z-Tests and t-Tests for One Sample - Rutgers Universitymmm431/quant_methods... · t-Tests for One Sample & Two Related Samples ... – For large df, the t distribution will be nearly

Comparing Means:

t-Tests for One Sample & Two Related

Samples

01:830:200:01-05 Fall 2014

t-Tests for One Sample & Two Related Samples

Using the z-Test: Assumptions

• The z-test (of a sample mean against a population mean) is

based on the assumption that the sample means are normally

distributed.

• For this to be true, one of the following must also be true:

– The underlying population (e.g., of test scores) is normally distributed

– The sample size (n) is sufficiently large to approximate normality by way

of the central limit theorem

• Additionally, we must know the population standard deviation

(σ)

01:830:200:01-05 Fall 2014


The t-Statistic

• What if we don’t know σ ?

– In most real-world situations in which we want to test a hypothesis, we

do not know the population standard deviation σ

– If we don’t know σ we can compute a test statistic using s, but this

statistic will no longer be normally distributed, so we can no longer use

the z test statistic

• Why?

– s is variable across samples and its sampling distribution is not normally

distributed

• s2 is distributed as a chi-square distribution, which we’ll talk about near the

end of the semester

01:830:200:01-05 Fall 2014


The Sampling Distributions of s2 and s

Student Score

155 74

120 67

66 69

216 77

188 84

Student Score

237 68

192 76

101 78

109 69

180 76

Student Score

221 65

85 70

223 71

48 63

40 69

Sample 1

Sample 2

Sample 3

74.2M

73.4M

67.6M

2 45.70

6.76

s

s

2 20.80

4.56

s

s

2 11.80

3.44

s

s

7

5n

01:830:200:01-05 Fall 2014


The t-Statistic

• If we compute something like z, but using s instead of σ, we

get a statistic that follows the t distribution

,M

zM

,

M

tM

s

Remember: Similarly,

where Mn

where Ms

n

s

01:830:200:01-05 Fall 2014


Distribution of the t-Statistic

(n = 5)

01:830:200:01-05 Fall 2014


Distribution of the t-Statistic & Sample Size

• You can think of the t statistic as an "estimated z-score."

• The estimation comes from the fact that we are using the sample variance to estimate the unknown population variance.

• The value of degrees of freedom, df = n - 1, determines how well the distribution of t approximates a normal distribution and how well the t statistic represents a z-score.

– For large samples (large df), the estimation is very good and the t statistic

will be very similar to a z-score.

– For small samples (small df), the t statistic will provide a relatively poor estimate of z.

– For large df, the t distribution will be nearly normal, but for small df, the t distribution will be flatter and more spread out than a normal distribution.

01:830:200:01-05 Fall 2014



The shape of the t-distribution depends on the number of degrees of freedom

01:830:200:01-05 Fall 2014



Smaller samples (with fewer df) require greater t values to reject H0

t(4), α = 0.05

t(200), α = 0.05

01:830:200:01-05 Fall 2014


Level of significance for one-tailed test

0.25 0.2 0.15 0.1 0.05 0.025 0.01 0.005 0.0005

Level of significance for two-tailed test

df 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 0.001

1 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 636.619

2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 31.599

3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 12.924

4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 8.610

5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 6.869

6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.959

7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 5.408

8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 5.041

9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.781

10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.587

11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.437

12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 4.318

13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 4.221

14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 4.140

15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 4.073

16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 4.015

17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.965

18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.922

19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.883

20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.850 21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.819

22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.792

23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.768

24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.745

25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.725 26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.707

27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.690

28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.674

29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.659

30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.646 40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.551

50 0.679 0.849 1.047 1.299 1.676 2.009 2.403 2.678 3.496

100 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.390

t-Distribution Table

Two-tailed test

One-tailed test

α

t

α/2 α/2

t -t

01:830:200:01-05 Fall 2014


The One-Sample t-Test: Example

• Research Hypothesis H1: µDr.M ≠ µAVG

• Null Hypothesis H0: µDr.M = µAVG

• We sample 5 (i.e., n=5) students from Dr. M’s class,

administer the test and find that their average score is 75, with

a sample standard deviation of 7.0. Do we accept or reject

the null hypothesis?

– Assume a two-tailed test, with α = 0.05

70.0

?

