10.2 Two Population Means with Known Standard Deviations
recommended:
31-45, 93-98
Even though it is not common to know the population standard deviations without
knowing the population means, we will use this case as another example of using
a \(z\)-test.
We can use a \(z\)-test to compare population means and perform tests on
population means after we state the test statistic for the distribution of
the difference in sample means \(\bar{X}_{1}-\bar{X}_{2}\). We will use the two
given population standard deviations \(\sigma_{1}\) and \(\sigma_{2}\) to estimate
the population standard deviation of \(\bar{X}_{1}-\bar{X}_{2}\).
Given two samples from two independent normally distributed populations.
sample means: \(\bar{x}_{1}\), \(\bar{x}_{2}\)
population standard deviations: \(\sigma_{1}\), \(\sigma_{2}\)
sample sizes: \(n_{1}\), \(n_{2}\)
We define
the standard error (estimated standard deviation):
\(\displaystyle{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{1}}}}\)
the test statistic:
\(z=\displaystyle{\frac{(\bar{x}_{1}-\bar{x}_{2})-(\mu_{1}-\mu_{2})}{\displaystyle{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{1}}}}}}\)
10.3 Comparing Two Independent Population Proportions
recommended:
46-62, 99-114
We can use a \(z\)-test to compare and perform tests on two independent
population proportions after we state the test statistic for the distribution
of the difference in proportions \(P'_{1}-P'_{2}\). We will use the two sample
population proportions, \(p_{1}\) and \(p_{2}\), to construct a pooled
proportion, \(p_{c}\), for the distribution of the difference in proportions
\(P'_{1}-P'_{2}\).
Given two samples from independent normally distributed populations.
sample proportions: \(p_{1}\), \(p_{2}\)
sample sizes: \(n_{1}\), \(n_{2}\)
the number of successes and the number of failures for both samples
must be at least \(5\), that is,
\(n_{1}p_{1}, n_{1}q_{1}, n_{2}p_{2}, \mbox{and } n_{2}q_{2} \ge 5\)
the sample sizes cannot be more than \(5\%\) of the individual populations
from which they are taken
We define
the null hypothesis: \(H_{0}: p_{1}=p_{2}\)
the distribution for the difference in proportions:
\(\displaystyle{P'_{1}-P'_{2} \sim N\left(0,\sqrt{p_{c}(1-p_{c})\left(\frac{1}{n_{1}}+\frac{1}{n_{2}}\right)}\right)}\)
the test statistic: \(z=\displaystyle{\frac{(p'_{1}-p'_{2})-(p_{1}-p_{2})}{\displaystyle{\sqrt{p_{c}(1-p_{c})\left(\frac{1}{n_{1}}+\frac{1}{n_{2}}\right)}}}}\)
10.4 Matched or Paired Samples
recommended:
63-77, 115-123
We can use a \(t\)-test to compare means and perform tests matched pairs, or
paired samples
the sample sizes are often small
two samples are drawn from the same pair of individuals or objects
the differences are calculated for each matched pair and the differences
form a sample from the distribution, \(\bar{X}_{d}\), of the sample
means of the differences
we need to assume that either \(\bar{X}_{d}\) is a normal distribution,
or that the number of differences is sufficiently large that
\(\bar{X}_{d}\) is approximately normal
Given two matched samples
mean of the sample of differences: \(\bar{x}_{d}\)
standard deviation of the sample of differences: \(s_{d}\)
mean of the population of differences: \(\mu_{d}\)
sample size: \(n\)
the \(t\)-distribution \(t_{n-1}\) with \(n-1\) degrees of freedom
We define
the test statistic: \(t=\displaystyle{\frac{\bar{x}_{d}-\mu_{d}}{s_{d}/\sqrt{n}}}\)