# STAT（Experience of design) involved the ANOVA/Trend/F-test

I’m studying for my Statistics class and need an explanation.

there are 4 questions.

1.Twenty overweight men and women agree to participate in a study of the effectiveness of four diet regiments. The researcher first compares each subject’s weight with the “ideal weight” and then calculates how many pounds overweight each is.

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(a)Compute a 95% confidence interval of the mean number of pounds above the ideal weight for the study participants.

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Next, the subjects are grouped into five blocks of four subjects each by excess weight, and the four treatments A, B, C, and D are assigned at random, each to one subject in each block.

(b)Name this design: ______________________________________________

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(c)How many experimental units are there?

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(d)How many factors?

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(e)How many treatments?

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(f)Arrange the subjects in terms of increasing weight, then group the four least overweight, the next four, and so on. These groups are blocks. List the blocks of participants below.

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 Block 1 Block 2 Block 3 Block 4 Block 5

(g)The following table shows the random arrangement of the diet regiments within blocks, together with the final weight loss of the subjects in pounds (within parentheses).

 Block 1 A (10) C (12) D (18) B (15) Block 2 B (16) C (14) A (9) D (20) Block 3 C (17) D (17) A (13) B (13) Block 4 A (13) D (16) B (16) C (18) Block 5 D (17) A (12) C (15) B (11)

(h)Analyze this design correctly using an ANOVA table. Give the ANOVA table below.

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(i)Is the mean weight loss the same under the four regiments? Answer this question using significance level 5%.

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(j)What is the power of the test in part (j) above?

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(k)Draw a graph of the treatment by block interaction and interpret it.

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(l) How would you improve this design? State the most important change you could make that would improve this design the most.

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1.Consider a completely randomized design with 5 treatments and 6 observations (or replications) for each treatment. Suppose the treatments represent increasing amounts of fertilizer applied to a certain crop. The following partial results are obtained.

 Amount of fertilizer 0 2 4 6 8 Mean crop amount produced 4.9 10 13.9 15.7 16.3 Standard deviation of crop amount 1 1.5 2.3 3.1 4.5

Using the method of orthogonal polynomials, investigate whether:

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Final answer: the linear trend is_____________

(ii)Investigate whether the first and second order terms provide an adequate fit to the data.

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2.Twenty high school-aged students are randomly selected from three different school districts: a district in the city, a district in the suburbs, and a district in a rural area. Each group of twenty students consisted of 10 boys and 10 girls. Each of the students was asked what price they paid for their last haircut. The data were entered in SPSS and the two-way ANOVA output is shown below:

(i)If the data were analyzed as a one-way ANOVA with six “treatments” corresponding to the six combinations of home region and gender, what would have been the value of the F statistic?

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 (ii). What does the graph above suggests about the experiment? [5 points] A) The assumption of equal standard deviation for each treatment combination is satisfied. B) Normality, a required assumption for ANOVA, is suspect. C) There would appear to be interaction between student gender and region. D) The small number of observations per cell makes the graph uninformative. E) The price of a haircut appears to be unaffected by the region, as the lines for male and female students are reasonably parallel.

3.

The following is a partial analysis of variance table for a full two-factor factorial experiment with fixed-effect factors A and B having four levels each, with two replicates of the experiment.

(i)Complete the ANOVA table below.

[9 points]

 Source df SS MS F-value Pvalue Factor A 75 Factor B 30 AxB error 10 Total 400

(ii)Which factor(s) or factor interaction is significant? Using a=0.05, give the critical value or the p-value for each F-test and state your conclusions.

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 (iii) Which of the following statements about a two-way analysis of variance model is (are) TRUE? [5 points] A) The population of interest is classified according to two categorical variables, or factors. B) An experiment involving the simultaneous use of two factors offers advantages over two one-way experiments with respect to such matters as efficiency and reduction of residual variation. C) The two-way type of experiment requires twice as many experimental observations as would be required in two one-way experiments of the same factors. D) An experiment involving the simultaneous study of two factors allows for the investigation of interactions between the factors. E) Only A, B, and D are true.