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A random variable X follows a normal distribution with mean 20 and standard deviation 5 - Leaving Cert Mathematics - Question 1 - 2011

Question 1

A random variable X follows a normal distribution with mean 20 and standard deviation 5. Find P(14 ≤ X ≤ 26).

Worked Solution & Example Answer:A random variable X follows a normal distribution with mean 20 and standard deviation 5 - Leaving Cert Mathematics - Question 1 - 2011

Step 1

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Answer

To find the probabilities corresponding to X = 14 and X = 26, we need to calculate the z-scores using the formula:

$z = \frac{x - \mu}{\sigma}$

For X = 14: $z_1 = \frac{14 - 20}{5} = -1.2$

For X = 26: $z_2 = \frac{26 - 20}{5} = 1.2$

Step 2

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Answer

The probability we are looking for can be expressed as:

$P(14 ≤ X ≤ 26) = P(-1.2 ≤ z ≤ 1.2)$

This can be found using the cumulative distribution function (CDF) for the standard normal distribution.

Step 3

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Answer

Using standard normal distribution tables (or a calculator), we find: $P(z ≤ 1.2) ≈ 0.8849$

Next, we also need to compute: $P(z ≤ -1.2) ≈ 0.1151$

Step 4

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Answer

Now we find:

$P(-1.2 ≤ z ≤ 1.2) = P(z ≤ 1.2) - P(z ≤ -1.2)$

This results in:

$P(-1.2 ≤ z ≤ 1.2) ≈ 0.8849 - 0.1151 = 0.7698$

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