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You can also test for differences (paired t-test)

• What if I have two sets of observations from the same group?
• Pretest/Post test
• I might want to know if they differ significantly
• Differences might be due to chance
• The null hypothesis for this test
• H₀:  m₁ - m₂ = 0
• H₁:  m₁ - m₂ < > 0
• Assume same standard deviation for both samples

Again, calculate the statistic

 
 
 
 
 

Testing and Caffeine

• Each person in a class takes two exams
• One after drinking coffee
• One after drinking decaf
• H₀: mu_coffee - mu_decaf = 0
• Mean (coffee) = 86
• Mean (decaf) = 83
• s.d. = 6.42
• 32 students
•  t(31)  =  (86 - 83)  / (6.42/32^1/2)
•   = 2.64
• p < .05

So, we now know one sample t-tests

• We can answer these questions:
• Is x bar different than zero (paired tests)?
• Is x bar different than some null hypothesis?
• We can also make confidence intervals.

What if we want to see if two samples are different?
The great questions of our time:

• Are boys smarter than girls?
• Beer before liquor or liquor before beer?
• Drug A or drug B for the rats?

• mu₁ equals mu₂

• Things get more complicated with two groups
• Must assume that the standard deviation is the same for both groups
• Actually can do other procedures
• See the book or us if interested
• Standard deviation is found by pooling

Calculating the t statistic

• Given the pooled variance, a t statistic can be found
• Degrees of freedom reflect that two means need to be known.

Here's an experiment

• Half a morning class is given coffee and the other half is given deceive for a semester.
• H₀:  mu_coffee = mu_decaf
• H₁:  mu_coffee < > mu_decaf
• Mean (coffee) = 87
• Mean (deceive) = 82
• var (coffee) = 6.5, n = 16
• var(deceive) = 7.1, n = 16
• pooled var. = 6.8
• pooled sd=2.61
• t(30)=(87-82)/(2.61*sqrt(1/16+1/16))=5.42
•  p < .05
• Reject H₀

t test is robust

• Test assumes that variances are the same
• Even if the variances are not the same, the test still works pretty well
• Test assumes data are drawn from a normally distributed population
• Even if the population is not normally distributed, the test still works pretty well.
• Of course, there are limits