Question

A particular fruit's weights are normally distributed, with a mean of 344 grams and a standard deviation of 10 grams. If you pick 10 fruit at random, what is the probability that their mean weight will be between 334 grams and 354 grams

Answer 1

0.9984 = 99.84% probability that their mean weight will be between 334 grams and 354 grams.

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Mean of 344 grams and a standard deviation of 10 grams.

This means that
`\mu = 344, \sigma = 10`

Sample of 10:

This means that
`n = 10, s = (10)/(√(10))`

What is the probability that their mean weight will be between 334 grams and 354 grams?

This is the p-value of Z when X = 354 subtracted by the p-value of Z when X = 334.

X = 354

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

By the Central Limit Theorem

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

`Z = (354 - 344)/((10)/(√(10)))`

`Z = 3.16`

`Z = 3.16` has a p-value of 0.9992.

X = 334

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

`Z = (334 - 344)/((10)/(√(10)))`

`Z = -3.16`

`Z = -3.16` has a p-value of 0.0008.

0.9992 - 0.0008 = 0.9984

0.9984 = 99.84% probability that their mean weight will be between 334 grams and 354 grams.