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Hypothesis Tests for Means. The context “Statistical significance” Hypothesis tests and confidence intervals The steps Hypothesis Test statistic Distribution Alpha, and the rejection region Result p-Values One-sided vs. two-sided tests Hypothesis tests for proportions.

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hypothesis tests for means
Hypothesis Tests for Means
  • The context
  • “Statistical significance”
  • Hypothesis tests and confidence intervals
  • The steps
  • Hypothesis
  • Test statistic
  • Distribution
  • Alpha, and the rejection region
  • Result
  • p-Values
  • One-sided vs. two-sided tests
  • Hypothesis tests for proportions
the context
The context
  • PARAMETERS
  •  = population mean (unknown)
  •  = population SD (might be known)
  • STATISTICS
  • n = sample size
  • x = sample mean
  • s = sample SD (using n-1)
  • ALSO
  • 0 = conjectured value of 
statistical significance
Statistical significance
  • We’re trying to decide whether  is equal to 0.
  • As usual we use x as an estimate of . Usually x is at least a little different from 0. But could the difference be due to random variation?
  • IF YES – then we DO NOT REJECT the hypothesis that  is really equal to 0. We say that x is notsignificantly different from 0.
  • IF NO – then we REJECT the hypothesis that  = 0. We say that x IS significantly different from 0.
hypothesis tests are just confidence intervals
Hypothesis tests are just confidence intervals
  • If we only cared about hypothesis tests for means, we could make this a lot simpler.
  • Just construct a confidence interval for ,
  • based on n, x, s (or ) and your favorite confidence level C.
  • If 0 is outside the confidence interval, then we reject the hypothesis that  = 0. The significance level is  = 1 – C.
  • That’s all there is to it. So why all the complex ritual of a hypothesis test?
  • Because there are other hypothesis tests, for other hypotheses (difference of two means, for example). For those tests, we need the ritual.
hypothesis test for
Hypothesis Test for 
  • Cookbook using rejection regions
  • 1. Choose hypotheses – H0 and HA.
  • 2. Define a test statistic.
  • 3. Predict the distribution of the test statistic,
  • assuming that H0 is true.
  • 4. Choose C and . Pick a rejection region.
  • 5. Look at the observed value of the test statistic.
  • Is it in the rejection region? If so, reject H0.
hypothesis test for1
Hypothesis Test for 
  • Cookbook using rejection regions
  • 1. Choose hypotheses – H0 and HA.
  • 2. Define a test statistic.
  • 3. Predict the distribution of the test statistic,
  • assuming that H0 is true.
  • 4. Choose C and . Pick a rejection region.
  • 5. Look at the observed value of the test statistic.
  • Is it in the rejection region? If so, reject H0.
choose hypotheses
Choose hypotheses
  • Two-sided test:
  • H0:  = 0 HA: 0
  • One-sided tests:
  • H0:  = 0 HA:  > 0
  • or
  • H0:  = 0 HA:  < 0
  • Working rule: Always use two-sided tests.
hypothesis test for2
Hypothesis Test for 
  • Cookbook using rejection regions
  • 1. Choose hypotheses – H0 and HA.
  • 2. Define a test statistic.
  • 3. Predict the distribution of the test statistic,
  • assuming that H0 is true.
  • 4. Choose C and . Pick a rejection region.
  • 5. Look at the observed value of the test statistic.
  • Is it in the rejection region? If so, reject H0.
define a test statistic
Define a test statistic
  • Choose
  • or
  • Do you know  ? Maybe it comes with the null hypothesis. If so, use it.
hypothesis test for3
Hypothesis Test for 
  • Cookbook using rejection regions
  • 1. Choose hypotheses – H0 and HA.
  • 2. Define a test statistic.
  • 3. Predict the distribution of the test statistic,
  • assuming that H0 is true.
  • 4. Choose C and . Pick a rejection region.
  • 5. Look at the observed value of the test statistic.
  • Is it in the rejection region? If so, reject H0.
distribution of the test statistic
Distribution of the test statistic
  • ASSUME H0 IS TRUE.
  • Then (if you know ) z has a STANDARD NORMAL distribution.
  • Or (if you’re using s) t has a “t” distribution with
  • n-1 degrees of freedom.
hypothesis test for4
Hypothesis Test for 
  • Cookbook using rejection regions
  • 1. Choose hypotheses – H0 and HA.
  • 2. Define a test statistic.
  • 3. Predict the distribution of the test statistic,
  • assuming that H0 is true.
  • 4. Choose C and . Pick a rejection region.
  • 5. Look at the observed value of the test statistic.
  • Is it in the rejection region? If so, reject H0.
standard normal case

