Normal mean 60, sd 5: value with 95% below it Step by Step

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The question

Question: X is normal, with mean 60 and SD 5: which value has 95% below it?

μ = 60, σ = 5: x with 95% below it?

A value with 95 percent below it is also called the 95th percentile.

95% below it = the 95th percentile

1 Shade the area

The bell, with the mean 60 in the middle.

x is the cut-off we want.

95 percent is below x: we shade it, on the left of x.

2 Area to the left

The z-table works with the area to the left of z.

z-table: area to the left of z

95 percent is 0.95.

It is already on the left.

area to the left = 0.95

Careful!

More than half the bell is on the left.

So x is above the mean, and z is positive.

Careful: more than half on the left: z is positive

3 Find it inside the table

Now we look for 0.9500 inside the table.

0.9500 is halfway between 0.9495 and 0.9505.

So z is halfway between 1.64 and 1.65: 1.645.

4 Back to x

Start at the mean, then go z SDs: x is the mean plus z times the SD.

x = μ + z × σ

1.645 times 5 is 8.225.

60 plus 8.225 is 68.225.

x = 60 + 1.645 × 5 = 60 + 8.225 = 68.225

5 Check

Let's check: the z of 68.225 is 68.225 minus 60, over 5.

That is 1.645.

z = 68.225 − 605 = 1.645

The table at 1.65 gives 0.9505: about 0.95 to the left.

So it is right!

table at z = 1.65: 0.9505 ≈ 0.95

Let's look back.

First: shade the area.

Then: area to the left.

Then: find it inside the table.

Then: back to x.

Last: check.

Answer