Question

A normal population has a mean of 22 and a standard deviation of 5.

1. Compute the z value associated with 25.
2. What proportion of the population is between 22 and 25?
3. What proportion of the population is less than 18?

Answer 1

Given Information:

Mean = μ = 22

Standard deviation = σ = 5

Required Information:

1. z-value = ?

2. P(22 < X < 25) = ?

3. P(X < 18) = ?

Answer:

1. z-value = 0.6

2. P(22 < X < 25) = 22.58%

3. P(X < 18) = 21.19%

Explanation:

What is Normal Distribution?

Normal Distribution is a continuous probability distribution and is symmetrical around the mean. The shape of this distribution is like a bell curve and most of the data is clustered around the mean. The area under this bell shaped curve represents the probability

1. We want to find out the z-value associated with 25

`P(X = 25) = P(Z = (x - \mu)/(\sigma) )\\\\P(X = 25) = P(Z = (25 - 22)/(5) )\\\\P(X = 25) = P(Z = (3)/(5) )\\\\P(X = 25) = P(Z = 0.6)\\\\`

Therefore, the z-value associated with X = 25 is 0.6

2. We want to find out the proportion of the population that is between 22 and 25.

`P(22 < X < 25) = P( (x - \mu)/(\sigma) < Z < (x - \mu)/(\sigma) )\\\\P(22 < X < 25) = P( (22- 22)/(5) < Z < (25 - 22)/(5) )\\\\P(22 < X < 25) = P( (0)/(5) < Z < (3)/(5) )\\\\P(22 < X < 25) = P( 0 < Z < 0.6 )\\\\P(22 < X < 25) = P( Z < 0.6 ) - P( Z < 0 ) \\\\`

The z-score corresponding to 0 is 0.50

The z-score corresponding to 0.6 is 0.7258

`P(22 < X < 25) = 0.7258 - 0.50 \\\\P(22 < X < 25) = 0.2258\\\\P(22 < X < 25) = 22.58 \%`

Therefore, the proportion of the population that is between 22 and 25 is 22.58%

3. We want to find out the proportion of the population that is less than 18

`P(X < 18) = P(Z < (x - \mu)/(\sigma) )\\\\P(X < 18) = P(Z < (18 - 22)/(5) )\\\\P(X < 18) = P(Z < (-4)/(5) )\\\\P(X < 18) = P(Z < -0.8)\\\\`

The z-score corresponding to -0.8 is 0.2119

`P(X < 18) = 0.2119\\\\P(X < 18) = 21.19 \%`

Therefore, the proportion of the population that is less than 18 is 21.19%

How to use z-table?

Step 1:

In the z-table, find the two-digit number on the left side corresponding to your z-score. (e.g 1.0, 2.2, 0.5 etc.)

Step 2:

Then look up at the top of z-table to find the remaining decimal point in the range of 0.00 to 0.09. (e.g. if you are looking for 0.6 then go for 0.00 column)

Step 3:

Finally, find the corresponding probability from the z-table at the intersection of step 1 and step 2.