hypothesis testing and comparison of two populations

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  • 7/30/2019 Hypothesis Testing and Comparison of Two Populations

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    Hypothesis Testing and

    Comparison of Two Populations

    Dr. Burton

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    If the heights of male teenagers are normally distributed with a mean of 60

    inches and standard deviation of 10, And the sample size was 25, what

    percentage of boys heights in inches would be:

    Between 57 and 63

    Lass than 58

    61 or larger

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    7.2a

    60

    0

    Height

    Z57 63

    %

    Z=x -s / n

    57 - 60

    10 / 25

    63 - 60

    10 / 25

    Z= -1.5 = .4332

    Z= 1.5 = .4332

    .8664 = 86.8%

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    7.2b

    60

    0Height

    Z

    58

    -1.0

    %

    Z= x -s / n

    58 - 6010 / 25

    Z = -1.0 = .5000 - .3413

    .1587 = 16%

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    7.2c

    60

    0 0.5Height

    Z

    61

    %

    Z= x -s / n

    61 - 6010 / 25

    Z = 0.50

    - .1915 = .3085 = 30.9%= 0.50

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    Hypothesis Testing

    Hypothesis: A statement of belief

    Null Hypothesis, H0: there is no differencebetween the population mean and the

    hypothesized value 0.Alternative Hypothesis, H

    a: reject the null

    hypothesis and accept that there is a

    difference between the population mean and the hypothesized value 0.

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    Probabilities of Type I and Type II

    errorsH0 True H0 False

    Accept H0

    Reject H0Type IError

    Type II

    ErrorCorrect

    results

    Correct

    results

    Truth

    Test

    result

    1 -

    1 -

    H0 True = statistically insignificant

    H0 False = statistically significant

    Accept H0 = statistically insignificant

    Reject H0 = statistically significant

    Differences

    a b

    c d

    http://en.wikipedia.org/wiki/False_positive

    http://en.wikipedia.org/wiki/False_positivehttp://en.wikipedia.org/wiki/False_positive
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    -3 -2 -1 0 1 2 3

    SE

    Probability Distribution

    for a two-tailed test

    SE

    Magnitude of (XE XC)

    1.96 SE

    XE < XC XE > XC

    = 0.05

    0.0250.025

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    -3 -2 -1 0 1 2 3

    SE

    Probability Distribution

    for a one-tailed test

    SE

    Magnitude of (XE XC)

    1.645 SE

    XE < XC XE > XC

    = 0.05

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    Box 10 - 5

    t =

    A

    Distance between the means

    Variation around the means

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    Box 10 - 5

    t =

    A

    B

    Distance between the means

    Variation around the means

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    Box 10 - 5

    t =

    A

    B

    C

    Distance between the means

    Variation around the means

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    t-Tests

    Students t -test is used if:

    two samples come from two different

    groups.

    e.g. A group of students and a group of

    professors

    Paired t -test is used if:

    two samples from the sample group.

    e.g. a pre and post test on the same

    group of subjects.

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    One-Tailed vs. Two Tailed

    Tests The Key Question: Am I interested in the

    deviation from the mean of the sample from the

    mean of the population in one or both

    directions. If you want to determine whether one mean is

    significantly from the other, perform a two-tailed

    test.

    If you want to determine whether one mean is

    significantly larger, or significantly smaller,

    perform a one-tailed test.

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    t-Test(Two Tailed)

    Independent Sample means

    xA - xB - 0

    t =

    Sp [ ( 1/NA ) + ( 1/NB) ]

    d f = NA + NB - 2

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    Independent Sample Means

    Sample A (A Mean)2

    26 34.3424 14.90

    18 4.58

    17 9.86

    18 4.58

    20 .02

    18 4.58Mean = 20.14

    A2 = 2913

    N = 7

    (A Mean)2 = 72.86

    Var = 12.14

    s = 3.48

    Sample B (B Mean)2

    38 113.8526 1.77

    24 11.09

    24 11.09

    30 7.13

    22 28.41

    Mean = 27.33

    B2 = 4656

    N = 6

    (B Mean)2 = 173.34

    Var = 34.67

    s = 5.89

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    Standard error of the difference

    between the means (SED)

