Mat 217 Normal Distribution and Multivariable Calculus 1. Suppose x is a normally distributed random variable with mean   10.4 and standard deviation   1.6 . Find each of the following probabilities: i) p(6.1  x  13.3) ii) p( x  11.8) iii) p( x  15.4) Do not forget to draw normal curve in each case. 2. The physical fitness of an athlete is often measured by how much oxygen the athlete takes in, measured in millimeters per kilogram ml/kg. The mean maximum oxygen uptake for elite athletes has been found to be 80 ml/kg with standard deviation of 9.2. Assume that the distribution is approximately normal. What is the probability that an athlete has maximum oxygen uptake of at least 75 ml/kg? 3. Healthy people have body temperatures that are normally distributed with mean value of 98.20 degrees F and a standard deviation of 0.62 degree F. a) if a healthy person is selected at random, what is the probability that he or she has a body temperature above 98.20 degrees F? b) A hospital wants to select a minimum temperature for requiring further medical tests. What should he temperature be, if we want only 1% of healthy people to exceed it? 4. IQ scores are normally distributed with mean of 100 and a standard deviation 15. Mensa is an international society that has one – and only one qualification for membership, a score in the top 3 on an IQ test. What IQ score should one have in order to be eligible for Mensa? Round your answer to two decimal places. 5. The A large class had a test whose scores were normally distributed with a mean score of 79 and a standard deviation of 5. A test is chosen at random. Find the following probabilities. [6 pts]. Draw normal curve in each case. a. p( X  80) . Solution: p( X  80)  0.5  normalcdf (79,80,79,5)  57.93% 79 80 b. p(65  X  85) Solution: p(65  X  85)  normalcdf (65,85,79,5)  88.24% c. 65 79 85 p( X  75) Solution: p( X  75)  0.5  normalcdf (75,79,79,5)  78.813% 75 79 6. Suppose x is normally distributed random variable with mean   10.5 and standard deviation   0.95 . Find each of the following probabilities. Show the normal curve along with shaded area. a) P(17  x  15.5) [ 7.094 108  0 ] b) P( x  16.5) [0.9999997128] c) P(18  x) [ 1.4689 1015  0 7. Healthy people have body temperatures that are normally distributed with mean of 98.2 degrees Fahrenheit and a standard deviation of 0.65 degree Fahrenheit. Determine the minimum temperature if we want only 20% of healthy people to exceed it, 20% of the healthy people have less than what temperature in degrees Fahrenheit? X  98.2  INVNORM (0.8)  x  98.747 , 0.65 X  98.2  INVNORM (0.2)  x  97.68 0.65 Solution: 8. Find the point(s) on f ( x, y )  2 x 2  2 xy  2 y 2  36 x  42 y  158 where you have maximum. [Suppose A  f xx , B  f yy , C  f xx f yy  ( f xy ) 2 then the function f ( x, y) has maximum when A  0, B  0, AB  C 2  0 .] Show complete work and box your answer. Answer: [MAX AT (5,8)] 9. Find the point(s) on f ( x, y )   x 3  xy  y 2 where you have maximum. [Suppose A  f xx , B  f yy , C  f xx f yy  ( f xy ) 2 then the function f ( x, y) has maximum when A  0, B  0, AB  C 2  0 .] Show complete work and box your answer 10. Find all the critical point(s) of f ( x, y )   x 2  3xy  y 2 . Also find the following: A  f xx , B  f yy , C  f xy . Show complete work and box your answer. 11. For the Given that Y  AK a Lb , find and simplify KYK  LYL . Show work and box your answer ln( xy) , find hx (1, 2) and hy (1, 2) . Show work and xy box your answer. Answer: [0.1534, 0.077] 12. For the given function h( x, y)  13. Use Lagrange multiplier method to find the point(s) on f ( x, y )  x 2  y 2  2 x  1 x 2  y 2  16 subject to for maximum. Lagrange method is Answer: [(-4, 0), 25] L( x, y)  f ( x, y)  [ g ( x, y)  k ] 14. If f ( x, y)  eax by then f xx f yy  f xy f yx  X , find X Answer: 0 15. Use Lagrange multiplier method to find the point(s) on f ( x, y )  x 2  y 2  2 x  1 subject to x 2  y 2  16 for maximum. Lagrange method is L( x, y)  f ( x, y)  [ g ( x, y)  k ]