Title: Statistical Inference: Hypothesis Test
1Statistical InferenceHypothesis Test
2Review 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?
3Review Confidence Intervals
- General formula for Confidence Interval
- If n is small, use t-distribution
4CI for small n t Distribution
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6Todays 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
7Hypothesis 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.
8Example
- 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.
9Elements 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
10Hypothesis 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
11Hypothesis 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
12Hypothesis 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
13Hypothesis 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
14Types 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
15Hypothesis 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
16One vs. Two-tailed Test
- Test statistics (Z or t) are different!
17One 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..
18One 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)
19One-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
20One-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.
21One-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
22Small 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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24Small 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
25Small 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).
26One 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
27Test 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
28One-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.
29Sample 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
30One-sample Tests
31Summary
- 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
32One-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
33One-sample Test of Mean in SPSS
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35SPSS Output
- T0.485lt1.96, fail to reject H0, cannot accept H1
- P0.630 gt 0.05, fail to reject H0 (compare Sig.
with a)
36How Did SPSS Do It?
37What 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)
38Hypothesis the average poverty rate is not 18
H0 m 18 H1 m ? 18