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CH12: Comparing two means
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Terms in this set (26)
paired design
both treatments are applied to every sampled unit; more powerful than unpaired designs; reduces effects of variation among sampling units
two sample design
each treatment group is composed of an independent random sample of units
paired measurements
converted to a single measurement by taking the difference between them; 20 individuals grouped into 10 pairs, a10 measurements
sample mean difference
d(bar)
confidence interval for the mean of a paired difference
same way as the CI for any other mean
d(bar) - t(a(2),df) SE < u(d) < d(bar) + t(a(2),df)
SE(d) = s(d)/√n
n = # of pairs
paired t-test
used to test a null hypothesis that the mean difference of paired measurements equals a specified value; tests differences when both treatments have been applied to every sampling unit and the data are therefore paired
Ho and Ha of a paired t-test
Ho: the mean change in antibody production after testosterone implants was zero (u(o) = 0)
Ha: the mean change in antibody production after testosterone implants was not zero (u(o) ≠ 0; two tailed)
calculating test statistic
t = d(bar) - u(d)o / SE(dbar)
Assumption of a paired t-test
1. sampling units are randomly sampled form the population
2. paired differences have a normal distribution
two-sample design
the two treatments are applied to separate, independent sampling units
difference between the sample means
Y(bar)1 - Y(bar)2
standard error of Y1-Y2
SE (Y1-Y2) = √ (s(p)^2 ((1/n1)+(1/n2))
pooled sample variance
s(p)^2; the average of the variances of the samples weighted by their degrees of freedom
s(p)^2 = (df1xs1^2 + df2s2^2) / (df1 + df2)
df1 = n1-1
df2 = n2-1
two sample t-test test statistic
t = (Y1-Y2)/(SE)
df = df1 + df2 = n1+n2-2
confidence interval for the difference between two populations means
(Y1-Y2) - t(a(2),df) SE < (u1-u2) < (Y1-Y2) + t(a(2),df) SE
Ho and Ha for two sample t-tests
Ho: u1 = u2
Ha: u1 ≠ u2
Assumptions for a two sample t-test
1. each of the two samples is a random sample from its populations
2. the numerical variable is normally distributed in each population
3. the standard deviation (and variance) of the numerical variable is the same in both populations
if there is more than a threefold difference in stand deviations, or if the sample sizes of the two groups are very different, two-sample t-test should not be used
Welch's t-test
compares the means of two groups and can be used even when the variances of the two groups are not equal
indirect comparison
comparisons between two groups should always be made directly, not indirectly by comparing both to the same null hypothesized value
Comparing variances
test the difference between populations in the variability of measurements:
1. F-test
2. Levene's test
F-test
evaluates whether two population variances are equal
Ho: o1^2 = o2^2
Ha: o1^2 ≠ o2^2
o1^1 = variance of population 1
o2^2 = variance of population 2
Test statistic = F
Test statistic F
F is calculated form the ratio of the two sample variances
F = s1^2/s2^2
If Ho were true, F should be near one, deviating from it only by chance
df for F-test
df = (n1-1, n2-1)
1st #: degrees of freedom of the numerator of the F-ratio
2nd #: degrees of freedom of the denominator
of the F-ratio
F-test assumption
the variable is normally distributed in both populations; highly sensitive (i.e. not robust); will often falsely reject the null hypothesis (i.e. high power) of equal variance if the distribution in one of the populations in not normal
Levene's test
used when F-test assumptions are violated (i.e. both distributions are normal)
Levene's test assumption
1. the frequency distribution of measurements is roughly symmetrical within all groups; performs better than F-test when assumption is violated; can be applied to more than two groups
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