Is this quarter fair?


• How could you determine this?

• You assume that flipping the coin a large number of times would result in heads half the time (i.e., it has a .50 probability)


• Say you flip it 100 times

• 52 times it is a head

• Not exactly 50, but its close– probably due to random error


• What if you got 65 heads?

• 70?

• 95?

• At what point is the discrepancy from the expected becoming too great to attribute to chance?

Example• You give 100 random students a questionnaire designed

to measure attitudes toward living in dormitories

• Scores range from 1 to 7 – (1 = unfavorable; 4 = neutral; 7 = favorable)

• You wonder if the mean score of the population is different then the population mean at Haverford (which is 4)

Hypothesis

• Alternative hypothesis– H1: sample = 4

– In other words, the population mean will be different than 4

Hypothesis


• Null hypothesis– H0: sample = 4

– In other words, the population mean will not be different than 4

Results

• N = 100

• X = 4.51

• s = 1.94

• Notice, your sample mean is consistent with H1, but you must determine if this difference is simply due to chance

Results

• N = 100

• X = 4.51

• s = 1.94

• To determine if this difference is due to chance you must calculate an observed t value

Observed t-value

tobs = (X - ) / Sx

Observed t-value

tobs = (X - ) / Sx

This will test if the null hypothesis H0: sample = 4 is true

The bigger the tobs the more likely that H1: sample = 4 is true

Observed t-value

tobs = (X - ) / Sx

Sx = S / N

Observed t-value

tobs = (X - ) / .194

.194 = 1.94/ 100

Observed t-value

tobs = (4.51 – 4.0) / .194

Observed t-value

2.63 = (4.51 – 4.0) / .194

t distribution

t distribution

tobs = 2.63

t distribution

tobs = 2.63

Next, must determine if this t value happened due to chance or if represent a real difference in means. Usually, we want to be 95% certain.

t critical

• To find out how big the tobs must be to be significantly different than 0 you find a tcrit value.

• Calculate df = N - 1

• Table D – First Column are df

– Look at an alpha of .05 with two-tails

t distribution

tobs = 2.63

t distribution

tobs = 2.63

tcrit = 1.98tcrit = -1.98

t distribution

tobs = 2.63


If tobs fall in critical area reject the null hypothesis

Reject H0: sample = 4

t distribution

tobs = 2.63


If tobs does not fall in critical area do not reject the null hypothesis

Do not reject H0: sample = 4

Decision

• Since tobs falls in the critical region we reject Ho and accept H1

• It is statistically significant, the average favorability of Villanova dorms is significantly different than the favorability of Haverford dorms.

• p < .05

p < .05

• We usually test for significance at the “.05 level”

• This means that the results we got in the previous example would only happen 5 times out of 100 if the true population mean was really 4

Hypothesis Testing

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Hypothesis Testing

• Basic Logic• 1) Want to test a hypothesis (called the research or

alternative hypothesis).– “Second born children are smarter than everyone else

(Mean IQ of everyone else = 100”)

• 2) Set up the null hypothesis that your sample was drawn from the general population– “Your sample of second born children come from a

population with a mean of 100”

Hypothesis Testing

• Basic Logic• 3) Collect a random sample

– You collect a sample of second born children and find their mean IQ is 145

• 4) Calculate the probability of your sample mean occurring given the null hypothesis– What is the probability of getting a sample mean of 145

if they were from a population mean of 100

Hypothesis Testing

• Basic Logic• 5) On the basis of that probably you make a

decision to either reject of fail to reject the null hypothesis.

– If it is very unlikely (p < .05) to get a mean of 145 if the population mean was 100 you would reject the null

– Second born children are SIGNIFICANTLY smarter than the general population

Example• You wonder if the average IQ score of

Villanova students is significantly different (at alpha = .05)than the average IQ of the population (which is 100). To determine this you collect a sample of 54 students.

• N = 54

• X = 130

• s = 18.4

The Steps

• Try to always follow these steps!

Step 1: Write out Hypotheses


• Null hypothesis– H0: sample = 100

Step 2: Calculate the Critical t

• N = 54

• df = 53 = .05

• tcrit = 2.0

Step 3: Draw Critical Region


Step 4: Calculate t observed

tobs = (X - ) / Sx


tobs = (X - ) / Sx

Sx = S / N


tobs = (X - ) / Sx

2.5 = 18.4 / 54


tobs = (X - ) / Sx

12 = (130 - 100) / 2.52.5 = 18.4 / 54

Step 5: See if tobs falls in the critical region




tobs = 12

Step 6: Decision

• If tobs falls in the critical region:

– Reject H0, and accept H1

• If tobs does not fall in the critical region:

– Fail to reject H0

Step 7: Put answer into words

• We reject H0 and accept H1.

• The average IQ of Villanova students statistically different ( = .05) than the average IQ of the population.

Practice• You wonder if the average agreeableness score of

Villanova students is significantly different (at alpha = .05) than the average agreeableness of the population (which is 3.8). You collect data from 31 people.

• N = 31

• X = 3.92

• s = 1.52

Step 1: Write out Hypotheses

• Alternative hypothesis– H1: sample = 3.8

• Null hypothesis– H0: sample = 3.8

Step 2: Calculate the Critical t

• N = 31

• df = 30 = .05

• tcrit = 2.042

Step 3: Draw Critical Region

tcrit = 2.042tcrit = -2.042


tobs = (X - ) / Sx


tobs = (X - ) / Sx

Sx = S / N


tobs = (X - ) / Sx

.27 = 1.52 / 31


tobs = (X - ) / Sx

.44 = (3.92 - 3.80) / .27


tcrit = 2.042tcrit = -2.042


tcrit = 2.042tcrit = -2.042

tobs = .44

Step 6: Decision

• If tobs falls in the critical region:

– Reject H0, and accept H1

• If tobs does not fall in the critical region:

– Fail to reject H0

Step 7: Put answer into words

• We fail to reject H0

• The average agreeableness score of Villanova students is not statistically different ( = .05) than the average agreeableness score of the population.

Is this quarter fair?

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