Question

The following questions were answered, Please create bell shaped graph to reflect answers.

Week 6 Chapter 6: The Normal Distribution For this week read Chapter 6 Answer the following question: Question 1 Given a normal distribution with µ=15 and σ = 5, what is the probability that X>20 = Z = X - µ        =    20 – 15         =   1            σ                         5          5 We have to find P( Z >1) P(Z >1) = 1 – P(Z >1) = = 1 – 0.8413 = 0.1587                                    (Using z table) X<20 = Z = X - µ        =    20 – 15         =   1            σ                         5          5 We have to find P( Z<1) P( Z<1) = 0.8413 (Using z table) We have to find P(X<20 or X>20) = P((X<20) + P(X>20) = 0.8143 + 0.1587 = 1.00 Question 2 Given a normal distribution with µ =15 and σ =5, what is the probability that 5% of the values are less than what X values? We have to find z corresponding to P(Z < z) = 0.05 z = -1.64 (Obtained using standard normal distribution table) z = (x - µ)/σ -1.64 = (x - 15)/5 x = 15 - 1.64*5 x = 6.8 5% of the values are less than 6.8 Between what two X values (symmetrically distributed around the mean) are 95 % of the values? For determining the two values of X (x1 and x2), first we need to determine corresponding z-values (z1 and z2). Following is the normal distribution graph that indicates z-values to be calculated. The area highlighted represent 95% of the values As we know that total area under normal distribution curve is 1. The area under highlighted portion represent 95% of the total area that is 0.95. Since distribution is symmetrical around the mean so the area under two portion that are not highlighted will be same = (1-0.95)/2 = 0.05/2 = 0.025. In terms of z, these two non-highlighted portions can be written as: P(Z < z1) = 0.025 and P(Z > z2) = 0.025 z1 = -1.96 and z2 = 1.96 (Obtained using standard normal table) Now, x values corresponding to these z values can be computed using following formula: z = (x-µ)/σ For x1 z1 = (x1-µ)/σ -1.96 = (x1-15)/5 x1 = 15-1.96*5 x1 = 5.2 For x2 z2 = (x2-µ)/σ -1.96 = (x2-15)/5 x2 = 15+1.96*5 x2 = 24.8 Hence, it can be said that the 95 % of the values lies between 5.2 and 24.8

Answer 1