Introduction to Hypothesis Testing The One-Sample z Test.

Introduction to Hypothesis Testing

The One-Sample z Test

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The One-Sample z Test

• Conditions of Applicability:

– One group of subjects

– Comparing to population with known mean and variance.

• Note: this is not a common situation in Psychology!

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Example: Finish times for the 2005 Toronto Marathon (Oct 16, 2005)

• Suppose your population of interest are women who ran the marathon (slightly artificial).

• You hypothesize that women in their early twenties (20-24) are faster than the average woman who ran the marathon.

• Here the ‘treatment’ is ‘youth’.

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Null Hypothesis Testing

• Largely due to English mathematician Sir R.A. Fisher (1890-1962)

• ‘Proof by contradiction’

• Suppose the null hypothesis is true– In our example, the null hypothesis is that the finishing times for young

women are drawn from the same distribution as for the rest of the female contestants.

– Knowing the mean and standard deviation of the population, we can compute the sampling distribution of the mean for a sample of size n. This is the null hypothesis distribution.

– The mean time for our sample of young women should be plausible under this sampling distribution.

– If it is not plausible, it suggests that the null hypothesis is false.

– This lends credence to our alternate hypothesis (that young women are faster).

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How do we judge the plausibility of the null hypothesis?

• The sample mean should be plausible under the sampling distribution of the mean.

( )p X

X

X

Highly plausible

Fairly plausible

XImplausible

X

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Plausibility of the null hypothesis

• The plausibility of the null hypothesis is judged by computing the probability p of observing a sample mean that is at least as deviant from the population mean as the value we have observed.

( )p X

X

Xp

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Plausibility of the null hypothesis

• This computation is simplified by converting to z-scores.

• Under the assumption of normality, we can determine this probability from a standard normal table.

( )p z

0

1

zp

X

Xz

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Results for 2005 Toronto Marathon

420

4hr 16min 256 min

33min

n

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Results for Random Sample of Women Under 25

38

4hr 9min 249 min

n

X

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Statistical Decisions

• We now know the probability that an observation like ours could have been drawn from the general female contestant population, i.e. that our ‘treatment of youth’ had no effect.

• This probability is pretty small. Should we reject the null hypothesis? This is the process of turning a continuous probability (a real number) into a binary decision (yes or no).

• If we reject the null hypothesis, there is a chance we will be wrong. We have to decide what chance we are willing to take, i.e. the maximum p-value we will accept as grounds for rejecting the null hypothesis.

• We call this probability threshold the alpha () level. A typical value is .05.

• The level must be decided prior to the experiment.

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Actual Situation

Researcher’s Decision Null Hypothesis is True Null Hypothesis is False

Accept the Null Hypothesis

Reject the Null Hypothesis

Type I and Type II Errors

• Type I Error: the null hypothesis is true and we reject it.

• Type II Error: the null hypothesis is false and we fail to reject it.

0 0(accept | true)p H H

0 0(reject | true)p H H

0 0(accept | f alse)p H H

0 0(reject | f alse)p H H

1

1 (power)

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Type I and Type II Errors

• Which is more serious?

– Type I can be bad, as rejecting the null hypothesis (e.g., ‘This stuff really works’), may cause actions to be taken that have no value.

– Type II may not be so bad, if it is understood that the treatment may still have an effect (we fail to reject the null hypothesis, but we do not reject the alternate hypothesis).

– But Type II may be bad if it leads to inaction when action would have produced good results (e.g., a cure for cancer).

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One-Tailed vs Two-Tailed Tests

• Our marathon hypothesis was one-tailed, because we made a specific prediction about the direction of the effect (young women are faster).

• Suppose we had simply hypothesized that young women are different.

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Two-Tailed Test

( )p z

0

1

z

p

X

Xz

z

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One-Tailed vs Two-Tailed Tests

• Use a one-tailed test when you have a specific reason to believe the effect will be in a particular direction, and you do not care if the effect is in the opposite direction.

• Otherwise, use a two-tailed test.

• One-tailed tests will always result in smaller p values, and hence a greater chance of reaching significance for your directional hypothesis.

• The decision of whether to perform one-tailed or two-tailed tests must be made prior to data collection.

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Basic Procedure for Statistical Inference

1. State the hypothesis

2. Select the statistical test and significance level

3. Select the sample and collect the data

4. Find the region of rejection

5. Calculate the test statistic

6. Make the statistical decision

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Step 1. State the Hypothesis

marathon times for young women are the same

as for the general female contest

Null hyp

ant popu

othesis:

lation.

youAlte ng wrnate hy omen arepo fthesis: aster.

00 :H

0:AH

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Step 2. Select the Statistical Test and the Significance Level

• We are comparing a sample mean to a population with known mean and standard deviation z-test

• p=.05 is probably appropriate.

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Step 3. Select the Sample and Collect the Data

• Ideally, we would randomly assign the treatment to a random sample of the population (Toronto Marathon women). Is this possible?

• Instead, we randomly sample female contestants under 25.

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Step 4. Find the Region of Rejection

• The z value defining the rejection region is called the critical value for your test, and is a function of the selected α-level. For this reason, we often denote the critical value as zα

( )p z

0

1

1.65z .05

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Step 5. Calculate the Test Statistic

X

Xz

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Step 6. Make the Statistical Decision

• p<Reject null hypothesis.

• p>Fail to reject null hypothesis.

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Example: Height of Female Psychology Graduate Students

Canadian Adult Female Population:

162.10 cm

6.55 cm

Sample: Female students enrolled in PSYC 6130C 2008-09

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Assumptions Underlying One-Sample z Test

• Random sampling

• Variable is normal

– CLT: Deviations from normality ok as long as sample is large.

• Dispersion of sampled population is the same as for the comparison population

– e.g. suppose means are the same, but dispersion of sampled population is greater than dispersion of comparison population.

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Limitations of the One-Sample Test

• Strongly depends on random sampling.

• Better to have two groups of subjects: test (treatment) group and control group.

• Problem of random sampling reduces to problem of random assignment to two groups: much easier!

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Reporting your results

• Express your result in evocative English, then include the required numbers.

• Follow APA style.

• Example:

– Young female runners were not found to be significantly faster than the general female contestant population, z=-1.31, p=0.095, one-tailed.

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More on Type I and Type II Errors

1

Total number of significant results

• Consistent use of a fixed alpha-level determines the proportion of null experiments that generate significant results.

• Don’t have enough information to know how many reported results are errors, because:

– Don’t know the relative proportion of cases where H0 is true and H0 is false.

– Don’t know the power of effective experiments.

– Typically only significant results are reported (publication bias).

0 is trueH 0 is f alseH