Unit 2 Revision 2

Unit 2 Revision 2 Continuation from Revision 4, but now, we’re doing the REVERSE process. Example 1: Find the value of k given that P(X < k) = 0.1986 and X ~ N (32,8) . Solution & Explanation: The concepts that we’ve discussed in Revision 4 earlier are to be utilised. 1. Area under the curve = 1 (That’s probability = 1, the sum of probabilities)

Area = 0.5

Area = 0.5

0 Values on this line are the z-values called critical values 2. The Standardized Normal Distribution Table gives only HALF the area, that is the area to the right of the middle line where mean = 0. 3. The formula for standardization is P ( X < x) = P ( Z <

x−µ ). σ

Referring to the question, we have P( X < k ) = P( Z <

k −µ k − 32 ) = P( Z < ) = 0.1986 σ 8

0.1986 is the area under the curve and it is less than 0.5, so this area is the area indicated in the graph below: Area = 0.3014

Area = 0.1986 z-value = – 0.85

z-value = 0.85

But, the Table does not allow us to find the z-value for this area. Referring to the Table for area of 0.3014 ( the nearest value is 0.3023), and the z-value is 0.85; 1

So, the corresponding z-value for the light blue area is – 0.85. Thus,

k − 32 = −0.85 8

⇒ k − 32= − 2.4⇒ k= 29.6

Example 2 If X ~ N (41,9) , find the value of k given that P ( X > k ) = 0.732 (i) P ( X < k ) = 0.386 (ii) Solution: (i) First standardised the expression,… k − 41   P( X > k ) = P  Z >  = 0.732 3   Area = 0.232 (Yellow) Area = 0.732 Area = 0.5 (Blue) Since the area is 0.732, the z-value falls in the negative region. To find the z-value, the Table gives “0.62”, since z-value is negative, we have k − 41 = −0.62 3 ⇒ k = −1.86 + 41 = 39.14 (ii)

k − 41   P( X < k ) = P  Z <  = 0.386 3  

Area = 0.386

Area = 0.114 Area = 0.386

The area lies in the negative region of z-value. The Table gives z-value as 0.29 (using Area = 0.114); thus we have k − 41 = −0.29 3

⇒ k=− 0.87 + =41 40.13

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Now, try these exercises: IF you’ve difficulties to imagine, always make a sketch to assist you. Exercises 1.

If X ~ N (80,16) , find the value of k given that P ( X > k ) = 0.643 (i) P ( X < k ) = 0.286 (ii)

2.

If X ~ N (38,9) , find the value of k given that P ( X > k ) = 0.828 (i) P ( X < k ) = 0.42 (ii) P ( X < k ) = 0.285 (iii)

Answers below……………..

1.

(i)

78.52

(ii)

77.72

2.

(i)

35.15

(ii)

37.4

(iii)

36.29

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