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.

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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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Extreme data and p-values

Example. Now suppose x = 2.1 as shown. Should we reject H0?

f (x|H0)

2.1 c

x

don't reject H0

reject H0

answer: Test statistic not in the rejection region: don't 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 don't reject H0.

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Critical values

The boundaries of the rejection region are called critical values. Critical values are labeled by the probability to their right. They are complementary to quantiles: cp = q1-p. Example: for a standard normal c0.025 = 1.96 and c0.975 = -1.96. In R, for a standard normal c0.025 = qnorm(0.975).

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Two-sided p-values

These are trickier: what does `at least as extreme' mean in this case? The p-value is a tool for deciding if the test statistic is in the region. If the null distribution is symmetric around zero then

p = 2min(left tail prob. of x, right tail prob. of -x)

f (x|H0)

c1-/2

x c/2

x

reject H0

don't reject H0

reject H0

x is outside the rejection region, so p > : do not reject H0

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