could mean something’s going on (____________). You’d make a mistake if you…

a. decide there’s a burglar when there is not. Type I Error

b. decide there’s no burglar when there is. Type II Error


Decision Errors – Example #2

• “Did the training work or is this group of people just more talented than usual?”

• You implement a training program to improve job performance, and then compare the performance of trainees to average performance. You’d make a mistake if you….

a. Conclude participants don’t differ from average, but in reality the training DOES improve performance.

Type II error

b. Conclude participants do better than average, but in reality the training does NOT improve performance.

Type I error


Graph of Type I Error – α

When rejecting Ho, you may commit a Type I error.(Wrongly concluding cars DO NOT average 55 mph.)

You guess this.Ha: μ>55

But this is actually true.

Ho: μ=55

tcrittcrit

αα

So α is the chance of making a Type I error.


Graph of Type II Error – β

When retaining Ho, you may commit a Type II error.

(In this case, assuming cars DO average 55 mph.)

So β is the chance of making a Type II error.

Ho: μ=55

You guess this…

tcrit

β

Ha: μ>55…but this is

actually true.


Effect-size statistic: d

• Statistical vs. Practical Significance– Statistical Sig: Decides if difference is reliable (e.g., t-test)

– Practical Sig: Decides if difference is big enough to be practically important

– So, only do tests for practical significance if you get statistical significance first (i.e., if you reject the H0

• Effect size (d)– Def: Impact of IV on DV in terms of standard deviation units.

– So, d=1 means the IV “raises” scores 1 full standard deviation.

– d = .2+ small effect size

– d = .5+ moderate effect size

– d = .8+ large effect size xs

xd

ˆ

This is standard deviation, not standard error


Practice: Meditation

• You suspect the anxiety level of people in your meditation class will differ from a score of 3 on a 1-5 anxiety self-assessment scale.

• #1: Do an SPSS analysis and then fill-in the following information:

x23432221

μ =

σ =

ŝx =

Ŝxbar=

M =

Mean Difference =

tcrit =

tobt =

pobt =

d =



8 2.38 .916 .324anxietyN Mean

Std.Deviation

Std. ErrorMean

One-Sample Test

-1.930 7 .095 -.625anxietyt df

Sig.(2-tailed)

MeanDifference

Test Value = 3


Practice: Meditation

• You suspect the anxiety level of people in your meditation class will differ from a score of 3 on a 1-5 anxiety self-assessment scale.

• #1: Do an SPSS analysis and then fill-in the following information:

x23432221

μ = 3

σ =???

ŝx = .916

Ŝxbar= .324

M = 2.38

Mean Diff. = -.625

tcrit = ± 2.365

tobt = -1.930

pobt = .095

d = inappropriate


• #2: Hypothesis Testing Steps


#2: Hypothesis Testing Steps

1. Cf. M and μ.

2. Ho: μ = 3 Ha: μ ≠ 3

3. 2-tailed, α = .05, df=7

4. tobt = -1.930, pobt = .095

5. Retain Ho. The hypothesis was not supported. The anxiety of those meditating (M=2.38) did not differ significantly from average anxiety (μ=3), t(7) = -1.930, n.s.


• #3 Sketch the distribution, including regions of rejection, tcritical and tobtained.

• #4 What type of decision error is possible here?

• #5 Pretend you had a significant result – calculate d.

Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 1. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 2 Overview...

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Transcript of Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 1. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 2 Overview...