# Eating Paint

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Eating paint

It has been estimated that lead poisoning resulting from an unnatural craving (pica) for substances such as paint may affect as many as a quarter of a million children each year, causing them severe, irreversible retardation. Explanations for why children voluntarily consume lead range from "improper parental supervision" to "a child's need to mouth objects." Some researchers, however, have been investigating whether the habit of eating such substances has a nutritional explanation.

One such study involved a comparison of a regular diet and a calcium-deficient diet on the ingestion of a lead acetate solution on rats. Each rat in a group of 20 was randomly assigned to either an experimental or a control group. Those in the control group received a normal diet, while those in the experimental group received a calcium deficient diet. Each of the rats occupied a separate cage and was monitored to observe the amount of a 0.15% lead-acetate solution consumed during the study period (ml).

Here is the data:

Control: 5.4 6.2 3.1 3.8 6.5 5.8 6.4 4.5 4.9 4.0

Exper : 6.8 7.5 8.6 7.6 5.5 4.9 5.4 4.5 8.5 6.3

Carefully describe how to randomize in this experiment and draw an possible experimental plan, i.e. show which rat is randomized to which trial, which rat is assigned to which cage, etc. You could present a table similar to:

Cage: 1 2 3 4 .....

Rat : 2 10 12 14 .....

Treatment: c e e e .....

Construct side-by-side dot and box plots. What do the graphs seem to indicate?

Find the simple summary statistics for each group. Find a standard error for each mean, and a confidence interval for the mean of each group. Interpret the se's and the confidence intervals.

Estimate the difference in the means and obtain an estimated se for the difference and a confidence interval for the difference. Interpret the se and the confidence interval.

Define the parameters of this experiment, and give the null and alternate hypothesis in words and symbols. [You may use either a one or two sided test.]

What is the p-value. Interpret the p-value.

What do you conclude?

What are the potential Type I and II errors? These should be expressed in terms of potential conclusions. For example, ``A Type I error would be to conclude blah-blah-blah when in fact blah-blah-blah'' where blah-blah-blah are replaced by your words of wisdom. It is not necessary to compute the probabilities of these events. Notice that only of tese error will be applicable to this case (depending if the hypothesis is rejected or not).

Suppose that a biologically significant difference is about 1.5. What is the estimated power at the current sample size if you test at = 0.05? [Yes, I know that this is a retrospective power analysis and is technically a ``no-no''.]

What sample size would be needed to be 80% confident of detecting half the difference above (i.e. a difference of 0.75) at = 0.05?

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Comparing yields of corn in a greenhouse

The yields of four varieties of corn are to be compared using a greenhouse study. Within the greenhouse, there are 32 tables available for use arranged in a rectangular pattern of 8 by 4 tables with the short side of the rectangle nearest the windows. Eight replications of each variety are planted. The varieties were assigned to the tables using a completely randomized design. After growth, the yields of each table are recorded as follows:

A: 2.5 3.6 2.8 2.7 3.1 3.4 2.9 3.5

B: 3.6 3.9 4.1 4.3 2.9 3.5 3.8 3.7

C: 4.3 xxx 4.5 4.1 3.5 3.4 3.2 4.6

D: 2.8 2.9 3.1 2.4 3.2 2.5 3.6 2.7

The xxx indicates a plot where no data could be collected because of a technical failure unrelated to the yield.

Draw a possible experimental plan, i.e. do the randomization. It is easiest if you draw the 8x4 grid and show which variety was grown in each grid location.

Create side-by-side confidence interval plots. What do these seem to indicate?

Find summary statistics (mean, std dev, and standard error for each group ). Find a confidence interval for the mean of each group. Interpret the se and the confidence intervals. Do these confirm your impressions from above?

Do a formal statistical test. Give the null and alternate hypotheses, the ANOVA table, the p-value, and your decision.

Use a multiple comparison procedure to see which means may be equal:

Estimate the power of this design to detect the observed differences in the means at = 0.05 with the existing sample size. [Yes, I know that this is a retrospective power analysis is a ``no-no''. No need to use the adjusted power from JMP. This exercise is just to get you practise in using the DOE sample size computations.]

What sample size would be required to be 80% confident of detecting a difference of .5 in the means at = 0.05? Be sure to be careful to indicate if these is the total sample size or the sample size for each treatment group.

Is there a better design? Explain why or why not and draw a new randomization using your proposed experimental design.

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Sexual activity and the lifespan of male fruitflies

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