Question

A sample size 25 is picked up at random from a population which is normally

distributed with a mean of 100 and variance of 36. Calculate-
(a) P-{ X<99}
(b) P{98X

Answer 1

a) P(X < 99) = 0.2033.

b) P(98 < X < 100) = 0.4525

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 estabilishes 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 100 and variance of 36.

This means that
`\mu = 100, \sigma = √(36) = 6`

Sample of 25:

This means that
`n = 25, s = (6)/(√(25)) = 1.2`

(a) P(X<99)

This is the pvalue of Z when X = 99. So

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

By the Central Limit Theorem

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

`Z = (99 - 100)/(1.2)`

`Z = -0.83`

`Z = -0.83` has a pvalue of 0.2033. So

P(X < 99) = 0.2033.

b) P(98 < X < 100)

This is the pvalue of Z when X = 100 subtracted by the pvalue of Z when X = 98. So

X = 100

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

`Z = (100 - 100)/(1.2)`

`Z = 0`

`Z = 0` has a pvalue of 0.5

X = 98

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

`Z = (98 - 100)/(1.2)`

`Z = -1.67`

`Z = -1.67` has a pvalue of 0.0475

0.5 - 0.0475 = 0.4525

So

P(98 < X < 100) = 0.4525