Download 1. (The Empirical Rule) A certain population of data is norhas mean

Practice Problems‐‐‐Normal distribution. 1. (The Empirical Rule) A certain population of data is norhas mean 0 and standard deviation 1. a. What percentage of the population falls between ‐1 and 1? That is, what percentage of a normal population lies within 1 standard deviation of the mean? b. Same question for 2 standard deviations. c. Same question for 3 standard deviations. 2. You often go to a certain restaurant for breakfast. You believe that the length of time X spent waiting to pick up your food on any given visit follows a normal distribution with mean 4 minutes and standard deviation 1.5 minutes. a. What is the probability you will wait 3 minutes or less on any given visit? b. What is the probability you will wait 5 minutes or longer on any given visit? c. You will become impatient if you have to wait too long for your food. In fact, you decide you will become impatient if your wait time falls in the worst 5% of the distribution. How long should you wait on any given visit before becoming impatient? d. At Christmastime, you decide you will give the cashier a $10 tip if you receive particularly fast service (in the best 1% of the distribution). To how long of a wait time does this correspond? 3. Scores on the math section of the SAT follow an (approximately) normal distribution with mean 500 and standard deviation 100. a. What percentage of the students of the students taking the test score 600 or more? Try to do this directly and using problem 2 above. b. A college will only admit those students whose SAT math score is in the top 20% of the population. What score would be needed for admission? 4. (The Normal Approximation to the Binomial Distribution) You toss a fair coin 150 times, and count the number of heads X. We know X has a binomial distribution with mean 150 . 5
75and standard deviation 150 . 5 . 5
6.12 a. Find the probability that the coin lands heads 80 or more times. b. Given a random variable that has a normal distribution with mean 75 and standard deviation 6.12, find the probability it takes on a value of 80 or greater. c. Given a random variable that has a normal distribution with mean 75 and standard deviation 6.12, find the probability it takes on a value of 79.5 or greater. Note: that (a) and (c) closely agree. This technique is called the normal approximation to the binomial distribution. You should also notice that the answer in part (b) does not quite agree with the ‘true’ answer in part (a). In part (c), 79.5 is said to be the ‘continuity corrected’ version of the point we are interested in. d. Use this idea to estimate the probability that a fair coin will land heads 520 times or more when tossed 1000 times.