Question

suppose a normally distributed random variable x has a mean of 100 and P(x < 90) = 0.40. what is the probability that X is between 90 and 110?​

Answer 1

Answer:

0.50

Step-by-step explanation:

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Answer 2

Probability that X is between 90 and 110 = 0.2 if  a normally distributed random variable x has a mean of 100 and P(x < 90) = 0.40.

Given:

• A normally distributed random variable x has a mean of 100
• P(x < 90) = 0.4

To Find:

• the probability that X is between 90 and 110

Solution:

Using Shortcut Method:

Mean is 100

Hence P (< 100) = 0.5  as Mean is centered

Hence

P  (90 < X < 100) = 0.5 - 0.4 = 0.1

Difference between 100 and 90 is 10

Difference between 100 and 110 is 10

and 100 is Mean

Hence P  (100 < X < 110) = P  (90 < X < 100) = 0.1

Adding Both

P ( 90 < X < 110) = 0.1 + 0.1 = 0.2

Probability that X is between 90 and 110 = 0.2

Detailed Method using z score table.

Z  = ( Value - Mean) /SD

Z score for 0.4 = -0.254

-0.253 = (90 - 100)/SD

SD ≈ 39.53

Z = ( 110 - 100)/39.53

=>  Z = 0.253

For z score  < 0.253 , probability is 0.6

Hence Probability in Between is 0.6 - 0.4 = 0.2

Probability that X is between 90 and 110 = 0.2

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