Null Hypothesis Signi cance Testing p-values, signi cance ...
Null Hypothesis Significance Testing p-values, significance level, power, t-tests
18.05 Spring 2018
NO CLASS Monday April 16 (Patriots' Day) Problem set due Wednesday April 18
Watch class web site for RESCHEDULED OFFICE HOURS
Understand this figure
f (x|H0)
reject H0
don't reject H0
x reject H0
x = test statistic f (x|H0) = pdf of null distribution = green curve Rejection region is a portion of the x-axis.
Significance = probability over the rejection region = red area.
April 14, 2018 2 / 26
Simple and composite hypotheses
Simple hypothesis: the sampling distribution is fully specified. Usually the parameter of interest has a specific value.
Composite hypotheses: the sampling distribution is not fully specified. Usually the parameter of interest has a range of values.
Example. A coin has probability of heads. Toss it 30 times and let x be the number of heads. (i) H: = 0.4 is simple. x binomial(30, 0.4). (ii) H: > 0.4 is composite. x binomial(30, ) depends on which value of is chosen.
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Extreme data and p-values
Hypotheses: H0, HA. Test statistic: value: x, computed from data. Null distribution: f (x|H0) (assumes null hypothesis is true) Sides: HA determines if the rejection region is one or two-sided. Rejection region/Significance: P(x in rejection region | H0) = .
The p-value is a tool to check if the test statistic is in the rejection region. It is also a measure of the evidence for rejecting H0. p-value: P(data at least as extreme as x | H0)
"Data at least as extreme" is defined by the sidedness of the rejection region.
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Extreme data and p-values
Example. Suppose we have the right-sided rejection region shown below. Also suppose we see data with test statistic x = 4.2. Should we reject H0?
f (x|H0)
c 4.2
x
don't reject H0
reject H0
answer: The test statistic is in the rejection region, so reject H0.
Alternatively: blue area < red area Significance: = P(x in rejection region | H0) = red area. p-value: p = P(data at least as extreme as x | H0) = blue area. Since p < we reject H0.
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