Question

Samples of Subsoil PH level in a location are approximately normally distributed with mean 6.15 and standard deviation 0.45

(1) Find the probability that randomly selected subsoil sample from this region where the PH level between 5.97 & 7.59

(2) Find a PH level such that 1% of all subsoil sample from this region will have more than this PH level.

(3) what is the probability that in the sample of 60 subsoil PH level , at most 22 will be between 5.97 & 7.59

Answer 1

Given that samples of Subsoil pH level in a location are approximately normally distributed with mean 6.15 and standard deviation 0.45.

Let,

X= Subsoil pH level in a location

Before we go on to solve the problems let us know a bit about normal distribution.

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:

Note:-

Coming back to our problem

X= Subsoil pH level in a location

(1) We need to find the probability that a randomly selected subsoil sample from this region where the pH level lies between 5.97 & 7.59.

Hence the probability that a randomly selected subsoil sample from this region where the pH level lies between 5.97 & 7.59 is 0.6547

(2) We need to find a pH level such that 1% of all subsoil sample from this region will have more than this pH level

Hence 7.1985 is the pH level such that 1% of all subsoil sample from this region will have more than this pH level.

(3) We need to find the probability that in the sample of 60 subsoil pH level , at most 22 will be between 5.97 & 7.59

From the (1) part we know that,

Let,

Y= Number of subsoil samples whose pH will be between 5.97 & 7.59 out of the 60 subsoil samples.

What is a binomial distribution?

A discrete random variable X is said to have a binomial distribution if its PMF(Probability Mass Function) is given by,

Now since n is large we can use the normal approximation to the binomial and also,

np=60*0.6547=39.282 [Which is greater than 5]

n*(1-p)=60*0.3453=20.718 [Which is greater than 5]

Hence we can use normal approximation to the binomial.

Normal approximation to the binomial

If Y~Binomial(n,p). Then,

Continuity Correction Factor

When a continuous distribution is used to approximate a discrete discrete we use the continuity correction factor.

Coming back to our problem

Y= Number of subsoil samples whose pH will be between 5.97 & 7.59 out of the 60 subsoil samples.

We need to find the probability that in the sample of 60 subsoil pH level , at most 22 will be between 5.97 & 7.59

[Applying Continuity Correction]

Hence the probability that in the sample of 60 subsoil pH level , at most 22 will be between 5.97 & 7.59 is approximately 0