Q1. The human heart beats about once a second. Take that as granted. And also note that it's more common to talk about beats per minute. a) Suppose you count your heart beats for one minute, and then for another minute, and then, again for another minute. Suppose you get the following data: 59, 60, 61. Note that, this set of measurements are as good as things can get, because the mean is 60 (i.e., what we expect), and the spread cannot be any less (for now, exclude the possibility of getting the same count more than once). What is the variance of this data? Do "by hand." b) Suppose, you don't have time to sit for a whole minute and count your heart beat. 50, instead you count your heart beats for 15 seconds, and then just multiply the count by 4 to get the count per minute. To follow the same best-scenario as in part a, suppose your data are as follows: 14, 15, 16. And, don't forget to multiply these numbers by 4, in order to get counts per minute. What is the variance of the data? This time you may do it by hand, or by R. Just make sure that the data you use are in counts *per minute". c) This time, suppose you measure your heart beat across only 5 seconds. In the best-scenario, i.e., the best mean and the least variance), what are the three observations you would get? Write them, and then nd their variance (again making sure your data are in counts *per minute") The moral of the above exercise is to see how heart beat results can become more or less reliable, independently of sample size (here, 3, in all three parts). Finally, I'd like to show you why in the denition of variance, the differences are squared. d) Return to part (a), and compute (by hand) the "variance" but forgetting to square the differences. What value do you get? e) You may think that the answer you got in part cl is specic to the example I cooked-up. Just to check, select any 4 numbers - yes, any 4 numbers! - and compute the "variance" but forgetting to square the differences. What value do you get? Hopefully, you'll see why it's necessary to square the differences