Today’s Blueprint
Last Class:

• Correlation (A Recap)
• Comparing Two Means
• The Logic Behind Means Difference Tests

• Comparing the Means of 2 Dependent Populations
• Comparing the Means of 2 Independent Populations
• Difference Tests
• Confidence Intervals (T-Interval)

• They are tests of significance
• They determine whether there is a significant difference between the mean values of two groups

• Two Types:
1. T-Tests for Independent Means (AKA: The Pooled T-Tests)
2. T-Tests for Dependent Means (AKA: The Paired T-Tests)

1. T-Tests for Independent Means = Tests the difference between the means of 2 unrelated groups

• Who is in the second sample doesn’t depend on who is in the first sample

2. T-Tests for Dependent Means = Tests the difference between the means of 2 related groups

• Who is in the second sample depends on who is in the first sample

• Allow researchers to compare groups:
• Determine whether one group is meaningfully different from another group
• In an experiment, researchers can determine whether differences between groups are due to the manipulation

• Data in each group follow a normal distribution
• For pooled test, the variances for each group are equal (no between group variation)
• The samples are independent

• But what if samples are not independent?!
• If samples are not independent, then we say that they are dependent, correlated, or paired
• Ways That Pairing Can Occur:
• When subjects in one group are “matched” with a similar subject in the second group.
• When subjects serve as their own control by receiving both of two different treatments.
• When, in “before and after” studies, the same subjects are measured twice.

• When doing an experiment
• When doing a comparative observational study

• Experiments = Manipulating one variable to observe its effect on another variable
• Example of a simple experiment:
• One dependent variable (Y)
• One independent variable (X) manipulated 2 ways
• Treatment group
• Control group

• Comparative Observation = A research study in which two or more groups are compared with respect to some measurement or response
• The groups, determined by their natural characteristics, are merely “observed”

Research Question: Do male and female drivers differ with respect to their fastest reported driving speed?

• Data:
• Sample of n1 = 17 males report average of 102.1 mph
• Sample of n2 = 21 females report average of 85.7 mph
• The difference in the sample means is 102.06 - 85.71 = 16.35 mph

Research Question (in statistical notation)

• Let  = the average fastest speed of all male students
• And let  = the average fastest speed of all female students
• Then we want to know whether  
• This is equivalent to wanting to know whether 

• In general, we can always compare two means by seeing how their difference compares to 0:

• Null hypothesis: 
• This is equivalent to saying that 
• Alternative hypothesis: 
• This is equivalent to saying that 

• Assume null hypothesis is true
• That is, assume 
• Or, equivalently, assume 

• P-value = “How likely is it that our sample means would differ by as much as 16.35 m.p.h. if the difference in population means really is 0?”
• We conventionally choose a significance level of 0.05

• The P-value is determined by standardizing, that is, by calculating the two-sample test statistic:

• …and comparing the value of the test statistic to the appropriate sampling distribution

• If variances of the measurements of the two groups are not equal, estimate the standard error of the difference as:

• Then the sampling distribution is an approximate t distribution with a complicated formula for d.f.

 
• If variances of the measurements of the two groups are equal...
• Note: Assume variances are equal only if neither sample standard deviation is more than twice that of the other sample standard deviation.

• Then the sampling distribution is a t distribution with n1+n2-2 degrees of freedom.

• Analyze ==> Compare Means ==> Independent-Samples T Test
• Select one or more continuous, numeric test variables. A separate t test is computed for each variable
• Example: Place the dependent variable (driving speed) and put it in the “Test variable” box
• Select a dichotomous grouping variable (a categorical variable that divides cases into two groups) 
• Example: Place the independent variable (gender) and put it in the “grouping variable” box
• Select your confidence interval (default is 95%) from the “options” box
• Click Define Groups and specify the values of the grouping variable that define the two groups
• Example: The grouping variable can be a string (alphanumeric) variable or a numeric variable that uses numeric codes to represent categories (e.g., 0=Male, 1=Female)

Gender       N      Mean     StDev   SE Mean
female      21     85.71      9.39       2.0
male        17     102.1      17.1       4.1

95% CI for mu (female) - mu (male): ( -25.9,  -6.8)
T-Test mu (female) = mu (male) (vs not =): T = -3.54
                                             P = 0.0017
                                             DF = 23

Determine Significance Test:

• Our obtained value (0.0017) is smaller than this (0.001)
• This tells us that our sample result is not likely if the null hypothesis is true
• Therefore, we can reject the null hypothesis

• There is sufficient evidence, at the 0.05 level of significance, to conclude that the average reported fastest driving speed of all male college students differs from the average reported fastest driving speed of all female students

• We can be “such-and-such” confident that the difference in the population means falls in the interval:

• Where the t* multiplier depends on the confidence level and is obtained either from the appropriate t distribution.
 
• Interpreting a confidence interval for the difference in 2 means:

• Data: 

• ...The Real question is whether males and females in the same job earn different average salaries.  Better then to compare the difference in salaries in “pairs” of males and females:

What is the P-Value?

• P-value = How likely is it that a paired sample would have a difference as large as $2,000 if the true difference were 0?

Hypotheses for Paired T-test

• Does the average difference of the population,  , differ from 0?

Data analyzed as Paired T-Test

Paired T for M - F

                  N      Mean     StDev   SE Mean
M                 4      41.5      26.2      13.1
F                 4      39.5      26.1      13.1
Difference        4     2.000     0.816     0.408

95% CI for mean difference: (0.701, 3.299)
T-Test of mean difference = 0 (vs not = 0):
                  T-Value = 4.90  P-Value = 0.016

Interpret Results

• P = 0.016.  This is smaller than 0.05
• Therefore, reject null 
• Sufficient evidence to conclude that average starting salaries differ between males and females