01:830:200:01-05 Fall 2014


The One-Sample t-Test: Steps

1. Use t distribution table to find critical t-value(s) representing

rejection region (denoted by tcrit or tα)

2. Compute t-statistic

– For data in which I give you raw scores, you will have to compute the

sample mean and sample standard deviation

3. Make a decision: does the t-statistic for your sample fall into

the rejection region?

01:830:200:01-05 Fall 2014



0.25 0.2 0.15 0.1 0.05 0.025 0.01 0.005 0.0005


df 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 0.001

1 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 636.619

2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 31.599

3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 12.924

4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 8.610

5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 6.869

6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.959

7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 5.408

8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 5.041

9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.781

10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.587

11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.437

12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 4.318

13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 4.221

14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 4.140

15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 4.073

16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 4.015

17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.965

18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.922

19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.883

20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.850 21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.819

22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.792

23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.768

24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.745

25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.725 26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.707

27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.690

28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.674

29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.659

30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.646 40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.551

50 0.679 0.849 1.047 1.299 1.676 2.009 2.403 2.678 3.496

100 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.390


Two-tailed test

One-tailed test

α

t

α/2 α/2

t -t

01:830:200:01-05 Fall 2014



0.25 0.2 0.15 0.1 0.05 0.025 0.01 0.005 0.0005


df 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 0.001

1 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 636.619

2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 31.599

3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 12.924

4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 8.610

5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 6.869

6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.959

7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 5.408

8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 5.041

9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.781

10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.587

11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.437

12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 4.318

13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 4.221

14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 4.140

15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 4.073

16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 4.015

17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.965

18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.922

19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.883

20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.850 21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.819

22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.792

23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.768

24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.745

25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.725 26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.707

27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.690

28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.674

29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.659

30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.646 40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.551

50 0.679 0.849 1.047 1.299 1.676 2.009 2.403 2.678 3.496

100 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.390


Two-tailed test

One-tailed test

α

t

α/2 α/2

t -t

01:830:200:01-05 Fall 2014


Compute t-Statistic

.05

70.0

7.0

2.776

5

75.0

s

n

M

t

M

dfs

t

n

01:830:200:01-05 Fall 2014


Compute t-Statistic

.05

70.0

7.0

2.776

5

75.0

s

n

M

t

75 701

7

5

54 1.60

3.13

Mdf

s

n

t n

t

t

01:830:200:01-05 Fall 2014


µ from H0

0 t

critical region

(extreme 5%)

reject H0 middle 95%

retain H0

t(4), α= 0.05, 2-tailed test

-2.78 2.78

M

t = 1.60 retain H0:

no difference

01:830:200:01-05 Fall 2014


The One-Sample t-Test: A Full Example

Moon illusion example: how large must moon be at zenith to appear equivalent in size to moon at horizon?

• x = sizezenith /sizehorizon

• Null Hypothesis H0: µ = ?

• Research Hypothesis H1: µ ≠ ?

• x = {1.73,1.06,2.03, 1.40, 0.95,1.13,1.41,1.73,1.63,1.56}

• Do we accept or reject the null hypothesis? – Assume a two-tailed test, with α = 0.05

01:830:200:01-05 Fall 2014


The One-Sample t-Test: A Full Example

Moon illusion example: how large must moon be at zenith to appear equivalent in size to moon at horizon?

• x = sizezenith /sizehorizon

• Null Hypothesis H0: µ = 1

• Research Hypothesis H1: µ ≠ 1

• x = {1.73,1.06,2.03, 1.40, 0.95,1.13,1.41,1.73,1.63,1.56}


01:830:200:01-05 Fall 2014



0.25 0.2 0.15 0.1 0.05 0.025 0.01 0.005 0.0005


df 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 0.001

1 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 636.619

2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 31.599

3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 12.924

4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 8.610

5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 6.869

6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.959

7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 5.408

8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 5.041

9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.781

10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.587

11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.437

12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 4.318

13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 4.221

14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 4.140

15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 4.073

16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 4.015

17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.965

18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.922

19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.883

20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.850 21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.819

22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.792

23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.768

24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.745

25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.725 26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.707

27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.690

28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.674

29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.659

30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.646 40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.551

