Is this quarter fair Is this quarter fair

Is this quarter fair?

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)

Is this quarter fair? • Say you flip it 100 times • 52 times it is a head • Not exactly 50, but its close – probably due to random error

Is this quarter fair? • 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 – H 1: sample = 4 – In other words, the population mean will be different than 4

Hypothesis • Alternative hypothesis – H 1: sample = 4 • Null hypothesis – H 0: 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 H 1, 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 value t

Observed t-value tobs = (X - ) / Sx

Observed t-value tobs = (X - ) / Sx This will test if the null hypothesis H 0: sample = 4 is true The bigger the tobs the more likely that H 1: sample = 4 is true

Observed t-value tobs = (X - ) / Sx Sx = S / N

Observed t-value tobs = (X - ) /. 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 tcrit = -1. 98 tcrit = 1. 98 tobs = 2. 63

t distribution tcrit = -1. 98 tcrit = 1. 98 tobs = 2. 63

t distribution tcrit = -1. 98 tcrit = 1. 98 tobs = 2. 63 If tobs fall in critical area reject the null hypothesis Reject H 0: sample = 4

t distribution tcrit = -1. 98 tcrit = 1. 98 tobs = 2. 63 If tobs does not fall in critical area do not reject the null hypothesis Do not reject H 0: sample = 4

Decision • Since tobs falls in the critical region we reject Ho and accept H 1 • 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 • 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 probability 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

Test 2 • Test 1 • Mean = 90 / SD = 5. 54 • Test 2 • Mean = 85 / SD = 8. 23

Test 2 • r =. 52 • Y = 27 + (. 65)TEST 1