Question

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.8 millimeters (mm) and a standard deviation of 1.7 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

Solution:

Given: Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.8 millimeters (mm) and a standard deviation of 1.7 mm.

That is: X ~ Normal Distribution ( Mean = , SD = )

Part a) Find probability that the thickness is less than 3.0 mm
That is : P( X < 3.0 ) = ........?

Find z score for x = 3.0

Thus we get:

P( X < 3.0) = P( Z < -1.06)

Look in z table for z =-1.0 and 0.06 and find area , which gives P( Z < -1.06)= 0.1446

Thus

P( X < 3.0) = P( Z < -1.06)

P( X < 3.0) = 0.1446

Part b) Find probability that the thickness is more than 7.0 mm
That is : P( X > 7.0 ) = ........?

Find z score for x = 7.0

Thus we get :

P( X > 7.0) = P( Z > 1.29)

P( X > 7.0) = 1 - P( Z < 1.29)

Look in z table for z = 1.2 and 0.09 and find area, which gives P( Z < 1.29) = 0.9015

Thus

P( X > 7.0) = 1 - P( Z < 1.29)

P( X > 7.0) = 1 - 0.9015

P( X > 7.0) =0.0985

Part c) Probability that the thickness is between 3.0 mm and 7.0 mm

That is:

P( 3.0 < X < 7.0) = .........?

P( 3.0 < X < 7.0) =P( X < 7.0) - P( X < 3.0)

P( 3.0 < X < 7.0) =P( Z < 1.29) - P( Z < -1.06)

From part a) P( Z < -1.06) =0.1446 and from part b) P( Z < 1.29) = 0.9015

Thus

P( 3.0 < X < 7.0) = 0.9015 - 0.1446

P( 3.0 < X < 7.0) = 0.7569