Page Title

PSY 305Module 3

Introduction to Hypothesis TestingZ-tests

Five steps in hypothesis testing

• State the research and null hypothesis• Determine characteristics of comparison distribution

Five steps in hypothesis testing

• Determine the cut-off score at which null will be rejected

• Calculate statistic• Make decision

Normal Distribution

• Properties – we know everything about this distribution!– Mean = 0; Standard Deviation = 1– Area under the curve always equals 1.0, and we

know all proportions under the curve

Normal Distribution

• Bell-shaped• Symmetrical

– The upper half is a mirror image of the lower half• Values of the mean, median, and mode are the same

Normal Distribution

• Each point along the x-axis corresponds with something called a z-score.

• We can make our scores (or observations) map onto this normal curve by transforming them into z-scores

Normal Distribution

Z scores

• IQ scores– Let’s say you have an IQ score of 132– Is this good or bad?– Reference group

Z scores

• A z-score indicates how many standard deviations an observation is above or below the mean.– Also called a standard score

Z Tests

• A z-score simply compares one score to the population.• A z-test actually compares a sample mean to the

population.

Calculating Z Tests

– z = the z-score you’re calculating– X = sample mean– µ = mean– σX

X

Xz

Standard error of the mean

The standard deviation of the sampling distribution of means is called the standard error of the mean. The formula for the true standard error of the mean is

NX

X

Let’s practice

• Let’s practice using our IQ example…– Mean IQ = 132– µ = 100, σ = 16– n = 10

• What is the z obtained?– 6.324

Your turn

• The sample mean is 95• The population mean is 90 with a standard deviation

of 10.• n = 8• What is the z obtained?

Answer

• 95-90 = 5• Standard error of mean

– 10/√8 = 3.536• 5/3.536 = 1.41

• Z obtained = 1.41

Evaluating the tail of the distribution

• Decision Rules– Assess the null hypothesis– Can directly test the probability of chance events– Cannot test the probability of the alternative

hypothesis

Decision rules

• If the obtained probability is equal to or less than a critical value, we reject the null.

• The critical value is called the alpha (α) level.• Indirectly accept the alternative hypothesis.

– Reject Ho, say results are significant

Decision rules

• If the obtained probability |≤| α, reject H0

• If the obtained probability > α, fail to reject H0 or retain H0.

• In psychology, we often use α = .05 or α = .01

Decision rules

• So, if we set α = .05, we are willing to limit the chance of rejecting the null when it is true to 5 times out of 100.

• Decrease our chances of making Type 1 error.

Correct Decisions, Type I, and Type II errors

• When making a decision, four possible outcomes• Correct Decision (Two Types)

– You said there was no effect and you were right.– You said there was an effect and you were right.

Type I and Type II errors

• Type I Error– You said there was a significant effect when there

really wasn’t.– You rejected the null hypothesis when you should

have retained it.

Type I and Type II errors

• Type II Error– you said there was no effect when there really

was.– You retained the null hypothesis when you should

have rejected it.

Evaluating Tale of the Distribution

• We test the tail of the distribution beginning with the obtained results

• If nondirectional- evaluate both tails.• If directional- evaluate only the tail that is in the

direction of alternative hypothesis.

One and Two Tailed Tests

• Must decide if test is one or two tailed before setting alpha level.

• Always use a two-tailed test unless we plan to retain the null hypo when results are extreme in the direction opposite to the predicted direction

Two tailed probability

• Outcomes we evaluate occur under both tails of the distribution

• Set .05 as our alpha but we have to divide it between the two tails.

• So, with 5% significance, are actually looking at 2.5% at each tail

Two tailed probability

• Our cut-off at the 5% level is -1.96 and +1.96• Our cut-off at the 1% level is -2.58 and +2.58

• These will always be used as the cut-off z-scores so be sure you learn these values!!

Two tailed probability

.025.025

One tailed probability

• All outcomes are under one tail of the distribution• So, set alpha at .05

• For 5% chance, the cut-off is +1.64 or -1.64• For 1% chance, the cut-off is +2.33 or -2.33

Critical Values

• Where do these cut-off values come from?– Our z-tables in the back of the book– Look at column C to find percentage of scores

beyond z– This tells us which z-score we need to use for our

cut-off

Previous Examples

• Recall example where • Mean IQ = 132

– µ = 100, σ = 16– n = 10

• z obtained = 6.324• Is this z significant at p < .05 for a two- tailed test?

Previous Examples

• z obtained = 6.324• z critical for p < .05, two tailed = +/-1.96• 6.325 > 1.96• Yes, the sample’s IQ is different from the population.

Previous Examples

• Recall example where • Your sample mean = 95• The population mean is 90 with a standard deviation of

10.• n = 8• Z obtained = 1.41• Is this z significant?

Previous Examples

• z obtained = 1.41• z critical for p < .05, two tailed = +/-1.96• 1.41 < 1.96• No, the sample is not significantly different from the

population

Note

• Draw your normal distribution.• Mark the critical region and cut-off values.• Determine if the z is significant.

Statistical significance

• Does not tell us if effect is important– Effect size tells us this

• Does not mean the effect will replicate• Does not mean effect will generalize to other populations

Statistical significance

• It tells you that difference is large enough that it would not occur by chance more than some probability– We usually set p < .05– A difference that large will occur 5% of the time