Normal mean 70, sd 5: probability that x is more than 78 Step by Step

Step by step

The question

Question: X is normal, with mean 70 and SD 5: what is the probability that X is more than 78?

P(X > 78): μ = 70, σ = 5

μ is the mean.

σ is the standard deviation, the SD.

μ = the mean, σ = the SD

1 Find z

z counts how many SDs x is from the mean: x minus the mean, over the SD.

z = x − μσ

78 minus 70 is 8.

Divided by 5, that is 1.6.

z = 78 − 705 = 85 = 1.6

2 Draw and shade

The bell: the mean, 70, in the middle.

78 is right of it.

We want X more than 78.

We shade that part.

3 Read the table

The z-table gives the area to the left of z.

Row 1.6, column .00.

For z = 1.6: the row 1.6 and the column .00 meet at 0.9452.

4 The area we want

Careful!

0.9452 is the area to the left.

We want the right side.

Careful: not 0.9452 ✗

The whole bell is 1.

The right side is 1 minus the left: 1 minus 0.9452 is 0.0548.

P(X > 78) = 1 − 0.9452 = 0.0548

5 Check

Let's check another way.

The bell is the same on both sides.

So the area right of 1.6 is the area left of −1.6.

right of 1.6 = left of −1.6

The table at −1.6 gives 0.0548.

The same!

So it is right!

table at −1.6: 0.0548

Let's look back.

First: find z.

Then: draw and shade.

Then: read the table.

Then: the area we want.

Last: check.

Answer