Question

A particular fruit's weights are normally distributed, with a mean of 214 grams and a standard deviation of 21 grams. If you pick 2 fruit at random, what is the probability that their mean weight will be between 187 grams and 198 grams

Answer 1

Given that a particular fruit's weight are normally distributed with a mean of 214 grams and a standard deviation of 21 grams.

Now 2 fruit at random are picked we need to find the probability that their mean weight will be between 187 grams and 198 grams.

Before we go on to solve the problems let us know a bit about Normal Distribution and its properties

Normal Distribution

A continuous random variable X is said to have a normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Notation:

Standard Normal Distribution

A continuous random variable X is said to have a standard normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Exact evaluation of ?(x) is not possible but numerical method can be applied. The values of ?(x) has been tabulated extensively in Biometrika Volume I.

Notation:

Some Properties of Normal Distribution

1. If X~Normal(μ,σ²)

2. If X~Normal(μ_i,σ_i²), i=1(1)n independently then

This is called the reproductive property of the Normal Distribution.

3.If X₁,X₂,...,X_n~Normal(μ,σ²) identically and independently then using property 2 we can say that,

Coming back to our problem,

Given in our problem that two random fruits are picked,

Therefore the probability that their mean weight will be between 187 grams and 198 grams is 0.1057