Stat 322 – HW 4

Due Friday, Feb. 9

Always include relevant computer output and details of any calculations conducted by hand.

1) problem 3 (p. 420), using Table A.9 or Minitab (Calc > Probability Distributions > F) to estimate/determine the p-value. Include a well-labeled sketch of this F distribution with the p-value shaded.

2) problem 7 (p. 421) -- just fill in the table

3) Use Minitab to answer problem 6 (p. 421) and problem 15 (p. 427). You should also include boxplots and discussion of technical conditions, and summarize the results of your statistical analysis in English.

4) Use Minitab to answer problem 18 (p. 428)

5) Porosity of metal, an important determinant of strength and other properties, can be measured by looking at cross sections of the metal under a microscope. The pore (volid) is dark, the metal is light and the boundary between these two phases is often clearly delineated. If a grid is lad in the microscopic field, the number of intersections between the grid lines and the pore boundaries is proportional to the length of that boundary per unit area (and hence is proportions to the pore surface area per unit volume of metal). Such counts were made for samples of antimony, Linde copper, and electrolytic copper and are in porecounts.mtw.

(a) Check the technical conditions for ANOVA with these data. Which condition appears most seriously violated? Include graphical support for your conclusions.

(b) Apply a square root transformation to the data and see if these data are appropriate for ANOVA? Conduct an ANOVA for these transformed data and state your conclusions.

(c) If discussed in class, why is the square root transformation suggested for these data?

Writing Assignment 4 – due Monday, Feb. 12

We briefly discussed that the ANOVA procedure is preferable to running several two-sample comparisons in order to control the Type I Error Rate. Similarly, when conducting multiple comparisons, we need to worry about the overall type I error rate.

(a) Find an expression for P(at least one Type I error in k tests) if each test has a = P(Type I error) = .05.

(b) Suppose k tests were performed, using your expression in (a) to find the overall Type I error rate. How does it compare to 5%?

A very basic type of adjustment is the “Bonferroni Adjustment” where each test uses a/k where k is the number of pairwise comparisons.

(c) What error rate does this suggest for the individual tests when k = 10?

(d) Suggest a downside to using such a small Type I error rate with the individual tests.

(e) Show that this adjustment (using a/k) ensures P(at least one Type I error in k tests) a.

To do this, you will probably want to make use of the complement rule and Boole’s Inequality: P(È A_i) SP(A_i) or a related mathematical relationship between (1-x)ⁿ and nx when x is small.