Question

Let x be a continuous random variable that is normally distributed with a mean of 70 and a standard deviation of 14. Find to 4 decimal places the probability that x assumes a valuea. less than 55b. greater than 90c. greater than 51d. less than 78

Answer 1

In order to determine the required probabilities it is necessary to calculate the z-score for each case and then we can use a normal distribution table to find the associated probability for a certan value of the z-score.

a) For P(X < 55), that is the probability for x is less than 55.

`Z=(X-\mu)/(\sigma)=(55-70)/(14)\approx-1.07`

Then P(X<55) = P(Z<-1.07). It means that we have to searh in a normal distribution table the previous value of Z to determine the probability. For instance, in this case:

The intersection of the two rectangles gives us the probability.

By searhing in a normal distribution table you have:

P(Z<-1.07) = 0.1423

The same procedure is applied to the rest of the points in this problem.

b) For P(Z>90):

`Z=(X-\mu)/(\sigma)=(90-70)/(14)\approx1.43`

Then P(X>90) = 1 - P(Z>1.43). This is used because of the organization of standard normal distribution tables.

P(Z>1.43) = 1 - 0.9236 = 0.0764

c) For P(X>51):

`Z=(51-70)/(14)\approx-1.36`

P(X>51) = 1 - P(Z>-1.36) = 1 - 0.0869 = 0.9130

d) For P(X<78):

`Z=(78-70)/(14)=0.57`

P(X<78) = P(Z<0.57) = 0.7156