Question

An apple juice producer buys all his apples from a conglomerate of apple growers in one northwestern state. The amount of juice obtained from each of these apples is approximately normally distributed with a mean of 2.25 ounces and a standard deviation of 0.15 ounce. What is the probability that a randomly selected apple will contain more than 2.50 ounces? .9525 .4525 .0475 .5474

Answer 1

The probability that randomly selected apple will contain more than $2.50 \text{ }\text{ounce}$ juice is $\fbox{0.0475}$.

Further explanation:

Given:

The mean $\mu$ of apple juice is $2.50 \text{ }\text{ounce}$.

The standard deviation $\sigma$ of apple juice is $0.15 \text{ }\text{ounce}$.

Calculation:

The random variable $\bar{X}$ follow Normal distribution with parameter mean and variance.

Normal distribution is symmetric and it bell shaped curve.

$\bar{X} \sim \text{Normal}(\mu,\sigma^{2} )$

Variance is square of standard deviation.

Variance = $\sigma^{2}$

Variance = $0.15^{2}$

Variance = $0.0225$

$\bar{X}\sim \text{Normal}(\mu,\sigma^{2} )$

Substitute $2.25$ for $\mu$ and $0.0225$ for $\sigma^{2}$

$\bar{X}\sim\text {Normal}(2.25,0.0225)$

The probability that randomly selected apple will contain more than $2.50$ can be calculated by applying the central Limit Theorem as,

$\text{Probability}=P(\bar{X}>2.50)}\\\text{Probability}=P\left(\dfrac{{\bar{X}-\mu}}{\sigma}>\dfrac{{2.50-2.25}}{0.15}\right)\\\text{Probability}=P\left(Z}>\dfrac{{0.25}}{0.15}\right)\\ \text{Probability}=P\left(Z}>\dfrac{{5}}{3}\right)\\\text{Probability}=P(Z}>1.67})$

The Normal distribution is symmetric.

Therefore, the probability of greater than $1.67$ is equal to the probability of less than $1.67$.

$P(Z>1.67})=1-P(Z1.67})=1-0.9525\\P(Z>1.67})=0.0475$

Hence, the probability that randomly selected apple will contain more than $2.50 \text{ }\text{ounce}$ juice is $\fbox{0.0475}$.

Learn More:

1. Learn more about Unit conversion brainly.com/question/4837736

2. Learn more about Range of a function brainly.com/question/1435353

3. Learn more about Graph representation brainly.com/question/2491745

Answer Details:

Grade: College Statistics

Subject: Mathematics

Chapter: Probability and Statistics

Keywords:

Probability, Statistics, Apple juice, Normal distribution, Normal approximation, Central Limit Theorem, Z-table, Mean, Standard deviation, Symmetric, Variance, Conglomerate of Apple, Northwestern.