Finding  and/or  if one/two probabilities are known 50 marks is the pass mark20% of students fail 85 marks is a grade A5% of students get A‘s P(Z<-0.842) = % 5% Z = Z = P(Z<-1.645) = 0.95 If 5% get more than 85 then 95% get less than 85 From invnormal (0.95)  (1.645) = % 5% So the cumulative probability up to Z = is 0.95 If 20% get less than 50 then From the invnormal (0.2)  (-0.842) = 0.2 So the cumulative probability up to Z = is 0.2 Remember: draw a diagram of the original normal data followed by a diagram of the standardised data

Using Z = If x = 50 marks = Solving simultaneously  and  = 14 marks If x = 85 marks =  +  =  +  = 85

Solution: Using InvNormal(0.15) and InvNormal(0.75), e.g. 2 Find  and  if and We have and N.B. is negative So, and

We must solve simultaneously: and We can change the signs in the 1 st equation and add:Adding: Substitute into either equation to find  : e.g.

Solution: Let X be the random variable “length of rod (cm)” e.g.3 The lengths of a batch of rods follows a Normal distribution. 10% of the rods are longer than 101 cm. and 5% are shorter than 95 cm. Find the mean length and standard deviation. and So, 050)( 1  zZP

and Solving simultaneously: We can change the signs in the 2 nd equation and add: Substitute into either equation to find  : Adding: e.g.3 The lengths of a batch of rods follows a Normal distribution. 10% of the rods are longer than 101 cm. and 5% are shorter than 95 cm. Find the mean length and standard deviation.

Exercise and 1.Find the values of  and  if 2. A large sample of light bulbs from a factory were tested and found to have a life-time which followed a Normal distribution.  of the bulbs failed in less than  hours and  lasted more than  hours. Find the mean and standard deviation of the distribution.

Solution: Using the Percentage Points of the Normal Distribution table, We have and So, and 1. and and

Solving simultaneously: and We can change the signs in the 1 st equation and add: Adding: Substitute into either equation to find  : e.g. and

Solution: Let X be the random variable “life of bulb (hrs)” So, 2. A large sample of light bulbs from a factory were tested and found to have a life-time which followed a Normal distribution.  of the bulbs failed in less than  hours and  lasted more than  hours. Find the mean and standard deviation of the distribution. We know that and So, and

Using the InverseNormal(0.25) and InverseNormal(0.85), So, and z 1 is negative Adding:

Solution: and where Using the InverseNormal(0.8), So, It doesn’t matter whether we find the z value first or use the standardising formula first. e.g.1 Find the values of  in the following:  (0.8416) = % 20% –1 –1 –2 –2 –3 –3 –4 –4 50 means Remember: draw a diagram of the original normal data followed by a diagram of the standardised data Finding the mean or S.D

Tip: It’s easy to make a mistake and add instead of subtract or vice versa so check that your answer is reasonable by comparing with the information in the question. We had The mean is clearly less than 50 so the answer is reasonable.

and where Using the InvNormal(0.95), So, Find the values of the unknown in the following: Solution:  (1.6449) = means Remember: draw a diagram of the original normal data followed by a diagram of the standardised data

Exercise 1. Find the values of the unknowns in the following: (a) and (b) and

Solution: (a) and where Using the InvNormal(0.84), So,

Solution: where Using InvNormal(0.97), So, (b) and