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The random variable X has a normal distribution with mean 20 and standard deviation 4 - Edexcel - A-Level Maths Statistics - Question 6 - 2007 - Paper 2

Question 6

The random variable X has a normal distribution with mean 20 and standard deviation 4. (a) Find P(X > 25). (b) Find the value of d such that P(20 < X < d) = 0.4641... show full transcript

Worked Solution & Example Answer:The random variable X has a normal distribution with mean 20 and standard deviation 4 - Edexcel - A-Level Maths Statistics - Question 6 - 2007 - Paper 2

Step 1

Find P(X > 25)

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Answer

To find P(X > 25), we start by standardizing the variable X using the Z-score formula:

$Z = \frac{X - \mu}{\sigma}$

where $\mu$ = 20 (mean) and $\sigma$ = 4 (standard deviation).

Substituting the values:

$Z = \frac{25 - 20}{4} = \frac{5}{4} = 1.25$

Using the standard normal distribution table or a calculator, we find:

$P(Z > 1.25) = 1 - P(Z \leq 1.25) \approx 1 - 0.8944 = 0.1056$

Thus,

$P(X > 25) \approx 0.1056.$

Step 2

Find the value of d such that P(20 < X < d) = 0.4641

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Answer

We want to find the value of $d$ such that:

$P(20 < X < d) = 0.4641.$

First, we know that:

$P(X < d) = P(X < 20) + P(20 < X < d) = 0.50 + 0.4641 = 0.9641.$

Now, we standardize the upper bound $d$ :

$P(Z < z) = 0.9641$

Using the Z-table, we find that:

$z \approx 1.80.$

Now, we can use the Z-score formula to find $d$ :

$1.80 = \frac{d - 20}{4}$

Solving for $d$ , we have:

$d = 1.80 \times 4 + 20 = 27.2.$

Thus, the value of $d$ is:

$d = 27.2.$

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