01 Aug STATS PROBLEMS 3
1. Many studies have suggested that there is a link between exercise and healthy bones. Exercise stresses the bones and this causes them to get stronger. One study examined the effect of jumping on the bone density of growing rats. There were three treatments: a control with no jumping, a low-jump condition (the jump height was 30 centimeters), and a high-jump condition (60 centimeters). After 8 weeks of 10 jumps per day, 5 days per week, the bone density of the rats (expressed in mg/cm3 ) was measured. Here are the data. data27.dat
(a) Make a table giving the sample size, mean, and standard deviation for each group of rats. Consider whether or not it is reasonable to pool the variances. (Round your answers for x, s, and to one decimal place.)
| Group | n | s | ||
| Control | ||||
| Low jump | ||||
| High jump |
(b) Run the analysis of variance. Report the F statistic with its degrees of freedom and P-value. What do you conclude? (Round your test statistic to two decimal places and your P-value to three decimal places.)
| F | = |
| P | = |
Conclusion: There is statistically significant difference between the three treatment means at the α = .05 level.
2. Different varieties of the tropical flower Heliconia are fertilized by different species of hummingbirds. Over time, the lengths of the flowers and the form of the hummingbirds’ beaks have evolved to match each other. Here are data on the lengths in millimeters of three varieties of these flowers on the island of Dominica. data44.dat Do a complete analysis that includes description of the data and a significance test to compare the mean lengths of the flowers for the three species. (Round your answers for x to four decimal places, s to three decimal places, and to three decimal places. Round your test statistic to two decimal places. Round your P-value to three decimal places.)
| flower type | n | s | ||
| H. bihai | ||||
| H. caribaea red | ||||
| H. caribaea yellow |
F = P =
3. 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. data456.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 | ||
| Aluminum | ||||
| Clay | ||||
| Iron |
(b) Run the analysis of variance. Report the F statistic with its degrees of freedom and P-value. What do you conclude? (Round your test statistic to two decimal places. Round your P-value to three decimal places.) F = P = Conclusion: There is statistically significant difference between the three treatment means at the α = .05 level.
4. Do left-handed people live shorter lives than right-handed people? A study of this question examined a sample of 949 death records and contacted next of kin to determine handedness. Note that there are many possible definitions of “left-handed.” The researchers examined the effects of different definitions on the results of their analysis and found that their conclusions were not sensitive to the exact definition used. For the results presented here, people were defined to be right-handed if they wrote, drew, and threw a ball with the right hand. All others were defined to be left-handed. People were classified by gender (female or male) and handedness (left or right), and a 2 × 2 ANOVA was run with the age at death as the response variable. The F statistics were 22.36 (handedness), 37.44 (gender), and 2.10 (interaction). The following marginal mean ages at death (in years) were reported: 77.39 (females), 71.32 (males), 75.00 (right-handed), and 66.03 (left-handed).
(a) For each of the F statistics given, find the degrees of freedom and an approximate P-value. Summarize the results of these tests. (b) Using the information given, write a short summary of the results of the study.
5. A study of undergraduate computer science students examined changes in major after the first year. The study examined the fates of 256 students who enrolled as first-year students in the same fall semester. The students were classified according to gender and their declared major at the beginning of the second year. For convenience we use the labels CS for computer science majors, EO for engineering and other science majors, and O for other majors. The explanatory variables included several high school grade summaries coded as 10 = A, 9 = A-, etc. Here are the mean high school mathematics grades for these students.
| Major | |||
| Gender | CS | EO | O |
| Males | 8.68 | 8.35 | 7.65 |
| Females | 9.11 | 9.36 | 8.04 |
Describe the main effects and interaction using appropriate graphs and calculations.
6. A survey looked at the value of recreational sports on college campuses. One of the questions asked each student to rate the importance of recreational sports to college satisfaction and success. Responses were on a 10-point scale with 1 indicating total lack of importance and 10 indicating very high importance. The following table summarizes these results:
| Class | n | Mean score |
| Freshman Sophomore Junior Senior | 722 535 592 437 | 7.6 7.6 7.5 7.3 |
(a) To compare the mean scores across classes, what are the degrees of freedom for the ANOVA F statistic?
| numerator df | |
| denominator df |
(b) The MSG = 11.706. If sp = 2.15, what is the F statistic? (Round your answer to two decimal places.) F = (c) Give an approximate (from Table E) or exact (from software) P-value. (Enter your answer to four decimal places.)
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