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 0.7 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

Answer 1

0.0051 = 0.51% probability that the thickness is less than 3.0 mm

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean of 4.8 millimeters (mm) and a standard deviation of 0.7 mm.

This means that
`\mu = 4.8, \sigma = 0.7`

(a) Probability that the the thickness is less than 3.0 mm

pvalue of Z when X = 3. So

`Z = (X - \mu)/(\sigma)`

`Z = (3 - 4.8)/(0.7)`

`Z = -2.57`

`Z = -2.57` has a pvalue of 0.0051

0.0051 = 0.51% probability that the thickness is less than 3.0 mm