Question

Whe weights of United States pennies are distributed approximently normal with a mean of 2.5 grams and a standard deviation of .03 grams. What is the propability that a randmoly chosen penny weighs less than 2.49 grams

Answer 1

Probability that a randomly chosen penny weighs less than 2.49 grams is 0.48803.

Explanation:

We are given that the weights of United States pennies are distributed approximately normal with a mean of 2.5 grams and a standard deviation of 0.03 grams.

Let X = weights of United States pennies

So, X ~ N(
`\mu=2.5,\sigma^(2) = 0.03^(2)`)

The z score probability distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = population mean = 2.5 grams

`\sigma` = standard deviation = 0.03 grams

So, probability that a randomly chosen penny weighs less than 2.49 grams is given by = P(X < 2.49 grams)

P(X < 2.49) = P(
`(X-\mu)/(\sigma)` <
`(2.49 -2.5)/(0.03)` ) = P(Z < -0.33) = 1 - P(Z
`\leq` 0.03)

= 1 - 0.51197 = 0.48803

Therefore, probability that a randomly chosen penny weighs less than 2.49 grams is 0.48803.

Answer 2

37.07% probability that a randmoly chosen penny weighs less than 2.49 grams

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 = (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 = 2.5, \sigma = 0.03`

What is the probability that a randmoly chosen penny weighs less than 2.49 grams

This is the pvalue of Z when X = 2.49. So

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

`Z = (2.49 - 2.5)/(0.03)`

`Z = -0.33`

`Z = -0.33` has a pvalue of 0.3707

37.07% probability that a randmoly chosen penny weighs less than 2.49 grams