Problem 10: One sample \(t\) -test. You are in charge of conducting clinical trials for a drug intended to lower cholesterol levels. You measure reduction of cholesterol levels in

Complete the following table for three imaginary cholesterol drug trials. \(\bar{X}\) represents the mean cholesterol level change in the experimental subjects treated with each drug. \(s\) is the standard deviation in cholesterol level change. \(s_{\bar{X}}\) is the standard deviation of \(\bar{X}\). Calculate \(s_{\bar{X}}\) and \(t\) scores for the cholesterol experiments for trials 1 3 and classify the level of significance of the results as NS (not significant), 0.05, 0.01 or 0.001.

11. In the blank provided, stipulate for each of the following sets of hypotheses whether a one- or two-tailed test should be conducted.

1. \(H_{0} \mu_{0}=\mu_{1} ; H_{1} \quad \mu_{0} \neq \mu_{1}\)
2. \(H_{0} \mu_{0}=\mu_{1} ; H_{1} \quad \mu_{0}>\mu_{1}\)
3. \(H_{0} \mu_{0} \leq \mu_{1} ; H_{1} \quad \mu_{0}>\mu_{1}\)
   Problem 12: Consider the following two hypotheses:
   
   Suppose we have a sufficiently large sample size to use the \(Z\) -test and that \(s_{\bar{X}}=1\). Suppose further that the real mean of the population is \(\mu=3\) (i.e. that \(H_{1}\) is actually true). Our \(Z\) statistic will be:
   \(Z=\frac{\bar{X}-0}{s_{\bar{X}}}=\frac{\bar{X}}{s_{\bar{X}}}\) For significance \(Z\) should exceed the \(\alpha=0.05\) one-tailed \(Z\) threshold. Hence, we can reframe our hypothesis test in terms of \(\bar{X}\) and calculate an \(\bar{X}\) rejection threshold which is:
   \(\bar{X}>Z_{\alpha(1)=0.05 S_{\bar{X}}}\)
4. (2 points) What is the numerical value of the \(\bar{X}\) threshold rejection threshold?
5. (6 points) The shaded areas above correspond to probabilities. What probabilities do the areas of the: 1) grey, 2) turquoise and 3 ) grey + yellow regions (together) correspond to?

13. (10 points) For each of the following: 1) classify the factors as fixed or random effects and 2) state whether a Model I (fixed factors), II (random factors) or III (mixed, both fixed and random factors) ANOVA should be used for analysis.

1. (3 points) You study four chemical and four radiation cancer treatment therapies and measure their reduction of cancer.

1. Chemical: effect (fixed or random).
2. Radiation: effect (fixed or random).
3. Model (I, II or III).
   b. (3 points) Suppose you are investigating sales in a chain of department stores that you own in three cities in the U.S. You want to build a new store and want to decide which of three candidate cities would be best for building the new store, based on likely sales. So, you select a representative sample of three stores in each city and then compare sales among stores and cities with ANOVA.
4. City: effect (fixed or random).
5. Store: effect (fixed or random).
6. Model (I, II or III).
   c. (3 points) You investigate three types of teaching methods in a representative sample of public schools chosen randomly from those in NY. All teaching methods are tested in each school.
7. Teaching Methods: effect (fixed or random).
8. Schools: effect (fixed or random).
9. Model: (I, II or III).

d. (1 point) One of the three examples 6. a - c above should be analyzed not only as a Model III ANOVA but also as a randomized complete block design. Which one?

14. (11 points) The file Exam IA.sav contains data on clutch size (the number of eggs laid per nesting attempt) for birds from two types of land mass (islands and mainlands) and from two types of habitat (temperate and tropical). Assume both of these are fixed factors. Conduct an analysis of variance and fill in the table below (1 point each):

f. (3 points) Is there a significant effect of Habitat on clutch size? (yes/no). Of Land Mass? (yes/no). Is there a significant interaction? (yes/no)

g. (2 points) The graph below plots mean clutch size for the two treatments Habitat and Land Mass. interpret the graph in light of the results presented above in \(\mathbf{a}\) and the table.

15. (12 points) Each of the graphs below plot means of a dependent variable \(Y\) as a function of two fixed effects \(A\) and \(B\) in a two-way factorial analysis of variance, each of which has two levels 1 and 2. Interpret both graphs. For each graph state whether there is an effect on \(Y\) of: 1) the levels of treatment A, 2) the levels of treatment \(B\) and 3) the interaction between \(A\) and \(B\).

16. (10 points) The file Exam IB.sav contains measurements of the strength of cotton fibers (the dependent variable) grown under five levels of \(\mathrm{K}_{2} \mathrm{O}\) (potassium oxide) fertilization treatment (the independent variable). The treatments were replicated in three agricultural plots (blocks). Analyze the data as a randomized complete block design and report the following.

h. (2 points) The \(\mathrm{K} 2 \mathrm{O}\) treatment is significant at the \(\mathrm{NS} \ldots, 0.05 \ldots, 0.01\) or 0.001 _ level (check one)?

1. (2 points) The block is significant at the NS ________, 0.05 ____, 0.01 _____ or 0.001 level (check one)?

j. (2 points) Why is there no $F$ test for a Block $\times$ Treatment interaction?

17. (10 points) A recent study suggested that exposure to peppermint can improve concentration. Ten students were asked to read a paragraph and were then given a reading comprehension test. On the following day, the same students were asked to eat two Altoids 'curiously strong' peppermint candies and then to read a second paragraph and take a second comprehension test. All students read the same two paragraphs and took the same comprehension tests, but half read one paragraph first the other half read the second paragraph first (i.e. so there was no effect of the order the paragraphs were read - this doesn't affect the analysis you will do, but is mentioned because it is good experimental design). Use the appropriate \(t\) test to test the control and peppermint reading comprehension scores to determine whether the peppermint scores are significantly higher.

18. The table below contains data on fat absorption by donuts in the process of cooking. Each cell is the amount of fat absorbed by 24 donuts during frying. Four types of fat were used (1-4).

1. (5 points) Do a one-way (single factor) ANOVA on these data to determine if there is a significant difference in the amount of fat absorbed among the four fats tested.
2. (5 points) Also conduct the post-hoc Tukey multiple comparisons test and show which pairs of means are significantly different.

19. (5 points) Explain why a factorial ANOVA is more efficient than sequential oneway (single factor) ANOVAs.

20. (5 points). Why should we perform a one-way ANOVA rather than multiple \(t\) tests when means are being compared among more than two groups?

21. (5 points). What two assumptions must data conform to for the analysis of variance to be appropriate?

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