Anthony Greene 1

Simple Hypothesis TestingDetecting Statistical Differences In The Simplest Case:

and are both known

I The Logic of Hypothesis Testing:

The Null Hypothesis

II The Tail Region, Critical Values: α

III Type I and Type II Error

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The Fundamental Idea1. Apply a treatment to a sample

2. Measure the sample mean (this means using a sampling distribution) after the treatment and compare it to the original mean

3. Remembering differences always exist due to chance, figure out the odds that your experimental difference is due to chance.

4. If its too unlikely that chance was the reason for the difference, conclude that you have an effect

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Null and Alternative Hypotheses

Null hypothesis: A hypothesis to be tested. We use the

symbol H0 to represent the null hypothesis.

Alternative hypothesis: A hypothesis to be considered as

an alternate to the null hypothesis. We use the symbol Ha to

represent the alternative hypothesis.

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The Distribution of Sample Means As The Basis for Hypothesis TestingThe set of potential samples is divided into those that are likely to be obtained and those that are

very unlikely if the null hypothesis is true.

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The Logic of the Hypothesis Test

1. We start with knowledge about the distribution given no effect (e.g., known parameters or a control group) and the data for a particular experimental treatment

2. Begin with the assumption that there is no experimental effect: this is the null hypothesis

3. Compute the probability of the observed data given the null hypothesis

4. If this probability is less than (usually 0.05) then reject the null hypothesis and accept the alternative hypothesis

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The Logic of the Hypothesis TestThe critical region (unlikely outcomes) for = .05.

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Avoiding Confusion About zzcrit vs. zobs

zcrit

zobs

21.2

z

Mz

M

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Air Puff to Eyeblink Latency (ms)

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95% of all samples of 25 eyeblinks have mean within 1.96 standard deviations of

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Probability that the sample mean of 450 ms is a chance difference from the null-hypothesis mean of 454 ms

z = -2.56

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Using More Extreme Critical Values The locations of the critical region boundaries for three different levels of significance: = .05,

= .01, and = .001.

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Test Statistic, Rejection Region, Nonrejection Region, Critical ValuesTest statistic: The statistic used as a basis for decidingwhether the null hypothesis should be rejected.

Rejection region: The set of values for the test statistic thatleads to rejection of the null hypothesis.

Nonrejection region: The set of values for the test statisticthat leads to nonrejection of the null hypothesis.

Critical values: The values of the test statistic that separatethe rejection and nonrejection regions.

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Rejection regions for two-tailed, left-tailed, and right-tailed tests

While one-tailed tests are mathematically justified, they are rarely used in the experimental literature

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Graphical display of rejection regions for two-tailed, left-tailed, and right-tailed tests

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α for 1 and 2-tailed tests

α/2 α α

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α for 1 and 2-tailed tests for α = 0.05

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Correct and incorrect decisions for a hypothesis test

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Correct and incorrect decisions for a hypothesis test

+

=

+

=

1.00 1.00

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Type I and Type II Errors

Type I error: Rejecting the null hypothesis when it is in fact true.

Type II error: Not rejecting the null hypothesis when it is in fact false.

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Significance Level

The probability of making a Type I error, that is, of rejecting a true null hypothesis, is called the significance level, , of a hypothesis test.

That is, given the null hypothesis, if the liklihood of the observed data is small, (less than ) we reject the null hypothesis. However, by rejecting it, there is still an (e.g., 0.05) probability that rejecting the null hypothesis was the incorrect decision.

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Relation Between Type I and Type II Error Probabilities

For a fixed sample size, the smaller we specify the significance level, , (i.e., lower probability of type I error) the larger will be the probability, , of not rejecting a false null hypothesis.

Another way to say this is that the lower we set the significance, the harder it is to detect a true experimental effect.

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Possible Conclusions for a Hypothesis Test

• If the null hypothesis is rejected, we conclude that the alternative hypothesis is true.

• If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alternative hypothesis.

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Critical Values, α = P(type I error)

Suppose a hypothesis test is to be performed at a specified significance level, . Then the critical value(s) must be chosen so that if the null hypothesis is true, the probability is equal to that the test statistic will fall in the rejection region.

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Some important values of z

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Power

The power of a hypothesis test is the probability of not making a Type II error, that is, the probability of rejecting a false null hypothesis. We have

Power = 1 – P(Type II error) = 1 –

The power of a hypothesis test is between 0 and 1 and measures the ability of the hypothesis test to detect a false null hypothesis. If the power is near 0, the hypothesis test is not very good at detecting a false null hypothesis; if the power is near 1, the hypothesis test is extremely good at detecting a false null hypothesis.

For a fixed significance level, increasing the sample size increases the power.

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Basic Idea

H0: Parent distribution for your sample

if there IS NO effect

μ0 = 40

Zcrit = 1.64

M = 42

Fail to reject

M = 48

Reject null hypothesis

Conclude Effect

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Basic Idea

H0: Parent distribution for your sample

if there IS NO effect

Ha: Parent distribution for your sample

if there IS an effect

μ0 = 40 μa = ?

Zcrit = 1.64

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Basic Idea

H0: Parent distribution for your sample

if there IS NO effect

μ0 = 40

Zcrit = 1.64

α

1 - α

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Basic Idea

Ha: Parent distribution for your sample

if there IS an effect

μa = ?

Zcrit = 1.64

β 1 - β

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Basic Idea

H0: Parent distribution for your sample

if there IS NO effect

Ha: Parent distribution for your sample

if there IS an effect

μ0 = 40 μa = ?

Zcrit = 1.28

We can move zcrit

Zcrit = 2.58

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Basic Idea

H0: Parent distribution for your sample

if there IS NO effect

Ha: Parent distribution for your sample

if there IS an effect

μ0 = 40 μa = ?

Zcrit = 1.64

We can increase n

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The one-sample z-test for a population mean (Slide 1 of 3)

Step 1 The null hypothesis is H0: = 0 and the alternative hypothesis is one of the following:

Ha: 0 Ha: < 0 Ha: > 0 (Two Tailed) (Left Tailed) (Right Tailed)

Step 2 Decide on the significance level, Step 3 The critical values are

±z/2 -z +z

(Two Tailed) (Left Tailed) (Right Tailed)

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The one-sample z-test for a population mean (Slide 2 of 3)

α/2 α α

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The one-sample z-test for a population mean (Slide 3 of 3)

Step 4 Compute the value of the test statistic

Step 5 If the value of the test statistic falls in the rejection region, reject H0, otherwise do not reject H0.

n

Mz

/0

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Synopsis

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P-Value

To obtain the P-value of a hypothesis test, we compute, assuming the null hypothesis is true, the probability of observing a value of the test statistic as extreme or more extreme than that observed. By “extreme” we mean “far from what we would expect to observe if the null hypothesis were true.” We use the letter P to denote the P-value. The P-value is also referred to as the observed significance level or the probability value.

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P-value for a z-test

•Two-tailed test: The P-value is the probability of observing a value of the test statistic z at least as large in magnitude as the value actually observed, which is the area under the standard normal curve that lies outside the interval from –|z0| to |z0|,

•Left-tailed test: The P-value is the probability of observing a value of the test statistic z as small as or smaller than the value actually observed, which is the area under the standard normal curve that lies to the left of z0,

•Right-tailed test: The P-value is the probability of observing a value of the test statistic z as large as or larger than the value actually observed, which is the area under the standard normal curve that lies to the right of z0,

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Guidelines for using the P-value to assess the evidence against the null hypothesis