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 H1: sample = 4 In other words, the population mean will be different than 4
Hypothesis Alternative hypothesis Null hypothesis H1: 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 tcrit = -1.98 tcrit = 1.98 tobs = 2.63
t distribution tcrit = -1.98 tcrit = 1.98 tobs = 2.63
t distribution Reject H0: sample = 4 tcrit = -1.98 tcrit = 1.98 tobs = 2.63 If tobs fall in critical area reject the null hypothesis Reject H0: sample = 4
t distribution Do not reject H0: sample = 4 tcrit = -1.98 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 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