Question

If X follows a normal distribution with µ = 100 and σ = 10, what is the probability that X > 90?99.37%97.72%84.13%69.14%

Answer 1

The probability that X > 90 is 84.13%

SOLUTION

Problem Statement

The question requires us to find the probability that X is greater than 90 if X is a variable from a normal distribution with a population mean (µ) of 100 and population standard deviation(σ) of 10.

Method

- The normal distribution gives the probability density curve that helps us predict the probability of a certain event happening. This probability is represented by the area under this normal distribution curve.

- The event in question here is when X > 90. We need to find the area under the curve where this inequality is true; this area under the curve will give us the probability that X > 90 as requested in the question. This area is depicted below:

- To find the area under the curve (or Probability), we need to:

1. Find the Z-score associated with the inequality X >90 given a normal distribution of population mean of 100 and population standard deviation of 10. We use the formula below to find the Z-score:

`\begin{gathered} Z=(X-\mu)/(\sigma) \\ \text{where,} \\ \mu=\text{population mean} \\ \sigma=\text{population standard deviation} \end{gathered}`

2. Convert the Z-score to a probability using a Z table or a Z-score calculator. This probability is the probability we are looking for

Implementation

1. Find the Z-score:

2. Convert Z-score to probability:

Let us check the Z-distribution table to find the probability that corresponds to the Z-score.

In the Z-score table, we have:

This Z-score represents P(Z< -1) = 0.1587 (according to the table itself). But because we are looking for P(Z > -1), we need to subtract our answer from 1 to get the area under the curve to the right of X > 90.

That is:

`\begin{gathered} P(Z>-1)=1-P(Z<-1) \\ P(Z>-1)=1-0.1587 \\ \\ \therefore P(Z>-1)=0.84134 \\ \\ \text{ The probability is given in percentage, so:} \\ P(Z>-1)=0.84134*100\text{ \%} \\ \\ P(Z>-1)=84.134\text{ \% }\approx84.13\text{ \%} \end{gathered}`

Final Answer

The probability that X > 90 is 84.13%