Rejection region consists of two parts, each with probability /2.

z*/2

- z*/2

(Standard normal case)
  • The rejection region is a range (or double-range) of values of the test statistic that are
  • (a) UNLIKELY if H0 is true
  • (b) roughly consistent with the alternative HA.
  • The rejection region should have probability  (given H0).
  • Two-sided case:
predicting the distribution
Predicting the distribution
  • If you’re using t, just use t-critical values.
  • For the one-sided case:

Rejection region probability , all in one tail.

z*

chance of a type i error
Chance of a Type I error
  • Note:
  • IF H0 is actually true, then there is still a probability of  that you will reject the null hypothesis.

z*/2

- z*/2

chance of a type i error1
Chance of a Type I error
  • There are two possible bad results:
  • TYPE I ERROR (“act of commission”) – reject H0, when H0 is actually true.
  • The probability of a Type I error is 
  • (given that H0 is true)
  • TYPE II ERROR (“act of omission”) – don’t reject H0, when H0 is actually false.
  • The probability of a Type II error depends
  • on the actual value of 
hypothesis test for5
Hypothesis Test for 
  • Cookbook using rejection regions
  • 1. Choose hypotheses – H0 and HA.
  • 2. Define a test statistic.
  • 3. Predict the distribution of the test statistic,
  • assuming that H0 is true.
  • 4. Choose C and . Pick a rejection region.
  • 5. Look at the observed value of the test statistic.
  • Is it in the rejection region? If so, reject H0.
tradeoff
Tradeoff
  • High  (say, 10%) then you have a good chance of having a statistically significant result, but it won’t impress anyone.
  • MORE TYPE I ERRORS
  • Low  (say, 1%) then your significant results are more convincing, but you’ll have fewer of them.
  • MORE TYPE II ERRORS
  • Is there a way to avoid choosing  in advance?
determine p value
Determine p-value
  • The “p-value” is the answer to this question:
  • What fraction of x ‘s are more extreme than the one you actually obtained?
  • If HA: 0 this means, what fraction are further from zero than the value you obtained?
  • If HA:  > 0 this means, what fraction are more than the value you obtained?
  • If HA:  < 0 this means, what fraction are less than the value you obtained?
determine p value1

tail: 0.0107

z=2.30

Determine p-value
  • Example:
  • Do a test of H0:  = 0 vs. HA:   0 .
  • Get test statistic z = 2.30.
  • What’s the p-value?
  • Probability of seeing 2.30 OR MORE: 0.0107
  • Probability of seeing 2.30 OR MORE EXTREME: 0.0214
  • p-value for 2-sided test: 0.0214
determine p value2
Determine p-value
  • Keep it simple?
  • p-value =
  • (for 1-sided test with z) = 1 - NORMSDIST ( |z| )
  • (for 2-sided test with z) = 2 × (1-NORMSDIST(|z|))
  • (for 1-sided test with t) = TDIST ( |t|, n-1, 1 )
  • (for 2-sided test with t) = TDIST ( |t|, n-1, 2 )

