Today’s Agenda
- Two Means
- Two Proportions
Difference of means
\[\bar{X_1} - \bar{X_2} \sim N \left(
\mu_1-\mu_2, \sqrt{\frac{(\sigma_1)^2}{n_1} + \frac{(\sigma_2)^2}{n_2}}
\right) \]
- A popcorn company uses two machines to add butter to their microwave
popcorn bags. Both machines are supposed to add \(5\)g of butter to each bag. One machine is
newer and has a standard deviation of \(1\)g of butter. The older machine is less
consistent and dispenses butter with a standard deviation of \(1.2\)g. A quality control employee at the
company wants to test that the machines are dispensing the same amount
of butter on average. They decide to do a simple random sample of \(20\) bags from each machine and obtain
sample mean \(5.1\) and standard
deviation \(.9\) from the new machine
and mean \(4.6\) and standard deviation
\(1.1\) from the old machine. Does this
provide statistically significant evidence that the two machines are
dispensing a different average amount of butter?
When the population standard deviation is unknown for both groups, we
use the sample standard deviation and the \(t\)-distribution with degrees of freedom
equal to the smaller of \(n_1-1\) and
\(n_2-1\).
- A scientific experiment measured change in blood pressure due to a
medication in a control and treatment group. In their measurements
negative data indicates a decrease in blood pressure. The control group
had an average decrease of \(-1.4\) and
the treatment group had an average decrease of \(-4\). With \(9\) people in each group and sample
standard deviations \(5.2\) and \(2.4\) in the control and treatment
respectively, does this data provide statistically significant evidence
of the effectiveness of the medication?
\(H_0: \mu_T = \mu_C\) or \(\mu_T - \mu_C = 0\)
\(H_A: \mu_T < \mu_C\) or \(\mu_T-\mu_C <0\)
- It is thought that middle school age boys and girls spend an equal
time on average watching tv. A study is done for \(25\) randomly selected children. The study
had \(16\) boys and \(9\) girls. The \(16\) boys watched tv for an average of
\(3.22\) hours per day with a sample
standard deviation of \(1\). The \(9\) girls watched an average of two hours
of television per day with a sample standard deviation of \(.866\). Does the study suggest a
statistically significant difference in the two population means using a
significance level of \(.05\)?