Question

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.9 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 .087 Correct: Your answer is correct. (b) the thickness is more than 7.0 mm

Answer 1

Answer:

a) 0.0869 = 8.69% probability that the thickness is less than 3.0 mm

b) 0.0668 = 6.68% probability that the thickness is more than 7.0 mm

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 4.9, \sigma = 1.4$

(a) the thickness is less than 3.0 mm

This is the pvalue of Z when X = 3.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3 - 4.9}{1.4}$

$Z = -1.36$

$Z = -1.36$ has a pvalue of 0.0869

0.0869 = 8.69% probability that the thickness is less than 3.0 mm

(b) the thickness is more than 7.0 mm

This is 1 subtracted by the pvalue of Z when X = 7. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{7 - 4.9}{1.4}$

$Z = 1.5$

$Z = 1.5$ has a pvalue of 0.9332

1 - 0.9332 = 0.0668

0.0668 = 6.68% probability that the thickness is more than 7.0 mm