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Two Sample Inference for Means

Farrokh Alemi Ph.D

Kashif Haqqi M.D.

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Summary Slide

• Review

• F test

• Test of two means– Small sample, equal variance– Small sample, unequal variance– Small dependent sample

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Review

• Frequency distributions and descriptive statistics.

• Comparing an observation to a distribution.

• Comparing two distributions.

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Objectives

• To learn how to compare two distributions.

• No need to know the formulas, focus on assumptions and interpretations.

• Be able to do the calculations using excel functions.

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Which Test Is Right?

Z tes t

K n ow n varian ce

U n eq u a l va rian ce t tes t

U n eq u a l va rian ceIn d ep en d en t sam p les

E q u a l va rian ce t tes t

E q u a l va rian ceIn d ep en d en t sam p les

P a ired t tes t

R ep eated g rou pD ep en d en t sam p les

V arian ce es tim atedfrom sm all sam p le

Z tes t

V arian ce es tiam tedfrom la rg e sam p le

C om p are twod is trib u tion s

C om p are ob serva tionto a d is trib u tion

N orm al p op u la tionR an d om S am p les

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F Test

• Used to test if two population variances are equal.• Assumes independent, random samples from

populations with normal distributions.• Test is conducted by taking the ratio of the

variances (square of standard deviations). If the two variances are equal the ratio will be one. The larger value is always on top.

• Critical test values are determined using number of observations minus one for each sample.

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Example

• Are nurses in government owned hospitals paid less than privately owned hospitals?

Private GovernmentAverage 26000 25400Standard deviation 600 450Number of observations 10 8

From Bluman A. Elementary statistics. McGraw Hill, 1998.

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Solution

• Hypothesis: variances are equal.• Alternative hypothesis: variance are

unequal.• Critical value for two tailed F test at 9 and 7

degrees of freedom is 8.51.• The F statistic is equal to 600*600/450*450

= 1.7.• The null hypothesis is not rejected.

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Which Test Is Right?

Z tes t

K n ow n varian ce

U n eq u a l va rian ce t tes t

U n eq u a l va rian ceIn d ep en d en t sam p les

E q u a l va rian ce t tes t

E q u a l va rian ceIn d ep en d en t sam p les

P a ired t tes t

R ep eated g rou pD ep en d en t sam p les

V arian ce es tim atedfrom sm all sam p le

Z tes t

V arian ce es tiam tedfrom la rg e sam p le

C om p are twod is trib u tion s

C om p are ob serva tionto a d is trib u tion

N orm al p op u la tionR an d om S am p les

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Test of Two MeansSmall Sample, Equal Variance

• Normal population.

• Independent sample observations.

• Random sample.

• Unknown variance.

• Two distributions have same variance, as per F test.

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Test of Two Means (Cont.)Small Sample, Equal Variance

• Test value is always calculated as: (observed value minus expected value) / standard deviation.

• In this case the observed value is the difference between two means.

• The expected value is zero as the two means are expected to be equal.

• What is the standard deviation of the difference?

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Standard Deviation of Difference

Equal Variance• Sd = square root {[(n1 -1)s1*s1 + (n2 -1)s2*s2)] /

n1 +n2-2)]}*square root (1/ n1 +1/ n2).• Sd is standard deviation of the difference of

means.• n1 is sample size and s1 is standard deviation

in 1st distribution.• n2 is sample size and s2 is standard deviation

in 2nd distribution.

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Test of Two Means (Cont.)Small Sample, Equal Variance

• Decide if one tail or two tailed test.

• Critical values depend on sample sizes and are calculated at n1 +n2-2 degrees of freedom.

• The hypothesis is rejected if the test value is larger than positive critical value or smaller than negative critical value.

Do this in Excel

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Which Test Is Right?

Z tes t

K n ow n varian ce

U n eq u a l va rian ce t tes t

U n eq u a l va rian ceIn d ep en d en t sam p les

E q u a l va rian ce t tes t

E q u a l va rian ceIn d ep en d en t sam p les

P a ired t tes t

R ep eated g rou pD ep en d en t sam p les

V arian ce es tim atedfrom sm all sam p le

Z tes t

V arian ce es tiam tedfrom la rg e sam p le

C om p are twod is trib u tion s

C om p are ob serva tionto a d is trib u tion

N orm al p op u la tionR an d om S am p les

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Test of Two MeansSmall Sample, Unequal Variance

• Normal population.

