Question

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.5 millimeters (mm) and a standard deviation of 1.5 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.)

(a) the thickness is less than 3.0 mm


(b) the thickness is more than 7.0 mm


(c) the thickness is between 3.0 mm and 7.0 mm

Answer 1

answers:

a 0.1587

b 0.0478

c 0.7935

Answer 2

a) 0.1587 (or 15.87%)

b) 0.0478 (or 4.78%)

c) 0.7935 (or 79.35%)

Explanation:

Find the following probabilities:

(a) the thickness is less than 3.0 mm

We need to find the area under the Normal curve with a mean of 4.5 mm and a standard deviation of 1.5 mm to the left of 3 mm

(See picture 1 attached)

We can do that with the help of a calculator or a spreadsheet.

In Excel use

NORMDIST(3,4.5,1.5,1)

In OpenOffice Calc use

NORMDIST(3;4.5;1.5;1)

and we get the value 0.1587 (or 15.87%)

(b) the thickness is more than 7.0 mm

Now we need the area to the right of 7 (1 - the area to the left of 7)

(See picture 2 attached)

In Excel use

1-NORMDIST(7,4.5,1.5,1)

In OpenOffice Calc use

1-NORMDIST(7;4.5;1.5;1)

and we get the value 0.0478 (or 4.78%)

(c) the thickness is between 3.0 mm and 7.0 mm

We are looking for the area between 3 and 7

(See Picture 3)

Since the area under the Normal equals 1, we have

Area to the left of 3 + Area between 3 and 7 + Area to the right of 7 = 1

Hence,

0.1587 + Area between 3 and 7 + 0.0478 = 1

and

Area between 3 and 7 = 1 - 0.1587 - 0.0478 = 0.7935 (or 79.35%)