Chapter 2: Problem 49
Draw any dotplot to show a dataset that is Clearly skewed to the left.
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Chapter 2: Problem 49
Draw any dotplot to show a dataset that is Clearly skewed to the left.
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Two variables are defined, a regression equation is given, and one data point is given. (a) Find the predicted value for the data point and compute the residual. (b) Interpret the slope in context. (c) Interpret the intercept in context, and if the intercept makes no sense in this context, explain why. Study \(=\) number of hours spent studying for an exam, Grade \(=\) grade on the exam. \(\widehat{\text { Grade }}=41.0+3.8\) (Study); data point is a student who studied 10 hours and received an 81 on the exam.
Multiple studies \(^{61}\) in both animals and humans show the importance of a mother's love (or the unconditional love of any close person to a child) in a child's brain development. A recent study shows that children with nurturing mothers had a substantially larger area of the brain called the hippocampus than children with less nurturing mothers. This is important because other studies have shown that the size of the hippocampus matters: People with large hippocampus area are more resilient and are more likely to be able to weather the stresses and strains of daily life. These observations come from experiments in animals and observational studies in humans. (a) Is the amount of maternal nurturing one receives as a child positively or negatively associated with hippocampus size? (b) Is hippocampus size positively or negatively associated with resiliency and the ability to weather the stresses of life? (c) How might a randomized experiment be designed to test the effect described in part (a) in humans? Would such an experiment be ethical? (d) Can we conclude that maternal nurturing in humans causes the hippocampus to grow larger? Can we conclude that maternal nurturing in animals (such as mice, who were used in many of the experiments) causes the hippocampus to grow larger? Explain.
If we have learned to solve problems by one method, we often have difficulty bringing new insight to similar problems. However, electrical stimulation of the brain appears to help subjects come up with fresh insight. In a recent experiment \({ }^{17}\) conducted at the University of Sydney in Australia, 40 participants were trained to solve problems in a certain way and then asked to solve an unfamiliar problem that required fresh insight. Half of the participants were randomly assigned to receive non-invasive electrical stimulation of the brain while the other half (control group) received sham stimulation as a placebo. The participants did not know which group they were in. In the control group, \(20 \%\) of the participants successfully solved the problem while \(60 \%\) of the participants who received brain stimulation solved the problem. (a) Is this an experiment or an observational study? Explain. (b) From the description, does it appear that the study is double-blind, single-blind, or not blind? (c) What are the variables? Indicate whether each is categorical or quantitative. (d) Make a two-way table of the data. (e) What percent of the people who correctly solved the problem had the electrical stimulation? (f) Give values for \(\hat{p}_{E},\) the proportion of people in the electrical stimulation group to solve the problem, and \(\hat{p}_{S},\) the proportion of people in the sham stimulation group to solve the problem. What is the difference in proportions \(\hat{p}_{E}-\hat{p}_{S} ?\) (g) Does electrical stimulation of the brain appear to help insight?
From the StudentSurvey dataset, select any categorical variable and select any quantitative variable. Use technology to create side-by-side boxplots to examine the relationship between the variables. State which two variables you are using and describe what you see in the boxplots. In addition, use technology to compute comparative summary statistics and compare means and standard deviations for the different groups.
Use the \(95 \%\) rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about \(95 \%\) of the data values. A bell-shaped distribution with mean 10 and standard deviation 3.
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