Question

Assume that x has a normal distribution. the mean is muμequals=137.0 and the standard deviation is sigmaσequals=5.3. find the probability that x is between 134.4 and 140.1.

Answer 1

X follows normal distribution with mean μ =137 and standard deviation σ =5.3

We have to find here probability that x is between 134.4 and 140.1

P(134.4 < X < 140.1) = P(X < 140.1) - P(X < 134.4)

To find probability first we have to convert x value into z score values using formula

Z = $\frac{x - mean}{standard deviation}$

For x =140.1 z score value is

Z = $\frac{140.1 - 137}{5.3}$

Z = 0.5849 ~ 0.0.58

Hence P(X < 140.1) = P(Z < 0.58)

Now z score for x= 134.4

Z = $\frac{134.4 - 137}{5.3}$

Z = -0.49

P(X < 134.4) = P(Z < -0.49)

Hence the above probability setup becomes

P(134.4 < X < 140.1) = P(X < 140.1) - P(X < 134.4)

= P(Z < 0.58) - P(Z < -0.49)

Using z score table to find probability corresponding to z= 0.58 and z=-0.49 we get

P(Z < 0.58) = 0.719

P(Z < -0.49) = 0.3121

Using these probabilities into above equation

P(134.4 < X < 140.1) = P(Z < 0.58) - P(Z < -0.49)

= 0.719 - 0.3121

P(134.4 < X < 140.1) = 0.4069

The probability that x is between 134.4 and 140.1 is 0.4069