Question

Thickness measurement ancient prehistoric Native American Pot Shards discovered in Hopi Village are approximately normally distributed with the mean of 5.1 millimeters and standard deviation of 0.9 millimeters. For a randomly found shard, What is the probability that the thickness is: a) Less than 3.0 millimeters b) More than 7.0 millimeters Present your answer in three decimal places. Present your answer: answer for "a",answer for "b" Group of answer choices

Answer 1

Answer:

a) $P(X$

And we can find this probability using the normal standard table and we got:

$P(z$

b) $P(X>7)=P(\frac{X-\mu}{\sigma}>\frac{7-\mu}{\sigma})=P(Z>\frac{7-5.1}{0.9})=P(z>2.11)$

And we can find this probability using the complement rule and the  normal standard table and we got:

$P(z>2.11)=1-P(Z$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

$X \sim N(5.1,0.9)$  

Where $\mu=5.1$ and $\sigma=0.9$

Part a

We are interested on this probability

$P(X$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X$

And we can find this probability using the normal standard table and we got:

$P(z$

Part b

We are interested on this probability

$P(X>7)$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X>7)=P(\frac{X-\mu}{\sigma}>\frac{7-\mu}{\sigma})=P(Z>\frac{7-5.1}{0.9})=P(z>2.11)$

And we can find this probability using the complement rule and the  normal standard table and we got:

$P(z>2.11)=1-P(Z$