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Statistical Inference: Hypothesis Test

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If p alpha level, results are significant at alpha level, we reject H0, and accept H1 ... P 0.05; Not significant at 0.05, 0.1 level, fail to reject H0, and ... – PowerPoint PPT presentation

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Title: Statistical Inference: Hypothesis Test


1
Statistical InferenceHypothesis Test
2
Review The Z-distribution
  • The Z-distribution is a normal distribution with
    a mean of 0, and standard deviation of 1
  • Also called the standard normal distribution
  • It is a probability distribution that tells you
    the of cases falling within a particular number
    of S.Ds around the mean
  • How to find Z-score with a known probability, and
    probability with a known z-score?

3
Review Confidence Intervals
  • General formula for Confidence Interval
  • If n is small, use t-distribution

4
CI for small n t Distribution
5
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6
Todays topic Hypothesis Tests
  • What is hypothesis test?
  • Elements, steps, types of hypothesis test
  • Significance test for a mean
  • One vs. two-tailed test
  • Small vs. large sample
  • Significance test for proportion

7
Hypothesis Test
  • Hypothesis a statement/claim based on theory,
    insights, observation, data analysis,
  • Hypothesis testing a formal language and method
    for examining claims (true or not) using
    inferential statistics
  • The logic
  • We cannot prove if the claim is true. So, we
    will cast doubt on other claims, thus indirectly
    support our own
  • The strategy
  • We first state an opposing claim If we can
    cast sufficient doubt on it, we have to accept
    our own claim.

8
Example
  • Suppose we wish to argue that women are paid less
    than men.
  • Hypotheses
  • First the opposite women are paid NO less than
    men.
  • Next the alternative women are paid less than
    men
  • If statistical analysis shows that the first
    claim is highly improbable, we can reject it,
    thus we accept the second claim that women are
    paid less than men.

9
Elements of Hypothesis Test
  • Hypotheses
  • Null hypothesis (H0) counter to our claim,
    directly tested, no effect
  • Alternative hypothesis (H1, or Ha) the one we
    hope to claim, contradicts the null hypothesis.
  • Test statistic
  • Reflects on the probability (P-value) of H0 being
    true rather than H1
  • P-value
  • Large P-value means data are more consistent with
    H0, small P-value means the data contradict H0
  • The smaller , the better!
  • Significance level (a)
  • a pre-determined probability threshold for
    rejecting the null hypothesis.
  • 0.1, 0.05, 0.001 usually 0.05

10
Hypothesis Testing Example
  • H0 women are not paid less than men
  • Gender has no effect on wage
  • H1 Women are paid less than men
  • If evidence suggests that H0 is highly
    improbable, we reject it and we accept H1
  • So, typically we
  • Reject H0, accept H1
  • Fail to reject H0, do not find support for H1 ,
    cannot accept H1

11
Hypothesis Test Steps
  • 1. Assumption
  • Type of data (quantitative, or qualitative)
    Method of sampling (random) Sample size (gt30 or
    not)
  • 2. State the research hypothesis (H1) and null
    hypothesis (H0)
  • 3. Calculate the test statistic
  • 4. P-value associated with the test statistic
  • 5. Compare p-value with a-level
  • Alpha level typically .05, sometimes .10 or .01
  • If pltalpha level, results are significant at
    alpha level, we reject H0, and accept H1
  • If pgtalpha level, we fail to reject H0

12
Hypothesis Test Alternative Steps
  • 1. Assumption
  • 2. State H0 and H1
  • 3. Choose a a-level
  • typically .05, sometimes .10 or .01
  • 4. Look up value of test statistic corresponding
    to the a-level (called the critical value)
  • 5. Calculate the relevant test statistic
  • 6. Compare test statistic to the critical value
  • If test statistic is larger, we reject H0, accept
    H1
  • If it is smaller, we cannot reject H0

13
Hypothesis Test Alternative Steps
  • 1. Assumption
  • 2. State H0, H1
  • 3. Choose an alpha-level
  • 4. Get software to conduct relevant statistical
    test.
  • Software will compute test statistic and provide
    a probability the probability of observing a
    test statistic of a given size (P-value).
  • If pltalpha, reject H0

14
Types of Hypothesis Tests
  • Data type
  • Quantitative test for mean (e.g. age, income)
  • Qualitative test for proportion (e.g. race,
    religion)
  • Sample size
  • Large sample, use z test, calculate z-score
  • Small sample, use t test or binomial test
  • Number of samples
  • One-sample test
  • Independent two-sample test
  • Dependent/paired two-sample test (e.g. time
    series husband-wife)
  • Hypothesis one-tailed vs. two-tailed
  • Different combinations

15
Hypothesis Testing one vs. two tailed test
  • Two-tailed test A hypothesis test in which the
    a-area of interest falls in both tails of a Z or
    t distribution.
  • One-tailed test A hypothesis test in which the
    a-area of interest falls in just one tail of a Z
    or t distribution.
  • called a directional hypothesis test

16
One vs. Two-tailed Test
  • Test statistics (Z or t) are different!

17
One vs. Two-tailed Test
  • A one-tailed test H0 mgt4 H1 m lt 4
  • Entire a-area is on left, as opposed to half
    (a/2) on each side..

