Question

A population of values has a normal distribution with μ = 180.2 and σ = 17.4 . You intend to draw a random sample of size n = 235 . Find the probability that a single randomly selected value is between 177.2 and 177.7. P(177.2 < X < 177.7) = Find the probability that a sample of size n = 235 is randomly selected with a mean between 177.2 and 177.7. P(177.2 < M < 177.7) =

Answer 1

First let see what are given to us:

• There is a population of values that follow a normal distribution with μ=180.2 and σ=17.4
• We intend to draw a random sample of size n=235

First we need to find the probability that a single randomly selected value is between 177.2 and 177.7

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 given problems,

a.

We need to find the probability that a single randomly selected value is between 177.2 and 177.7

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