One way & Two way ANOVA

1. Two aptitude tests are currently being used to screen applicants for a certain position within a company. The question arose as to whether the two tests are comparable, i.e., whether they yield the same results. Six applicants were selected at random to take both tests (in a random order). The following scores were recorded:

Applicant Test A Test B
1 85 87
2 93 94
3 98 92
4 68 73
5 76 73
6 83 85

a. What statistical test do you use in order to determine whether there is a difference in scores between the two aptitude tests?

b. Complete the table
Applicant Test A Test B Difference Rank
1 85 87
2 93 94
3 98 92
4 68 73
5 76 73
6 83 85

c. Compute the values for the test statistic. Use = 0.10.

d. State the rejection rule

e. Conclude about the differences in th tests scores.

2. A mechanical engineer at a manufacturing plant keeps a close watch on the performance and condition of the machines. The following data are the weight losses (in milligrams) of certain machine parts due to friction when used with three different lubricants.

Lubricant A Lubricant B Lubricant C
12 10 12
9 8 8
8 9 14
11 13 11
10 8 12
8 7 13
7 5 7
6 11 8

a. State the null and alternative hypotheses to test whether there is a significant difference in mean weight losses among the three lubricants.
b. How many degrees of freedom are associated with the F test statistic?
e. Set up the rejection decision rule for = 0.01.
f. What is the appropriate conclusion?

3. Ann May wishes to determine if the mean price of grocery items is the same for five supermarkets in her city. The same seven food (and brand) items were priced at the five stores.

a. Complete the partial ANOVA table below for testing the null hypothesis of no difference in the true mean price for the five stores in a completely randomized design. Can you reject the null hypothesis? Use = 0.05.

Source DF SS MS F
Store * * * *
Error * 249.1301 8.3043
Total * 252.0784

b. Complete the ANOVA table below where the grocery items were treated as blocks. Can you reject the null hypothesis of no difference between the true mean price for the five stores? Use  = 0.05.

Source DF SS MS F
Store * 2.9483 * *
Item * 247.3979 * *
Error * * *
Total * 252.0784

c. Why is it important to treat the food items as blocks? Compare parts (a) and (b).

4. Due to his high blood pressure, Sam watches the sodium content of the foods that he eats. Five samples for each of four brands of canned turkey (97% fat free) were tested for sodium content, measured in milligrams of sodium per 2-ounce serving.

Brand 1 Brand 2 Brand 3 Brand 4
250 175 175 200
251 185 180 210
260 175 180 210
255 180 170 195
245 165 190 205

The following summary table and ANOVA were generated by statistical software as shown below:
ANOVA TABLE
Source of Variation df SS MS F P-value F crit
Brand 3 18632.4 6210.8 134.871 0.000 3.23887
Error 16 736.8 46.05

Total 19 19369.2

a. State the hypotheses and use the p-value approach to conclude concerning mean amount of sodium in the four brands. Let = 0.05.

b. Using the following output generated by Minitab determine which of the means are different.

Tukey's pairwise comparisons
Family error rate = 0.0500
Individual error rate = 0.0113
Critical value = 4.05
Intervals for (column level mean) - (row level mean)

1 2 3
2 63.91
88.49
3 60.91 -15.29
85.49 9.29
4 35.91 -40.29 -37.29
60.49 -15.71 -12.71

See attached file for problems.
Please show all work.