t-test

Testing Inferences about Population Means

Learning Objectives Compute by hand and interpret

Single sample t Independent samples t Dependent samples t

Use SPSS to compute the same tests and interpret the output

Review 6 Steps for Significance Testing

1. Set alpha (p level).2. State hypotheses, Null and

Alternative.3. Calculate the test statistic (sample

value).

4. Find the critical value of the statistic.

5. State the decision rule.6. State the conclusion.

One Sample Exercise (1)

1. Set alpha. = .052. State hypotheses.

Null hypothesis is H0: = 1000. Alternative hypothesis is H1: 1000.

3. Calculate the test statistic

Testing whether light bulbs have a life of 1000 hours

Calculating the Single Sample t800750940970790980820760

1000860

What is the mean of our sample? = 867

What is the standard deviation for our sample of light bulbs?

SD= 96.73

35.459.30

1000867

XX S

Xt

59.3010

73.96

N

SDSE

X

Determining Significance

4. Determine the critical value. Look up in the table (Heiman, p. 708). Looking for alpha = .05, two tails with df = 10-1 = 9. Table says 2.262.

5. State decision rule. If absolute value of sample is greater than critical value, reject null. If |-4.35| > |2.262|, reject H0.

α(1 tail) 0.05 0.025 0.05 0.025 0.05 0.025 0.05 0.025 0.05 0.025

α(2 tail) 0.10 0.050 0.10 0.050 0.10 0.050 0.10 0.050 0.10 0.050

df df df df df

1 6.3138 12.707 21 1.7207 2.0796 41 1.6829 2.0196 61 1.6702 1.9996 81 1.6639 1.9897

2 2.9200 4.3026 22 1.7172 2.0739 42 1.6820 2.0181 62 1.6698 1.9990 82 1.6636 1.9893

3 2.3534 3.1824 23 1.7139 2.0686 43 1.6811 2.0167 63 1.6694 1.9983 83 1.6634 1.9889

4 2.1319 2.7764 24 1.7109 2.0639 44 1.6802 2.0154 64 1.6690 1.9977 84 1.6632 1.9886

5 2.0150 2.5706 25 1.7081 2.0596 45 1.6794 2.0141 65 1.6686 1.9971 85 1.6630 1.9883

6 1.9432 2.4469 26 1.7056 2.0555 46 1.6787 2.0129 66 1.6683 1.9966 86 1.6628 1.9879

7 1.8946 2.3646 27 1.7033 2.0518 47 1.6779 2.0117 67 1.6679 1.9960 87 1.6626 1.9876

8 1.8595 2.3060 28 1.7011 2.0484 48 1.6772 2.0106 68 1.6676 1.9955 88 1.6623 1.9873

9 1.8331 2.2621 29 1.6991 2.0452 49 1.6766 2.0096 69 1.6673 1.9950 89 1.6622 1.9870

10 1.8124 2.2282 30 1.6973 2.0423 50 1.6759 2.0086 70 1.6669 1.9944 90 1.6620 1.9867

11 1.7959 2.2010 31 1.6955 2.0395 51 1.6753 2.0076 71 1.6666 1.9939 91 1.6618 1.9864

12 1.7823 2.1788 32 1.6939 2.0369 52 1.6747 2.0066 72 1.6663 1.9935 92 1.6616 1.9861

13 1.7709 2.1604 33 1.6924 2.0345 53 1.6741 2.0057 73 1.6660 1.9930 93 1.6614 1.9858

14 1.7613 2.1448 34 1.6909 2.0322 54 1.6736 2.0049 74 1.6657 1.9925 94 1.6612 1.9855

15 1.7530 2.1314 35 1.6896 2.0301 55 1.6730 2.0041 75 1.6654 1.9921 95 1.6610 1.9852

16 1.7459 2.1199 36 1.6883 2.0281 56 1.6725 2.0032 76 1.6652 1.9917 96 1.6609 1.9850

17 1.7396 2.1098 37 1.6871 2.0262 57 1.6720 2.0025 77 1.6649 1.9913 97 1.6607 1.9847

18 1.7341 2.1009 38 1.6859 2.0244 58 1.6715 2.0017 78 1.6646 1.9909 98 1.6606 1.9845

19 1.7291 2.0930 39 1.6849 2.0227 59 1.6711 2.0010 79 1.6644 1.9904 99 1.6604 1.9842

20 1.7247 2.0860 40 1.6839 2.0211 60 1.6706 2.0003 80 1.6641 1.9901 100 1.6602 1.9840

Stating the Conclusion 6. State the conclusion. We reject the

null hypothesis that the bulbs were drawn from a population in which the average life is 1000 hrs. The difference between our sample mean (867) and the mean of the population (1000) is SO different that it is unlikely that our sample could have been drawn from a population with an average life of 1000 hours.

