Question

Suppose that the random variable x is normally distributed with μ = 1,000 and standard deviation σ = 100.

Find each of the following probabilities. Round your z-score calculations to 2 decimal places. Provide your probability answers to 4 decimal places.

	z-score	probability
P( x > 1257)
P( x < 1035)
P( x ≤ 700)

	z-score	z-score	probability
P(1000 ≤ x ≤ 1200)
P(812 ≤ x ≤ 913)

Answer 1

Solution :

Given that ,

mean = = 1000

standard deviation = = 100

P(x >1257 ) = 1 - P(x <1257 )

= 1 - P((x - ) / < (1257-1000) / 100)

= 1 - P(z <2.57 )

= 1 - 0.9949   

= 0.0051

z-score = 2.57

Probability = 0.0051

P(x 700 ) = P((x - ) / (700-1000) /100 )

= P(z -3.00 )

= 0.0013 Using standard normal table

z-score = -3.00

Probability = 0.0013

P(1000 x 1200) = P((1000-1000 /100 ) (x - ) / (1200-1000 /100 ) )

P(1000 x 1200)  = P( 0.00 z 2.00 )

P(1000 x 1200) = P(z 2.00) - P(z 0.00 )

P(1000 x 1200) = 0.9772 - 0.5000

z-score = 2.00

z-score = 0.00

Probability = 0.4772

P(812 x 913) = P((812-1000 / 100) (x - ) / (913-1000 /100 ) )

P(812 x 913)  = P(-1.88 z -0.87 )

P(812 x 913) = P(z -0.87 ) - P(z -1.88 )

P(812 x 913) = 0.1922 - 0.0301

z-score = -0.87

z-score = -1.88

Probability = 0.1621