7.4 Exercises

  1. Random Numbers (**): R allows you to generate random numbers from a variety of distributions. For example, you can generate 100 normally distributed random numbers with \(\mu=70\) and \(\sigma = 10\) using the following command: rnorm(100,70,10). Generate three datasets (\(\mu=70\) and \(\sigma = 10\)) with the command rnorm() and name them \(x1\), \(x2\), and \(x3\). The data sets \(x1\) and \(x\) have 50 random numbers each and \(x3\) has 1000 random numbers. Calculate and report the mean of \(x1\), \(x2\), and \(x3\). Plot a histogram of all three datasets.
    1. Why do the histograms of \(x1\) and \(x2\) look different despite the fact that they were generated using the same command? Do they look normally distributed?
    2. The histogram of \(x3\) will look much more like a normal distribution. Why is that the case?
    3. Compare the three means and explain which statistical law should make the mean of \(x3\) always closer to 70 than for \(x1\) and \(x2\).