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 ) 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

a) 0.1587

b) 0.0475

c) 0.7938

Step-by-step 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)

Φ(-1) = 0.1587

b)

1 - Φ(1.666) = 1 - 0.9525 = 0.0475

c)

Φ(1.666) - Φ(-1) = 0.9525 - 0.1587 = 0.7938

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