50 0.679 0.849 1.047 1.299 1.676 2.009 2.403 2.678 3.496

100 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.390


Two-tailed test

One-tailed test

α

t

α/2 α/2

t -t

01:830:200:01-05 Fall 2014



0.25 0.2 0.15 0.1 0.05 0.025 0.01 0.005 0.0005


df 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 0.001

1 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 636.619

2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 31.599

3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 12.924

4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 8.610

5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 6.869

6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.959

7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 5.408

8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 5.041

9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.781

10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.587

11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.437

12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 4.318

13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 4.221

14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 4.140

15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 4.073

16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 4.015

17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.965

18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.922

19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.883

20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.850 21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.819

22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.792

23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.768

24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.745

25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.725 26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.707

27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.690

28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.674

29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.659

30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.646 40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.551

50 0.679 0.849 1.047 1.299 1.676 2.009 2.403 2.678 3.496

100 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.390


Two-tailed test

One-tailed test

α

t

α/2 α/2

t -t

01:830:200:01-05 Fall 2014


X X2

1.73 2.99

1.06 1.12

2.03 4.12

1.40 1.96

0.95 0.90

1.13 1.28

1.41 1.99

1.73 2.99

1.63 2.66

1.56 2.43

sum 14.63 22.45

i

x

Mn

2

2x

S xn

S

1s

SS

n

1. Compute sample mean and SD 2. Use these values to compute t-statistic

M

dfs

t

n

01:830:200:01-05 Fall 2014


X X2

1.73 2.99

1.06 1.12

2.03 4.12

1.40 1.96

0.95 0.90

1.13 1.28

1.41 1.99

1.73 2.99

1.63 2.66

1.56 2.43

sum 14.63 22.45

14.63

101.463i

x

Mn

2

22 14.63

22.45 1.04610

xxSS

n

1.046

1 90.116 0.341

SS

ns


1.463 11

0.341

10

0.4639 4.29

0.108

Mdf

s

t n

t

t

n

(Remember, µ=1 under H0)

0.05 2.262t

04.29 2.262; reject H

01:830:200:01-05 Fall 2014


Hypothesis Testing with the t-Statistic

• Both the sample size and the sample variance influence the

outcome of a hypothesis test.

• The sample size is inversely related to the estimated standard

error. Therefore, a large sample size increases the likelihood

of a significant test (for an existing effect).

• The sample variance, on the other hand, is directly related to

the estimated standard error. Therefore, a large variance

decreases the likelihood of a significant test (for an existing

effect).

01:830:200:01-05 Fall 2014


• The related-samples t-test allows researchers to evaluate the

mean difference between two treatment conditions using the

data from a single sample.

– This test can also be called the repeated-measures t-test, the matched-

samples t-test, or the paired-samples t-test

• In a repeated-measures design, a single group of individuals

is obtained and each individual is measured in both of the

treatment conditions being compared.

• Thus, the data consist of two scores for each individual.

t-Tests for Two Related Samples: Repeated Measures

01:830:200:01-05 Fall 2014


t-Tests for Two Related Samples: Matched Subjects

• The related-samples t test can also be used for a similar

design, called a matched-subjects design, in which each

individual in one treatment is matched one-to-one with a

corresponding individual in the second treatment.

• The matching is accomplished by selecting pairs of subjects

so that the two subjects in each pair have identical (or nearly

identical) scores on the variable that is being used for

matching.

01:830:200:01-05 Fall 2014


t-Statistic for Two Related Samples

• The repeated-measures t statistic allows researchers to test a

hypothesis about the population mean difference between two

treatment conditions using sample data from a repeated-

measures research study.

• In this situation it is possible to compute a difference score for

each individual:

difference score = D = x2 – x1

where x1 is the person’s score in the first treatment and x2 is the

score in the second treatment.

01:830:200:01-05 Fall 2014


Two Related Samples: Example

X1 X2 D = X2 –X1

83.80 95.20 11.40

83.30 94.30 11.00

86.00 91.50 5.50

82.50 91.90 9.40

86.70 100.30 13.60

79.60 76.70 -2.90

76.90 76.80 -0.10

94.20 101.60 7.40

73.40 94.90 21.50

80.50 75.20 -5.30

x1: weight before treatment

x2: weight after treatment

Family therapy for anorexic girls

01:830:200:01-05 Fall 2014



The sample of difference scores is used to test hypotheses

about the population of difference scores.

