Download - L10 2-Sample Tests

Transcript
  • 1

    Chapter 10

    Two-Sample Tests

    David Chow

    Nov 2014

  • 2

    Learning Objectives

    In this chapter, you learn how to use hypothesis testing to compare the:

    a. 1 = 2? Means of two independent populations

    b. 1 = 2? Proportions of two independent populations,

    c. Variances of two independent populations (F-test)

    Only part (c), F-test of two variances, is covered in the exam

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    Any Difference Between 1 & 2?

    Comparing Means: Want to know if 1 = 2

    The question can also be posed in the form of a one-

    tailed test such as 1 > 2

    Eg: TOFEL scores from HK and Singapore students

    Comparing Proportions: Test if 1 = 2

    Eg: Proportion of students in a relationship (yr1 vs yr4)

    Eg: Passing proportion in math (local vs int schools )

    Eg: Proportion of contact-lens users (male vs female)

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    Some Funny Findings DSME 2010, Fall 2011

    Sleeping duration

    related to societal activities, living in campus

    not related to gender, part-time, dating

    No. of FB friends

    not related to gender, majors

    related to no. of siblings

    More phone conversations

    related to smartphone users, gender

    Skipping classes 1:58 PM

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    Two-Sample Tests

    Independent Populations

    Independent

    Population Means

    1 and 2 known

    1 and 2 unknown

    Use a Z test statistic

    Use S to estimate unknown ,

    use a t test statistic

    Assumptions:

    population distributions are normal Or sample size 30 for each sample

    Samples are randomly and independently drawn Independence: sample selected from one

    population has no effect on the sample

    selected from the other population

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    F-Test of Population

    Variances

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    Testing Population Variances

    Purpose: To determine if two independent

    populations have the same variability.

    H0: 12 = 2

    2

    H1: 12 2

    2

    H0: 12 2

    2

    H1: 12 < 2

    2

    H0: 12 2

    2

    H1: 12 > 2

    2

    Two-tail test Lower-tail test Upper-tail test

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    Testing Population Variances

    2

    2

    2

    1

    S

    SF

    The test statistic is:

    = Variance of Sample 1 (usu the larger one)

    n1 - 1 = numerator degrees of freedom

    n2 - 1 = denominator degrees of freedom

    = Variance of Sample 2

    2

    1S

    2

    2S

    Assume each population distributions is normally distributed, then the ratio

    of the two sample variances follows

    the F distribution.

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    Testing Population Variances

    The critical F-value is found from the F table.

    There are two appropriate degrees of freedom:

    numerator degrees of freedom (column),

    denominator degrees of freedom (row).

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    Testing Population Variances

    0

    FL Reject H0 Do not reject H0

    H0: 12 2

    2

    H1: 12 < 2

    2

    Reject H0 if F < FL

    0

    FU Reject H0 Do not

    reject H0

    H0: 12 2

    2

    H1: 12 > 2

    2

    Reject H0 if F > FU

    Lower-tail test Upper-tail test

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    Testing Population Variances

    L2

    2

    2

    1

    U2

    2

    2

    1

    FS

    SF

    FS

    SF

    Rejection region for a two-tail test is:

    F 0 /2

    Reject H0 Do not reject H0 FU

    H0: 12 = 2

    2

    H1: 12 2

    2

    FL

    /2

    Two-tail test

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    Testing Population Variances

    To find the critical F values, FU and FL:

    1. Find FU from the F table for n1 1 numerator

    and n2 1 denominator degrees of freedom

    *U

    LF

    1F 2. Find FL using the formula:

    Where FU* is from the F table with n2 1 numerator and n1 1

    denominator degrees of freedom (i.e., switch the d.f. from FU)

    Check FU

    only if

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    Eg: Dividend Yield

    Is there a difference in the variances between the

    NYSE & NASDAQ return rates at the = 0.05 level?

    NYSE NASDAQ

    Number 21 25

    Mean 3.27 2.53

    Std dev 1.30 1.16

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    Example

    Form the hypothesis test:

    H0: 21

    22 = 0 (there is no difference between variances)

    H1: 21

    22 0 (there is a difference between variances)

    Numerator:

    n1 1 = 21 1 = 20 d.f.

    Denominator:

    n2 1 = 25 1 = 24 d.f.

    FU = F.025, 20, 24 = 2.33

    Numerator:

    n2 1 = 25 1 = 24 d.f.

    Denominator:

    n1 1 = 21 1 = 20 d.f.

    FL = 1/F.025, 24, 20 = 0.41

    FU: FL:

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    Testing Population Variances

    The test statistic is:

    256.116.1

    30.12

    2

    2

    2

    2

    1 S

    SF

    0

    /2 = .025

    FU=2.33 Reject H0 Do not

    reject H0

    FL=0.41

    /2 = .025

    Reject H0 F

    F = 1.256 is not in the

    rejection region, so we do

    not reject H0

    Conclusion: There is insufficient

    evidence of a difference in

    variances at = .05

  • Review Questions

    10.36 Find the upper critical F-value for a two-tailed test if

    a. = 0.01, n1=16, n2=21

    b. = 0.05, n1=16, n2=21

    c. = 0.10, n1=16, n2=21

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  • Review Questions

    10.46 At the = 0.05 level, is there a difference in the variances between the male & female anxiety level? The scale runs from 20 (no anxiety) to 100 (highest level of anxiety).

    Male FEMALE Number 100 72

    Mean 40.26 36.85

    Std dev 13.35 9.42

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