A normal distribution has a mean of 50 and a standard deviation of 4.
a) Compute the probability of a value between 44.0 and 55.0.
b) Compute the probability of a value greater than 55.0.
c) Compute the probability of a value between 52.0 and 55.0

Solution:

The mean of the Normal Population Distribution = 𝜇 = 50

Standard deviation = 𝝈 = 4

To calculate the probability, the standard error of the mean (z) has to be calculated. By looking up the value of z in the Normal distribution tables the relevant area under the curve will give the probability one is seeking.

z  =  (x - 𝜇)/𝝈 

a) Probability of a value lying between 44 and 55. The relevant curve for seeking the probability is shown below:

finding the P(44<X<55)




 

The z value for x = 44 is   (44 - 50)/4  = -3/2 = -1.5 (-ve indicates left hand side of the mean). The area under the curve upto z = 1.5 from the center  is 0.4332. 

The z value for x = 55 is (55 - 50)/4 = 5/4 = 1.25. The area under the curve upto z = 1.25 from the center is 0.3944.  

The shaded area is the probability that the value will lie between 44 and 55 and is given by the sum of the areas found out from the standard normal distribution tables.

The shaded area = 0.4332 + 0.3944= 0.8276 which implies that the probability of the population  value lying between 44 and 55 is 82.76% or 0.8276.

b) Probability of the value being greater than 55 is represented by the diagram given below:

finding the P(X>55)

The area of interest now is the area under the curve to the right of the vertical line (𝜇 = 55 or z = 1.25). We already know that area under the curve up to z = 1.25 is 0.3944.

We also know that the Normal Probability Distribution curve is symmetrical about the axis z = 0 and the area on either side is 0.5.

Therefore the area under the curve beyond the line z = 1.25 will be 0.5 - 0.3944 = 0.1056. This means that the probability the value will be greater than 55 is 10.56% or 0.1056.

c)The probability value will lie between 52 and 55. The diagram representing this situation is shown below:

finding the P(52<X<55)

The shaded area is the area of interest as it gives the probability that the value will lie between 52 and 55. The z value corresponding to x = 52 is calculated as:

z = (52 - 50)/4  = 0.667

The area under the curve up to x = 52 is 0.2486 (from the tables). The shaded area will be the area under the curve from the center to x = 55 minus the area under the curve from the center to x = 52.

Area of the shaded curve = 0.3944 - 0.2486 = 0.1458.

Therefore the probability that the value will lie between x = 52 and x = 55 is 14.58% or 0.1458.

A normal distribution has a mean of 50 and a standard deviation of 4.

Summary:

A normal distribution has a mean of 50 and a standard deviation of 4.

a) The probability of a value between 44.0 and 55.0 is 0.8276 or 82.76% 

b) The probability of a value greater than 55.0 0.1056 or 10.56 %

c) The probability of a value between 52.0 and 55.0 is 0.1458 or 14.58%


 

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