18
One vs. Two- Tailed Test
  • In many instances, you are more likely to reject
    the null hypothesis when utilizing a one-tailed
    test
  • Concentrating the alpha area in one tail reduces
    the critical T-value needed to reject H0
  • Implication If you have strong theoretical
    base/suspicions for a directional hypothesis,
    then use one-tailed test. It increases your
    chances of rejecting H0, thus accepting H1
    (research hypothesis)

19
One-sample Tests for Means Example
  • Age of graduate students in a class n 35, Y-bar
    23, s4.45 I hypothesize the mean age of
    graduate students at UAlbany is not 25.
  • Assumption
  • quantitative, large data, random sample
  • H0 Population mean m 25
  • H1 Population mean m ? 25 (two-tailed test)
  • If it is one-tailed test
  • H0 m lt25, H1 mgt25
  • Or
  • H0 mgt25, H1 mlt25

the equal sign usually stays in H0
20
One-sample Test for Mean (example)
  • H0 m 25 H1 m ? 25
  • Large sample, use z-test calculate the Z-score
  • P-value? 20.00390.0078
  • Conclusion
  • P lt0.01, significant at 0.01 level reject H0,
    and accept H1. The mean age for graduate students
    in this school is not 25.

21
One-sample Test for Mean
  • Alternatively
  • Choose a-level0.05
  • Critical value1.96 (two-tail)
  • Conclusion
  • 2.66gt1.96, significant at 0.05 level, reject H0,
    and accept H1.
  • Tip P the smaller, the better z the
    larger, the better

22
Small Sample Test for Mean
  • Suppose small sample n10 (nlt30), use t-test
  • Critical value for a-level0.05, df9? Two-tail
    test, but the t table is one-tail.

23
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24
Small Sample Test for Mean
  • Suppose small sample n10 (nlt30), use t-test
  • Critical value for a-level0.05, df9? Two-tail
    test, but the t table is one-tail.
  • The critical value of t 2.262
  • Conclusion
  • 1.42lt2.262, not significant at 0.05 level, fail
    to reject H0, cannot accept H1

25
Small Sample One-tail Test
  • Assumptions
  • n 10, Y-bar 23, s4.45
  • H0 m gt 25. H1 m lt 25
  • Calculate the t-score
  • Critical value for a-level0.05, df9?
  • 1.833
  • Conclusion
  • 1.41lt1.833, not significant at 0.05 level.
  • But significant at 0.1 level (critical value.
    1.383).

26
One sample test for proportions
  • Example Did you vote for G. Bush for President
    in 2004? (Yes vs. No)
  • Assumption qualitative data, categorical
  • Similar steps to test for means
  • H0 ??0 H1 ?? ?0 (two-tailed)
  • H0 ?gt?0 H1 ?lt ?0 OR H0 ?lt
    ?0, H1 ?gt ?0 (one-tailed)
  • Z-test if it is a large sample, binomial test for
    small sample

27
Test for Proportion Example
  • In a survey of 1227 individuals, 591 responded
    yes. 591/12270.482
  • Hypothesis Bush did not get 50 of votes.
  • H0 ?0.5 H1 ?? 0.5 (two-tailed)
  • Large sample, use z-test
  • P-value? 20.10030.2006 (two-tailed test)
  • Conclusion
  • Pgt0.05 Not significant at 0.05, 0.1 level, fail
    to reject H0, and cannot accept H1

28
One-tail Test for Proportion
  • Hypothesis Bush got less than 50 of votes.
  • H0 ?gt0.5 H1 ?lt 0.5
  • P-value?
  • 0.1003
  • Conclusion
  • Not significant at 0.1, 0.05 level, fail to
    reject H0, cannot accept H1.

29
Sample Size in Test for Proportion
  • Actual n depends on ?0, the sample proportion
  • If sample proportion ?0 is between 0.3 and 0.7,
    the usual rule of n 30 is good.
  • More generally, for large sample,
  • If ?00.5, ngt20 if ?00.1, ngt100.
  • The sampling distribution of ?0 is more skewed
    when ? is near 0 or 1, thus need a larger sample.
  • We will skip the test for proportions with small
    sample

30
One-sample Tests
31
Summary
  • Concepts associated with hypothesis test
  • Steps for hypothesis test
  • Types of hypothesis test
  • One-sample test
  • For mean, large vs. small sample
  • For proportion, large sample
  • One vs. two-tailed test
  • Next time
  • Two-sample test

32
One-sample Test of Mean in SPSS
  • Hypothesis the average crime rate is not 8
  • H0 m 8 H1 m ? 8
  • SPSS uses t-test for all samples regardless of
    sample size

33
One-sample Test of Mean in SPSS
34
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35
SPSS Output
  • T0.485lt1.96, fail to reject H0, cannot accept H1
  • P0.630 gt 0.05, fail to reject H0 (compare Sig.
    with a)

36
How Did SPSS Do It?
  • Table p0.3156

37
What if it is a one-tailed test?
  • T0.485lt1.645, fail to reject H0, cannot accept
    H1
  • P0.630/20.310 gt 0.05, fail to reject H0
    (compare Sig./2 with a)

38
Hypothesis the average poverty rate is not 18
H0 m 18 H1 m ? 18
  • Whats the conclusion?
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