## Question

1. Some general questions:

(a) 5 What is the most likely shape of your distribution if the median is higher than the mean? Explain briefly why.

(b) 10 Describe in your own words the terms: P-value, type-I error, type-II error. Describe also the difference between the last two terms.

(c) 10 Briefly describe the steps involved in a statistical test.

2. Keeping body balance. How difficult is it to maintain your balance while concentrating? It is more difficult when you are older? Nine elderly (6 men and 3 women) and eight young men were subjects in an experiment. Each subject stood barefoot on a “force platform” and was asked to maintain a stable upright position and to react as quickly as possible to an unpredictable noise by pressing a hand held button. The noise came at randomly generated times and the subject concentrated on reacting as quickly as possible. The platform automatically measured how much each subject swayed in millimetres in both the forward/backward and the side-to-side directions. The larger the sway, the more trouble the participant had in keeping his/her balance.

The variable Age has levels “Elderly” and “Young”, FBSway is the size of Sway in forward/backward direction and SideSway the Sway in side-to-side direction.

(a) 7 Give a box-and-whiskers plot of the observed Sway in forward/backward direction combined for each Age category in a single plot. Explain and comment on what you observe.

(b) 8 Determine the confidence interval for the difference in means. State what assumptions you are making. Do those assumptions seem reasonable?

(c) 10 Test the difference in some measure of centrality (mean/median) of the observed Sway in forward/backward direction for Elderly versus Young persons. State the null hypothesis, the choice of test statistic, the value of the test statistic, the p-value, your decision as well as the conclusion from the test.

3. Cholesterol. Use the nhanes data from the library library(survey). An important question in medical research is what factors determine high cholesterol in persons. Write a report to answer the question: “Is Age a factor that influences having high cholesterol?” Make sure that your report includes the following sections:

(a) 8 Introduction. Describe the research question and the design of the study, by studying help(nhanes) and the above website.

(b) 8 Exploratory analysis. Construct a relevant summary of the variable Age and Cholesterol that may help you to get an exploratory insight in the research question.

(c) 10 Formal analysis. Perform a test for the difference of the proportion of high variable Cholesterol and Age category.

(d) 6 Conclusion. How would you answer your research question?

(e) 8 Discussion. Provide two points of discussion. Argue what aspects of the design and/or the analysis could have been better.

## Solution Preview

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1. a) The distribution will be negative skew or left-skewed. The negative skew distribution has a longer tail to the decreasing direction implying that lowest 50 percentile has lower values than the highest 50 percentile. It follows that the mean will be smaller than the median.(b) A p-value is the probability that the event under the null hypothesis occurs or more extreme event under the null hypothesis occurs. A type-I error is that the statistical test rejects the null hypothesis when the null hypothesis is true. A type-I error is that the statistical test fails to reject the null hypothesis when the null hypothesis is false. A type I error is detecting an effect that is not actually present while a type II error is failing to detect an effect that is actually present.

(c) First, we want to identify what you want the null hypothesis to be, which is usually a status-quo assumption. The alternative is the negation of the null hypothesis.The simple and typical null hypothesis is the assumption about the parameter value for the particular distribution. Then, you decide your tolerance of how much your observation from data deviate from the null hypothesis by specifying the significance level and the rejection region...

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