Please explain clearly and draw simple graphs when needed. Thank you!
QUESTION I: RANDOM VARIABLES AND NORMAL DISTRIBUTION In statistics, events which depend on chance are called random variables. As an example, we can consider a coin toss. Before the coin toss, nobody knows whether the result will be heads or tails. Hence, the coin toss is a random variable with two possible outcomes (ie: realizations), beads or tails. There are two possible realizations in this case. Two is a nite, number. Random variables with a set of nite possible realizations are called discrete random variables. So the coin toss is a discrete random variable. Some random variables in nature have a continuum of possible outcomes, instead of just a nite number. As an example, you can consider the height of an individual. On one end of the spectrum, there are individuals like Taco Fall (NBA player, Boston Celtics Center) who is 7 ft 5 in tall. On the other hand of the spectrum, there are individuals like Peter Dinklage (the outstanding actor who played Tyrion Lannister in Game of Thrones) who is 4 4 in tall. In between, there are billions of people with every imaginable height. Hence, height as a random variable has a continuum of possible outcomes. Random variables with a continuum of possible outcomes are called continuous random variables. In nature, several continuous random variables follow what is called a normal distribution. Examples include height, weight, intelligence scores (IQ), blood pressure of humans, SAT scores and several more. The normal distribution can be found in the gure below: 3 4. 1 (iii: 3 4 . 1 CH3 13.6% 30 -20 -10 0 lo 20 30 Figure 1: Normal Distribution Figure 1: Normal Distribution The normal distribution is also known as the Gaussian Distribution, named after the famous German mathematician Carl Friedrich Gauss. You can nd a picture of Gauss on the back of the German 10-mark banknote (Deutsche Mark) down below: ti 2,2\" :2 . 1 122 22.1 . In NY Figure-2: The German Ill-Mark Banknote Please note the normal distribution on the le-hand side of the lO-mark banknote. The normal distribution has been extensively studied over the centuries and we know a great deal about it. For instance, we know that if a continuous random variable follows the normal distribution, 34.2% of all realizations will be between the mean and mean plus one standard deviation of the distribution. As an example, 34.2% of all individuals have a height between the mean height and mean plus one standard deviation in the United States. Referring back to Figure-1, you can see which percentage of the society will have a height between which numbers. Given the information above, please answer the following questions: a. What is your height? Which percentage of the height distribution do you happen to be in? Hint: The mean height for women in the US is 5 ft 4 in and the standard deviation of the height distribution for women is 2.5 inches. The mean height for men in the US is 5 ft 9 in and the standard deviation of the height distribution for men is 3 inches. b. Watch Peter Dinklage give an acting master class in the following video: https ://www.youtube. com/watch ?v=e4UqSOSZhUA Peter Dinklage won several awards for that great performance. In his own words, Dinklage's character was \". . .guilty of being a dwarf" and he was on trial for that reason alone. Remembering that Dinklage is 4 ft 4 inches, which percentage of the height distribution does he happen to be fall into? c. Your instructor's favorite NBA player is Jimmy Butler, aka Jimmy Buckets. Butler is the shooting guard for Miami Heat and 6ft 7in tall. Which percentage of the height distribution does Butler fall into? (1. Your instructor is 5 3 inches tall. What percentage of the height distribution does your instructor fall into? 6. The height distribution does not remain constant over time. The mean height for both women and men increases over decades. If we go back thousands of years in time, we can see that individuals used to be signicantly shorter. As an example, archeological remains from the Mesolithic (10,000 years ago) indicate that women's average height was around 5 1 inch and men's average height was around 5 ft 5 inches at the time. Suppose you nd a time machine and travel back in time 10 thousand years. What percentage of the height distribution would you fall into? (Hint: You can assume a standard deviation of 2.5 inches for women and 3 inches for men.) f. The height distribution is not the same across countries either. As an example, in Indonesia the average height of men is 5 ft 2 inches, and the average height of women is 4 ft 10 inches. Suppose you travel to Jakarta, capital of Indonesia. What percentage of the height distribution would you fall into? (Hint: You can assume a standard deviation of 2.5 inches for women and 3 inches for men.)
