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# Testing of hypothesis problems

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4. Cancer of the colon and rectum is less common in the Mediterranean region than in other Western countries. The Mediterranean diet contains little animal fat and lots of olive oil. Italian researchers compared 1953 patients with colon or rectal cancer with a control group of 4145 patients admitted to the same hospitals for unrelated reasons. They estimated consumption of various foods from a detailed interview, then divided the patients into three groups according to their consumption of olive oil. Here are some of the data.

OLIVE OIL

Low Medium High Total
Colon Cancer 398 397 430 1225
Rectal Cancer 250 241 237 728
Controls 1368 1377 1409 4154

(a) Is this study an experiment? Explain your answer. (I can answer this probably, so do the other things more urgently).
(b) Is high olive oil consumption more common among patients without cancer than in patients with colon cancer or rectal cancer?
(c) Find the chi-square statistic X2 (this is supposed to be X squared, but I don't know how to get it to type on my computer). What would be the mean of X2 if the null hypothesis (no relationship) were true? What does comparing the observed value of X2 with this mean suggest? What is the P- value? What do you conclude?
(d) The investigators report that "less than 4% of cases or controls refused to participate." Why does this fact strengthen our confidence in the results? (This answer is also easy.. I don't have much problem with written and logic problems, but the calculations baffle me!)

In the following questions:

**** Describe situations in which we want to compare the mean responses in several populations. For each setting, identify the populations and the response variable. Then give I, the ni (this i is supposed to be small and below the n), and N. Finally, give the degrees of freedom of the ANOVA F test.

Do Problems 5 and 6 plus do the ANOVA for the data anyway, and state a conclusion based on the ANOVA p-value.

5. A maker of detergents wants to compare the attractiveness to consumers of six package designs. Each package is shown to 120 different consumers who rate the attractiveness of the design on a 1 to 10 scale.

6. How quickly do synthetic fabrics such as polyester decay in landfills? A researcher buried polyester strips in the soil for different lengths of time, then dug up the strips and measured the force required to break them. Breaking strength is easy to measure and is a good indicator or decay; lower strength means the fabric had decayed.

Part of the study buried 20 polyester strips in well-drained soil in the summer. Five of the strips, chosen at random, were dug up after 2 weeks; another 5 were dug up after 4 weeks, 8 weeks, and 16 weeks. Here are the breaking strengths in pounds.

2 weeks 118 126 126 120 129
4 weeks 130 120 114 126 128
8 weeks 122 136 128 146 140
16 weeks 124 98 110 140 110

(a) Find the mean strength for each group and plot the means against time. Does it appear that polyester loses strength consistently over time after it is buried?
(b) Find the standard deviations for each group. Do they meet our criterion for applying ANOVA?
(c) In Examples 17.2 and 17.3, we used the two-sample t test to compare the mean breaking strengths for strips buried for 2 weeks and for 16 weeks. The ANOVA F test extends the two samples t to more than two groups. Explain carefully why use of the two-sample t for two of the groups was acceptable but using the F test on all four groups is not acceptable.

DATA from examples:
17.2
2 weeks 118 126 126 120 129
16 weeks 124 98 110 140 110

Group Treatment n x bar s
1 2 weeks 5 123.80 4.60
2 16 weeks 5 116.40 16.09

17.3 test statistics (without formula) for null hypothesis result is: .9889