Today’s Agenda

  • Two Means
  • Two Proportions

Difference of means

\[\bar{X_1} - \bar{X_2} \sim N \left( \mu_1-\mu_2, \sqrt{\frac{(\sigma_1)^2}{n_1} + \frac{(\sigma_2)^2}{n_2}} \right) \]

  1. 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 \(n_2-1\).

  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 \(\mu_T-\mu_C <0\)

  1. 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 \(.05\)?