Question

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 5.1 millimeters (mm) and a standard deviation of 1.4 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

a)
Here, μ = 5.1, σ = 1.4 and x = 3. We need to compute P(X <= 3). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (3 - 5.1)/1.4 = -1.5

Therefore,
P(X <= 3) = P(z <= (3 - 5.1)/1.4)
= P(z <= -1.5)
= 0.0668

b)
z = (x - μ)/σ
z = (7 - 5.1)/1.4 = 1.36

Therefore,
P(X >= 7) = P(z <= (7 - 5.1)/1.4)
= P(z >= 1.36)
= 1 - 0.9131
= 0.0869

c)
z = (x - μ)/σ
z1 = (3 - 5.1)/1.4 = -1.5
z2 = (7 - 5.1)/1.4 = 1.36

Therefore, we get
P(3 <= X <= 7) = P((7 - 5.1)/1.4) <= z <= (7 - 5.1)/1.4)
= P(-1.5 <= z <= 1.36) = P(z <= 1.36) - P(z <= -1.5)
= 0.9131 - 0.0668
= 0.8463