Do piano lessons improve the spatial-temporal reasoning of preschool children? The data in the table below contain the change in spatial-temporal reasoning (after treatment minus before treatment) of 34 children who took piano lessons, 10 who took singing lessons, 20 who had some computer instruction, and 14 who received no extra lessons.
(a) Make a table giving the sample size, the mean, the standard deviation, and the standard error for each group.
(b) Analyze the data using one-way analysis of variance. State the null and alternative hypotheses, the test statistic with degrees of freedom, the P-value, and your conclusion.
Iron-deficiency anemia is the most common form of malnutrition in developing countries, affecting about 50% of children and women and 25% of men. Iron pots for cooking foods had traditionally been used in many of these countries, but they have been largely replaced by aluminum pots, which are cheaper and lighter. Some research has suggested that food cooked in iron pots will contain more iron than food cooked in other types of pots. One study designed to investigate this issue compared the iron content of some Ethiopian foods cooked in aluminum, clay, and iron pots. One of the foods was yesiga wet, beef cut into small pieces and prepared with several Ethiopian spices. The iron content of four samples of yesiga wet cooked in each of the three types of pots is given below. The units are milligrams of iron per 100 grams of cooked food. The data below MUST be separated into 3 columns of data: one for Aluminum, one for Clay, and one for Iron. Put these labels in the first row and the 4 observations of each beneath them (from the 'iron' content column). data369.dat
(a) Make a table giving the sample size, mean, and standard deviation for each type of pot. Is it reasonable to pool the variances? Note that with the small sample sizes in this experiment, we expect a large amount of variability in the sample standard deviations.
|Type of pot || n|| || s |
(b) Run the analysis of variance. Report the F
statistic with its degrees of freedom and P
-value. What do you conclude?F
Kudzu is a plant that was imported to the United States from Japan and now covers over seven million acres in the South. The plant contains chemicals called isoflavones that have been shown to have beneficial effects on bones. One study used three groups of rats to compare a control group with rats that were fed either a low dose or a high dose of isoflavones from kudzu. One of the outcomes examined was the bone mineral density in the femur (in grams per square centimeter). Here are the data.
|Treatment||Bone mineral density (g/cm2) |
|Control ||0.228 ||0.207 ||0.234 ||0.220 ||0.217 ||0.228 ||0.209 ||0.221 |
| ||0.204 ||0.220 ||0.203 ||0.219 ||0.218 ||0.245 ||0.210 |
|Low dose ||0.211 ||0.220 ||0.211 ||0.233 ||0.219 ||0.233 ||0.226 ||0.228 |
| ||0.216 ||0.225 ||0.200 ||0.208 ||0.198 ||0.208 ||0.203 |
|High dose ||0.250 ||0.237 ||0.217 ||0.206 ||0.247 ||0.228 ||0.245 ||0.232 |
| ||0.267 ||0.261 ||0.221 ||0.219 ||0.232 ||0.209 ||0.255 |
In this study there were three groups. Controls received a placebo, and the other two groups received either a low or a high dose of isoflavones from kudzu. You are planning a similar study of a new kind of isoflavone. Use the results of the study above to plan your study. Write a proposal explaining why your study should be funded.
The National Crime Victimization Survey estimates that there were over 400,000 violent crimes committed against women by their intimate partner that resulted in physical injury. An intervention study designed to increase safety behaviors of abused women compared the effectiveness of six telephone intervention sessions with a control group of abused women who received standard care. Fifteen different safety behaviors were examined. One of the variables analyzed was the total number of behaviors (out of 15) that each woman performed. Here is a summary of the means of this variable at baseline (just before the first telephone call) and at follow-up 3 and 6 months later.
|Group ||Baseline ||3 months ||6 months |
|Intervention||10.4 ||12.5 ||11.9 |
|Control ||9.6 ||9.9 ||10.4 |
(a) Find the marginal means. Are they useful for understanding the results of this study?
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(b) Plot the means. Do you think there is an interaction? Describe the meaning of an interaction for this study.
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(Note: This exercise is from a repeated-measures design, and the data are not particularly normal because they are counts with values from 1 to 10. Although we cannot use the methods in this chapter for statistical inference in this setting, the example does illustrate ideas about interactions.)