• The null hypothesis states that the population of difference scores has a

mean of zero:

• The alternative hypothesis states that there is a systematic difference

between treatments that causes the difference scores to be consistently

positive (or negative) and produces a non-zero mean difference between

the treatments:

0 2 1: 0DH

1 : 0DH

01:830:200:01-05 Fall 2014


• This should seem very familiar: the repeated-measures t statistic forms a ratio with exactly the same structure as the one-sample t statistic.

• The numerator of the t statistic measures the difference between the sample mean and the hypothesized population mean.

• The only differences are that: – the sample mean and standard deviation are computed for the difference scores D

– The population mean under H0 is always 0

– The number of degrees of freedom (df) is computed based on the difference scores


D

D D D

DM

D

M M

s s

n

t

1Ddf n

01:830:200:01-05 Fall 2014


The Repeated-Measures t-Test: Full Example

Does family therapy affect the weight gained by anorexic girls?

• x1 : weight before treatment

• x2 : weight after treatment

• D = x2 – x1

• Null Hypothesis H0: µD = 0

• Research Hypothesis H1: µD ≠ 0


01:830:200:01-05 Fall 2014



0.25 0.2 0.15 0.1 0.05 0.025 0.01 0.005 0.0005


df 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 0.001

1 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 636.619

2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 31.599

3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 12.924

4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 8.610

5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 6.869

6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.959

7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 5.408

8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 5.041

9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.781

10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.587

11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.437

12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 4.318

13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 4.221

14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 4.140

15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 4.073

16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 4.015

17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.965

18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.922

19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.883

20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.850 21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.819

22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.792

23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.768

24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.745

25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.725 26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.707

27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.690

28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.674

29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.659

30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.646 40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.551

50 0.679 0.849 1.047 1.299 1.676 2.009 2.403 2.678 3.496

100 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.390


Two-tailed test

One-tailed test

α

t

α/2 α/2

t -t

01:830:200:01-05 Fall 2014


71.50

17.15

0

iD

D

Mn

2

22 71.50

1108.05 596.8210

DSS D

n

569.826

16 8.1. 1

93 4

SS

ns


7.15 01

8.14

10

7.159 2.78

2.57

D

D

D

Mdf

n

n

t

ts

t

(Remember, µD=0 under H0) X1 X2 D = X2 –X1 D2

83.80 95.20 11.40 129.96

83.30 94.30 11.00 121.00

86.00 91.50 5.50 30.25

82.50 91.90 9.40 88.36

86.70 100.30 13.60 184.96

79.60 76.70 -2.90 8.41

76.90 76.80 -0.10 0.01

94.20 101.60 7.40 54.76

73.40 94.90 21.50 462.25

80.50 75.20 -5.30 28.09

Sum 71.50 1108.05

02.78 2.262; reject H

0.05 2.262t

01:830:200:01-05 Fall 2014


Measuring Effect Size for Mean Differences

• Because the significance of a treatment effect is determined

partially by the size of the effect and partially by the size of the

sample, you cannot assume that a significant effect is also a

large effect.

• Therefore, a measure of effect size is usually computed along

with the hypothesis test.

• Cohen’s d measures the size of the treatment effect in terms

of the standard deviation.

1 0d

Cohen’s d:

01:830:200:01-05 Fall 2014


Measuring Effect Size for Mean Differences

• Of course we usually do not have all of the population

parameters. Therefore, we usually compute an estimate of the

effect size.

• For z-tests:

• For one-sample t-tests:

• For related-samples t-tests:

0ˆ Md

0d̂s

M

0ˆ D D

D

M

sd

01:830:200:01-05 Fall 2014


Measuring Effect Size for the t-Statistic

ˆ Md

s

Cohen’s d:

For the moon illusion example:

1.0

0.341

10

1.463

s

n

M

01:830:200:01-05 Fall 2014


Measuring Effect Size for the t-Statistic

1.463 1ˆ 1.360.341

Md

s

Cohen’s d:

For the moon illusion example:

1.0

0.341

10

1.463

s

n

M

Documents

z-Tests and t-Tests for One Sample - Rutgers Universitymmm431/quant_methods... · t-Tests for One Sample & Two Related Samples ... – For large df, the t distribution will be nearly