One-Sample Statistics

10 867.0000 96.7299 30.5887BULBLIFEN Mean Std. Deviation

Std. ErrorMean

One-Sample Test

-4.348 9 .002 -133.0000 -202.1964 -63.8036BULBLIFEt df Sig. (2-tailed)

MeanDifference Lower Upper

95% ConfidenceInterval of the

Difference

Test Value = 1000

SPSS Results

Computers print p values rather than critical values. If p (Sig.) is less than .05, it’s significant.

t-tests with Two Samples

Independent Samples t-test

Dependent Samples t-test

Independent Samples t-test Used when we have two independent

samples, e.g., treatment and control groups.

Formula is: Terms in the numerator are the sample

means. Term in the denominator is the standard

error of the difference between means.

diffXX SE

XXt 21

21

Independent samples t-test

The formula for the standard error of the difference in means:

2

22

1

21

N

SD

N

SDSEdiff

Suppose we study the effect of caffeine on a motor test where the task is to keep a the mouse centered on a moving dot. Everyone gets a drink; half get caffeine, half get placebo; nobody knows who got what.

Independent Sample Data (Data are time off task)

Experimental (Caff) Control (No Caffeine)12 21

14 18

10 14

8 20

16 11

5 19

3 8

9 12

11 13

15

N1=9, M1=9.778, SD1=4.1164 N2=10, M2=15.1, SD2=4.2805

Independent Sample Steps(1)

1. Set alpha. Alpha = .052. State Hypotheses.

Null is H0: 1 = 2.

Alternative is H1: 1 2.

Independent Sample Steps(2)

3. Calculate test statistic:

93.110

)2805.4(

9

)1164.4( 22

2

22

1

21

N

SD

N

SDSEdiff

758.293.1

322.5

93.1

1.15778.921

diffSE

XXt

Independent Sample Steps (3)

4. Determine the critical value. Alpha is .05, 2 tails, and df = N1+N2-2 or 10+9-2 = 17. The value is 2.11.

5. State decision rule. If |-2.758| > 2.11, then reject the null.

6. Conclusion: Reject the null. the population means are different. Caffeine has an effect on the motor pursuit task.

α(1 tail) 0.05 0.025 0.05 0.025 0.05 0.025 0.05 0.025 0.05 0.025

α(2 tail) 0.10 0.050 0.10 0.050 0.10 0.050 0.10 0.050 0.10 0.050

df df df df df

1 6.3138 12.707 21 1.7207 2.0796 41 1.6829 2.0196 61 1.6702 1.9996 81 1.6639 1.9897

2 2.9200 4.3026 22 1.7172 2.0739 42 1.6820 2.0181 62 1.6698 1.9990 82 1.6636 1.9893

3 2.3534 3.1824 23 1.7139 2.0686 43 1.6811 2.0167 63 1.6694 1.9983 83 1.6634 1.9889

4 2.1319 2.7764 24 1.7109 2.0639 44 1.6802 2.0154 64 1.6690 1.9977 84 1.6632 1.9886

5 2.0150 2.5706 25 1.7081 2.0596 45 1.6794 2.0141 65 1.6686 1.9971 85 1.6630 1.9883

6 1.9432 2.4469 26 1.7056 2.0555 46 1.6787 2.0129 66 1.6683 1.9966 86 1.6628 1.9879

7 1.8946 2.3646 27 1.7033 2.0518 47 1.6779 2.0117 67 1.6679 1.9960 87 1.6626 1.9876

8 1.8595 2.3060 28 1.7011 2.0484 48 1.6772 2.0106 68 1.6676 1.9955 88 1.6623 1.9873

9 1.8331 2.2621 29 1.6991 2.0452 49 1.6766 2.0096 69 1.6673 1.9950 89 1.6622 1.9870

10 1.8124 2.2282 30 1.6973 2.0423 50 1.6759 2.0086 70 1.6669 1.9944 90 1.6620 1.9867

11 1.7959 2.2010 31 1.6955 2.0395 51 1.6753 2.0076 71 1.6666 1.9939 91 1.6618 1.9864

12 1.7823 2.1788 32 1.6939 2.0369 52 1.6747 2.0066 72 1.6663 1.9935 92 1.6616 1.9861

