Question

The weight of people in a small town in Missouri is known to be normally distributed with a mean of 186 pounds and a standard deviation of 29 pounds. On a raft that takes people across the river, a sign states, "Maximum capacity 3,417 pounds or 17 persons." What is the probability that a random sample of 17 persons will exceed the weight limit of 3,417 pounds?

Answer 1

Answer:

the probability that a random sample of 17 persons will exceed the weight limit of 3,417 pounds is 0.0166

Step-by-step explanation:

The summary of the given statistical data set are:

Sample Mean = 186

Standard deviation = 29

Maximum capacity 3,417 pounds or 17 persons.

sample size = 17

population mean =3417

The objective is to determine the probability  that a random sample of 17 persons will exceed the weight limit of 3,417 pounds

In order to do that;

Let assume X to be the random variable that follows the normal distribution;

where;

Mean $\mu$ = 186 × 17 = 3162

Standard deviation = $29* \sqrt{17}$

Standard deviation = 119.57

$P(X>3417) = P(\dfrac{X - \mu}{\sigma}>\dfrac{X - \mu}{\sigma})$

$P(X>3417) = P(\dfrac{3417 - \mu}{\sigma}>\dfrac{3417 - 3162}{119.57})$

$P(X>3417) = P(Z>\dfrac{255}{119.57})$

$P(X>3417) = P(Z>2.133)$

$P(X>3417) =1- 0.9834$

$P(X>3417) =0.0166$

Therefore; the probability that a random sample of 17 persons will exceed the weight limit of 3,417 pounds is 0.0166