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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

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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) =... 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)P(X > 25), we first standardize the variable XX. The Z-score is calculated as follows:

Z = rac{X - ext{mean}}{ ext{standard deviation}} = rac{25 - 20}{4} = 1.25

Next, we need to find P(Z>1.25)P(Z > 1.25) using the standard normal distribution table. The corresponding value is:

P(Z<1.25)≈0.8944P(Z < 1.25) \approx 0.8944

Therefore,

P(Z>1.25)=1−P(Z<1.25)=1−0.8944=0.1056P(Z > 1.25) = 1 - P(Z < 1.25) = 1 - 0.8944 = 0.1056

Thus,

P(X>25)≈0.1056P(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

For this part, we need to find the Z-score that corresponds to the probability of 0.9641 since:

P(20<X<d)=P(X<d)−P(X<20)=P(X<d)P(20 < X < d) = P(X < d) - P(X < 20) = P(X < d)

Given that P(X<20)=0.5P(X < 20) = 0.5, we need:

0.5+0.4641=0.96410.5 + 0.4641 = 0.9641

Finding the Z-score from standard normal distribution corresponding to 0.9641, we find:

Z≈1.80Z \approx 1.80

Now, we use the formula to convert it back to the XX variable:

Z = rac{X - ext{mean}}{ ext{standard deviation}}

Substituting the values,

1.80=d−2041.80 = \frac{d - 20}{4}

Solving for dd gives:

d−20=7.2d - 20 = 7.2

Thus,

d=20+7.2=27.2d = 20 + 7.2 = 27.2

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