13 1.7709 2.1604 33 1.6924 2.0345 53 1.6741 2.0057 73 1.6660 1.9930 93 1.6614 1.9858

14 1.7613 2.1448 34 1.6909 2.0322 54 1.6736 2.0049 74 1.6657 1.9925 94 1.6612 1.9855

15 1.7530 2.1314 35 1.6896 2.0301 55 1.6730 2.0041 75 1.6654 1.9921 95 1.6610 1.9852

16 1.7459 2.1199 36 1.6883 2.0281 56 1.6725 2.0032 76 1.6652 1.9917 96 1.6609 1.9850

17 1.7396 2.1098 37 1.6871 2.0262 57 1.6720 2.0025 77 1.6649 1.9913 97 1.6607 1.9847

18 1.7341 2.1009 38 1.6859 2.0244 58 1.6715 2.0017 78 1.6646 1.9909 98 1.6606 1.9845

19 1.7291 2.0930 39 1.6849 2.0227 59 1.6711 2.0010 79 1.6644 1.9904 99 1.6604 1.9842

20 1.7247 2.0860 40 1.6839 2.0211 60 1.6706 2.0003 80 1.6641 1.9901 100 1.6602 1.9840

Independent Samples Test

Levene's Test for Equality of

Variances t-test for Equality of Means

F Sig. t df

time Equal variances assumed .135 .718 2.755 17

Equal variances not

assumed 2.761 16.911

Independent Samples Test

t-test for Equality of Means

Sig. (2-tailed) Mean Difference

Std. Error

Difference

time Equal variances assumed .014 5.322 1.932

Equal variances not

assumed

.013 5.322 1.927

Independent Samples Test

t-test for Equality of Means

95% Confidence Interval of the

Difference

Lower Upper

time Equal variances assumed 1.247 9.398

Equal variances not

assumed

1.254 9.390

Using SPSS

Independent Samples Exercise

Experimental Control

12 20

14 18

10 148 2016

Work this problem by hand and with SPSS. You will have to enter the data into SPSS.

Group Statistics

5 12.0000 3.1623 1.4142

4 18.0000 2.8284 1.4142

GROUPexperimental group

control group

TIMEN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

.130 .729 -2.958 7 .021 -6.0000 2.0284 -10.7963 -1.2037

-3.000 6.857 .020 -6.0000 2.0000 -10.7493 -1.2507

Equal variancesassumed

Equal variancesnot assumed

TIMEF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

SPSS Results

Dependent Samples t-tests

Dependent Samples t-test Used when we have dependent samples –

matched, paired or tied somehow Repeated measures Brother & sister, husband & wife Left hand, right hand, etc.

Useful to control individual differences. Can result in more powerful test than independent samples t-test.

Dependent Samples tFormulas:

diffX SE

Dt

D

t is the difference in means over a standard error.

pairs

Ddiff

n

SDSE

The standard error is found by finding the difference between each pair of observations. The standard deviation of these difference is SDD. Divide SDD by sqrt(number of pairs) to get SEdiff.

Another way to write the formula

pairs

DX

nSD

Dt

D

Dependent Samples t example

Person TX (time in sec)

Placebo Difference

1 60 55 5

2 35 20 15

3 70 60 10

4 50 45 5

5 60 60 0

M 55 48 7

SD 13.23 16.81 5.70

Dependent Samples t Example (2)

55.25

70.5

pairs

diffn

SDSE

75.255.2

7

55.2

4855

diffSE

Dt

1. Set alpha = .052. Null hypothesis: H0: 1 = 2. Alternative

is H1: 1 2.

3. Calculate the test statistic:

Dependent Samples t Example (3)

4. Determine the critical value of t. Alpha =.05, tails=2 df = N(pairs)-1 =5-1=4. Critical value is 2.776

5. Decision rule: is absolute value of sample value larger than critical value?

6. Conclusion. Not (quite) significant. Painfree does not have an effect.

Using SPSS for dependent t-test

Paired Samples Statistics

Mean N Std. Deviation Std. Error Mean

Pair 1 TX 55.00 5 13.229 5.916

Placebo 48.00 5 16.808 7.517

Paired Samples Test

Paired Differences

Mean Std. Deviation Std. Error Mean

95% Confidence Interval of the

Difference

Lower Upper

Pair 1 TX - Placebo 7.000 5.701 2.550 -.079 14.079

Paired Samples Test

t df Sig. (2-tailed)

Pair 1 TX - Placebo 2.746 4 .052