Statistical Inference: Hypothesis Test

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Statistical InferenceHypothesis Test
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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?

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Review Confidence Intervals

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

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CI for small n t Distribution
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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

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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.

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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.

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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

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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

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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

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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

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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

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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

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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

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One vs. Two-tailed Test

• Test statistics (Z or t) are different!

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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..

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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)

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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
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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.

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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

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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.

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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

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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).

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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

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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

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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.

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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

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One-sample Tests
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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

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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

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One-sample Test of Mean in SPSS
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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)

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How Did SPSS Do It?

• Table p0.3156

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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)

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Hypothesis the average poverty rate is not 18
H0 m 18 H1 m ? 18

• Whats the conclusion?