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

a) 0.1587

b) 0.0475

c) 0.7938

Explanation:

Let's start defining our random variable.

X : ''Thickness (in mm) of ancient prehistoric Native American pot shards discovered in a Hopi village''

X is modeled as a normal random variable.

X ~ N(μ,σ)

Where μ is the mean and σ is the standard deviation.

To calculate all the probabilities, we are going to normalize the random variable X.

We are going to call to the standard normal distribution ''Z''.

[(X - μ) / σ] ≅ Z

We normalize by subtracting the mean to X and then dividing by standard deviation.

We can find the values of probabilities for Z in a standard normal distribution table.

We are going to call Φ(A) to the normal standard cumulative distribution evaluated in a value ''A''

a)

`P(X<3)=P((X-4.5)/(1.5)<(3-4.5)/(1.5))`

`P(Z<-1)=`Φ(-1) = 0.1587

b)

`P(X>7)=P((X-4.5)/(1.5)>(7-4.5)/(1.5))`

`P(Z>1.666)=1-P(Z\leq 1.666)=`

1 - Φ(1.666) = 1 - 0.9525 = 0.0475

c)

`P(3<X<7)=P((3-4.5)/(1.5)<(X-4.5)/(1.5)<(7-4.5)/(1.5))=`

`P(-1<Z<1.666)=` Φ(1.666) - Φ(-1) = 0.9525 - 0.1587 = 0.7938