3 questions of intro to probability statistics 1

3 questions of Intro to Probability Statistics


General instructions: If using R to answer questions, please include all code and output in your problem solution.

  1. Problem 12.5.3 in Rice, “Mathematical statistics and data analysis”, page 506.
  2. Let A1,A2,…,An denote n events. Prove Bonferroni inequality:n
    P ( ∪ ni = 1 A i ) ≤ ∑ P ( A i )i=1[Hint: Use induction. Show that the inequality above holds for 2 events; assume that it holds for n − 1 events, and show that it holds for n events. You can use Venn diagrams if helpful.]
  3. Consider the temperature reduction example that was presented in lecture. The data are as follows:Observation ( j)
    1 2 3 4 5 6 ȳi

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Placebo 98.3 Aspirin 96.0 Anacin 97.2 Tylenol 95.1

98.7 98.1 99.9 96.2 98.6 98.30 96.6 95.3 98.1 94.6 96.6 96.20 97.4 96.9 97.8 94.7 97.0 96.83 95.7 94.4 97.2 93.7 95.7 95.30 96.5 95.4 98.2 94.5 96.5 96.20

Bufferin 96.1
Suppose we wish to test whether there is any difference in mean temperature

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among patients receiving active treatments (i.e. not placebo).

(a) State an appropriate null and alternative hypothesis for addressing this question.

  1. (b) Carry out a test of the hypothesis stated in (a) by hand/calculator. In doing this, you should construct and fill in the ANOVA table.
    Report the p-value, state your decision and interpret the results of your hypothesis test in a way that would be understandable to a non- statistician.
  2. (c) Confirm your calculations in (b) by performing this hypothesis test using the aov function in R.
  3. (d) What assumptions are being made when conducting the above test?