    SED ofE - C =

    s A2Estimate of the s B

    2

    NA NB+

    SED ofxE - xC =

    A 2 B 2NA NB

    + Population

    Sample

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    Pooled estimate of the SED

    (SEDp)

    1Estimate of the 1

    NA NB+SEDp ofxA - xB = Sp

    s2(nA-1) + s2 (nB1)Sp =

    nA + n B - 2

    12.14 (6) + 34.67(5)Sp =

    7 + 6 - 2

    = 22.38 = 4.73

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    t-Test

    (Two Tailed)

    d f = NE

    + NC

    - 2 = 11

    xA - xB - 0

    t =

    Sp [ ( 1/NA ) + ( 1/NB) ]

    20.14- 27.33 - 0=

    4.73 ( 1/7 ) + ( 1/6)

    = -2.73

    Critical Value95%

    = 2.201

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    One-tailed and two-tailed t-tests

    A two-tailed test is generally

    recommended because differences in

    either direction need to be known.

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    Pairedt-test

    t paired = t p =d - 0

    Standard error of d

    = -------------d - 0

    S d2

    N

    df= N - 1

    d = D/N

    d2 = D2( D)2/ N

    S d2

    = d2

    / N - 1

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    Pre/post attitude assessment

    Student Before After Difference D squared

    1 25 28 3 9

    2 23 19 -4 16

    3 30 34 4 16

    4 7 10 3 95 3 6 3 9

    6 22 26 4 16

    7 12 13 1 1

    8 30 47 17 289

    9 5 16 11 121

    10 14 9 -5 25

    Total 171 208 D = 37 D2 = 511

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    Pre/post attitude

    assessmentStudent Before After Difference D squared

    Total 171 208 37 511

    t paired = t p =d - 0

    Standard error of d

    = -------------d - 0

    S d2

    N

    d = D/N

    N = 10

    d2= D2( D)2/ N

    S d2 = d2/ N - 1

    = 37/10 = 3.7

    = 511 - 1369/10 = 374.1

    = 374.1 / 101 = 41.5667

    = 3.7 / 2.0387

    = 1.815

    = 3.7 / 41.5667 / 10

    = 3.7 / 4. 15667

    df= N1 = 9

    0.05 > 1.833

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    Probabilities of Type I and Type II errors

    H0 True H0 False

    Accept H0

    Reject H0Type IError

    Type II

    ErrorCorrect

    results

    Correctresults

    Truth

    Test

    result

    1 -

    1 -

    H0 True = statistically insignificant

    H0 False = statistically significant

    Accept H0 = statistically insignificant

    Reject H0 = statistically significant

    Differences

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    Standard 2 X 2 table

    a = subjects with both the risk factor and the disease

    b = subjects with the risk factor but not the disease

    c = subjects with the disease but not the risk factor

    d = subjects with neither the risk factor nor the diseasea + b = all subjects with the risk factor

    c + d = all subjects without the risk factor

    a + c = all subjects with the disease

    b + d = all subjects without the disease

    a + b + c + d = all study subjects

    Present Absent

    Present

    Absent

    Disease status

    Risk

    Factor

    Status

    a b

    c d

    a + b

    c + d

    a + c b + d a+b+c+dTotal

    Total

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    Standard 2 X 2 table

    Sensitivity = a/a+c

    Specificity = d/b+d

    Present Absent

    Present

    Absent

    Disease status

    Risk

    Factor

    Status

    a b

    c d

    a + b

    c + d

    a + c b + d a+b+c+dTotal

    Total

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    Diabetic Screening Program

    Sensitivity = a/a+c = 100 X 5/6 = 83.3% (16.7% false neg.)

    Specificity = d/b+d = 100 X 81/94 = 86.2%(13.8% false pos.)

    Diabetic Nondiabetic

    >125mg/100ml