## Normal distributions -- basic calculations

The goal of this lab is to compute normal probabilities and inverse normal probabilities using Excel.

The Excel functions we consider are:

which computes the probability that=NormSDist(a)z≤ain the Normal Standard distribution

which computes=NormDist(a,m,s,0)Pr(in the normal distribution with meana)and standard deviationm,s

which computes the cumulative=NormDist(a,m,s,1)Pr(x≤in the normal distribution with meana)and standard deviationm, (Note: the cumulative probability is obtained when the 4th argument is set to 1 instead of 0.)s

gives the value of=NormInv(p,m,s)so thatxequalsNormDist(x,m,s,1)in the normal distribution with meanpand standard deviationm,sProblem 1. In this problem you will compute

Pr(a≤z≤b)for values of a and b using the Standard Normal distribution. The probability that z lies between a and b is found by substracting the cumulative probabilities, that is

=NormSDist(b)-NormSDist(a)

A.ComputePr(-1≤z≤1)by enteringin a cell. Label the probability.=NormSDist(1)-NormSDist(-1)

B.Now computePr(-2≤z≤2)in another cell. Label this probability as well. In a text box explain a "rule-of-thumb" suggested by these two calculations.

Although we used thefunction above the same process can be used to computeNormSDistPr(a≤z≤b)for general normal distributions using.NormDist(x,m,s,1)Problem 2. Suppose X is normal distribution with mean 100 and standard deviation 15.

A.ComputePr(x≤120)using appropriate values in.NormDist(x,m,s,1)

B.Use the transformationto convert 120 for the normal distribution to an appropriate z-value for the Standard Normal distribution. Label the x and z values.z=(x-m)/s

C.ComputePr(z≤b)whereis the z-value found in Part B. (Remember to use thebfunction for this.) Label the probability. Compare the results in Parts A and C and in a text box explain the significance.NormSDistProblem 3. In this problem you are to compare the shapes of two normal distributions with mean 100. One has standard deviation 10 and the other 5. In a new sheet enter numbers from 70 to 130 in column A using rows 3 to 63. For the first distribution enter

in cell B3. Enter a similar formula in cell C3 that gives the values of a distribution with standard deviation 5. Copy the formulas in cells B3 and C3 to apply to all values in the range A3:A63. Graph the two functions in columns B and C in the same chart using the values in column A as the x-values. The Chart type should be a "Scatter Diagram with points connected by Smooth lines Without Markers". In a text box compare the graphs of the two distributions.=NormDist(A3,100,10,0)Problem 4. In this problem you are to see how the

function works. Let X be a normal distribution with mean 100 and standard deviation 20.NormInv

A.We want to find thexso that 40% of the distribution values are below it, that is, we want to find the value that gives the 40th percentile. To find this x, enter the formulain a cell and label it. Next, check that it works by computing=NormInv(0.4,100,20)where x is the value just found. What does it give? Lable it.NormDist(x,100,20,1)

B.Find thexso that 20% of the distribution values exceed it. Label the value and check it.Problem 5. You now apply what you have learned. A company finds that the length of phone calls to its tech support staff is normally distributed with a mean of 9.6 minutes and standard deviation of 2.8 minutes. Find (and label your answers):

A.the probability a call lasts between 8 to 12 minutes.

B.the percent of call that last less than 5 minutes.

C.the percent of calls that last more than 15 minutes.

D.the length of time the longest 10% of the calls take.