Question

For a normal distribution with a mean of μ = 85 anda standard deviation of o= 20, find the proportion ofthe population corresponding to each of the following.a. Scores greater than 89b. Scores less than 72c. Scores between 70 and 100

Answer 1

a. 0.4207

b. 0.2578

c. 0.5468

Explanation:

Given:

μ = 85

σ = 20

Now, the probabilities are obtained, after obtaining the z-score for the given corresponding sample data points. We calculate the value of z as follows:

`z=(x-\mu)/(\sigma)`

We calculate for each case:

a. P( x > 89)

`\begin{gathered} P\left(x>89\right)=1-P\left(x<89\right) \\ \\ z=(89-85)/(20)=(4)/(20)=0.2 \end{gathered}`

We locate this value in the normal table:

Therefore:

`\begin{gathered} P(x\gt89)=1-P(x\lt89)=1-0.5793 \\ \\ P(x\gt89)=0.4207 \end{gathered}`

b. P (x < 72)

`\begin{gathered} P\left(x<72\right) \\ \\ z=(72-85)/(20)=(-13)/(20)=0.65 \end{gathered}`

We locate this value in the normal table:

Therefore:

`P(x<72)=0.2578`

c. P (70 < x < 100)