Statistics Help

Question

A distribution of values is normal with a mean of 240 and a standard deviation of 3.

Find the interval containing the middle-most 36% of scores: 

Enter your answer using interval notation. Example: (2.1,5.6)

Your numbers should be accurate to 1 decimal places.

Answer 1

Hi Jennifer,

First of all, we need to understand "middle 36%." That means the 36 values closest to the 50th percentile--on both the positive and negative side. So we want corresponding values for [(0.50 - 0.18), (0.50 + 0.18)]. Thus:

(0.32, 0.68). Now, with those probabilities in mind, this problem uses the classic equation in elementary statistics:

z=(x-mu)/sigma

Breaking this down:

z=z-score, obtained from z-table

x=value we are looking for

mu=mean

sigma=standard deviation

Now, for this problem, we were given a mean, standard deviation, and two probabilities (percentiles) 0.32 and 0.68. That means our first task is to go to the z-table and look those probabilities up in the interior. The closest to 0.32 is 0.3192, which corresponds to a z-score of -0.47. The closest to 0.68 is 0.6808 which corresponds to a z-score of positive 0.47.

Now, returning to our initial equation:

z=(x-mu)/sigma

We've got our two z-values and we need our two x-values, so I'll compute as x₁, x₂ and use z₁ and z_2.

We now have:

z₁=(x₁-mu)/sigma

z₂=(x₂-mu)/sigma

I'll start with x_1.

z₁= -0.47

x₁=x₁

mu=240

sigma=3

Substituting:

-0.47=(x₁-240)/3

Multiply both sides by 3:

-1.41=x₁ - 240

Add 240 on both sides:

x₁=238.59. to one decimal:

x₁₌238.6

Now, for x₂:

z₂=(x_2-mu)/sigma

Substituting:

z₂= 0.47

x₂=x₂

mu=240

sigma=3

0.47=(x₂-240)/3

Multiply both sides by 3:

1.41=x₂ - 240

Add 240 on both sides:

x₂=241.41 To one decimal:

x₂=241.4

Range in interval notation is (x_1, x₂), which is:

(238.6, 241.4)

I hope this helps.