df

number of tails

determine p value3
Determine p-value
  • The p-value is the border between ’s for which
  • we reject H0 and ’s for which we do not
  • reject H0.
  • REJECTION REGION VERSION: Pick , and the rejection region, in advance.
  • In this story, the p-value is an afterthought.
  • p-VALUE FIRST VERSION: Find the p-value first. Then if anyone has a favorite , you can…
  • Reject H0 if p < 
  • Do not reject if p > .
example 1969 draft lottery
Example: 1969 Draft Lottery
  • Null hypothesis (informally): The numbers for the second half of the year were drawn randomly from the population 1, 2, …, 366.
  • (Note: The mean of these numbers is 183.5, and
  • their standard deviation is 105.6547. )
  • Null hypothesis (formally): H0 :  = 183.5
  • (and this is one of those cases where  = 105.6547 comes with the null hypothesis)
  • Alternative: HA :   183.5
example 1969 draft lottery1
Example: 1969 Draft Lottery
  • H0 :  = 183.5 HA :   183.5
  • 0 = 183.5
  •  = 105.6547
  • Experiment: n = 184, x = _________
  • Test statistic:
  • p-value:
  • Conclusion: REJECT H0 (even at 1% significance level)

160.92

= - 2.898

0.00375

hypothesis tests for proportions
Hypothesis tests for proportions
  • PARAMETER
  • p = population proportion
  • STATISTICS
  • n = sample size
  • k = number of “hits”
  • p = k / n = sample proportion
hypothesis tests for proportions1
Hypothesis tests for proportions
  • Test statistic:
  • (Minor subtlety: The distribution of the test statistic is based on H0, so we use p0 in the formula for SE. This is different from what we do in confidence intervals, but not by much.)
another example
Another example
  • Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant?
another example1
Another example
  • Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant?
  • Choose:
  • H0: p = 0.50 HA: p  0.50
another example2
Another example
  • Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant?
  • Choose:
  • H0: p = 0.50 HA: p  0.50
  • Conditions? OK.
another example3
Another example
  • Suppose we have flipped 10000 coins, and obtained 5100 heads. Is this result statistically significant?
  • Choose:
  • H0: p = 0.50 HA: p  0.50
  • Conditions? OK.
  • Distribution of p^, given H0:
  • Normal, mean 0.50, SD=0.005
another example4
Another example
  • Our value of p^ is 0.51. That’s 2.0 SD’s above the mean.
  • What fraction of p^ values would be further from zero than 0.51 ?
another example5
Another example
  • Our value of p^ is 0.51. That’s 2.0 SD’s above the mean.
  • What fraction of p^ values would be further from zero than 0.51 ?
  • ABOUT 4.5%, counting both tails. So, P-value is 0.045.
result of test
Result of test
  • Is a P-value of 0.045 good enough to reject H0?
result of test1
Result of test
  • Is a P-value of 0.045 good enough to reject H0?
  • If we choose  = 0.05, then yes. But that’s a very mild test for such an extraordinary claim.
result of test2
Result of test
  • Is a P-value of 0.045 good enough to reject H0?
  • If we choose  = 0.05, then yes. But that’s a very mild test for such an extraordinary claim.
  • If we pick  = 0.05, then 5% of all our experiments will end in rejecting H0, even though H0 is true every time.
result of test3
Result of test
  • Is a P-value of 0.045 good enough to reject H0?
  • If we choose  = 0.05, then yes. But that’s a very mild test for such an extraordinary claim.
  • If we pick  = 0.05, then 5% of all our experiments will end in rejecting H0, even though H0 is true every time.
  • So we should choose a lower value of . In this case, our result isn’t really “statistically significant.”
result of test4
Result of test
  • Is a P-value of 0.045 good enough to reject H0?
  • If we choose  = 0.05, then yes. But that’s a very mild test for such an extraordinary claim.
  • If we pick  = 0.05, then 5% of all our experiments will end in rejecting H0, even though H0 is true every time.
  • So we should choose a lower value of . In this case, our result isn’t really “statistically significant.”
  • We need a bigger sample!