• Independent sample observations.

• Random sample.

• Unknown variance.

• Two distributions have different variance, as per F test.

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Test of Two Means (Cont.)Small Sample, Unequal Variance

• Test value is always calculated as: (observed value minus expected value) / standard deviation.

• In this case the observed value is the difference between two means.

• The expected value is zero as the two means are expected to be equal.

• What is the standard deviation of the difference?

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Standard Deviation of Difference

Unequal Variance• Sd = square root (s1*s1/n1 + s2*s2/n2).

• Sd is standard deviation of the difference of means.

• n1 is sample size and s1 is standard deviation in 1st distribution.

• n2 is sample size and s2 is standard deviation in 2nd distribution.

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Test of Two Means (Cont.)Small Sample, Unequal Variance

• Decide if one tail or two tailed test.

• Critical values depend on the smaller sample size minus one.

• The hypothesis is rejected if the test value is larger than positive critical value or smaller than negative critical value.

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Example

• Are nurses in government owned hospitals paid less than privately owned hospitals?

Private GovernmentAverage 26000 25400Standard deviation 600 450Number of observations 10 8

From Bluman A. Elementary statistics. McGraw Hill, 1998.

Do this in Excel

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Solution

• Hypothesis: 1 2.

• Alternative hypothesis: 1 2.

• Critical value for =0.01, one tailed test, with equal variances with 10+8-2 degrees of freedom is 2.583.

• Standard deviation of difference = 256.• Test value = 5.47.• Null hypothesis is rejected. Private hospitals do not

pay nurses less than or equal to government hospitals.

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Which Test Is Right?

Z tes t

K n ow n varian ce

U n eq u a l va rian ce t tes t

U n eq u a l va rian ceIn d ep en d en t sam p les

E q u a l va rian ce t tes t

E q u a l va rian ceIn d ep en d en t sam p les

P a ired t tes t

R ep ea ted g rou pD ep en d en t sam p les

V arian ce es tim atedfrom sm a ll sam p le

Z tes t

V arian ce es tiam tedfrom la rg e sam p le

C om p are tw od is trib u tion s

C om p are ob serva tionto a d is trib u tion

N orm a l p op u la tionR an d om S am p les

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Test of Two MeansSmall Dependent Sample

• Normal population.

• Dependent sample observations on same or matched case, before and after.

• Random selection of cases.

• Unknown variance.

• By definition, distributions before and after have same variance.

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Test of Two Means (Cont.)Small Dependent Sample

• Test value is always calculated as: (observed value minus expected value) / standard deviation.

• The observed value is the mean of paired differences.

• The expected value is zero as the mean of the paired differences is zero when the two means are the same.

• What is the standard deviation of the difference?

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Standard Deviation of Difference

Small Dependent Sample• Sd = square root [d2 – (d)2/n] /(n-1).

• Sd = standard deviation of differences.

• d = paired difference for one case.

• n = number of paired differences.

• SEd = standard error of differences.

• SEd = Sd / n.

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Test of Two Means (Cont.)Small Dependent Sample

• Decide if one tail or two tailed test.

• Critical values depend on the sample size minus one.

• The hypothesis is rejected if the test value is larger than positive critical value or smaller than negative critical value.

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Example

• Did clinician improve risk score for his patient after switching their medication (Higher scores are better scores)?

Patient12345678Before, 1210230182205262253219216After, 2219236179204270250222216

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Solution

• Hypothesis: mean 1 -2 is greater than or equal to zero.

• Alternative hypothesis mean of difference is less than zero.

• Critical value for a one tailed t-distribution at 8-1=7 degrees of freedom is –1.895.

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Solution: Compute Test Value

• Calculate sum of pair wise difference

• Calculate sum of squared pair wise differences

Patient Before After DifferenceSquare of difference

1 210 219 -9 812 230 236 -6 363 182 179 3 94 205 204 1 15 262 270 -8 646 253 250 3 97 219 222 -3 98 216 216 0 0

Total -19 209

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Solution: Computing Test Value Continued

• Compute mean as (d)/n.

• Compute standard deviation as Sd = square root [d2 – (d)2/n] /(n-1).

• Compute standard error as SEd = Sd / n.

• Computer test statistic as mean (minus expected mean of zero) divided by standard error.

Mean -2.375Standard deviation 4.84Standard error 1.711198Test statistic -1.38792

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•Hypothesis is not rejected.