The distances travelled to work, D km, by the employees at a large company are normally distributed with $D \sim N(30, 8^2)$. (a) Find the probability that a randomly selected employee has a journey to work of more than 20 km. (b) Find the upper quartile, $Q_3$, of D. (c) Write down the lower quartile, $Q_1$, of D. An outlier is defined as any value of D such that $D k$ where $h = Q_1 - 1.5(Q_3 - Q_1)$ and $k = Q_3 + 1.5(Q_3 - Q_1)$ (d) Find the value of h and the value of k. (e) An employee is selected at random. (f) Find the probability that the distance travelled to work by this employee is an outlier.

A-Level Edexcel Maths Statistics Normal Distribution: The distances travelled to work,

The distances travelled to work, D km, by the employees at a large company are normally distributed with $D \sim N(30, 8^2)$ - Edexcel - A-Level Maths Statistics - Question 7 - 2010 - Paper 2

Question 7

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The distances travelled to work, D km, by the employees at a large company are normally distributed with $D \sim N(30, 8^2)$. (a) Find the probability that a rand... show full transcript

Worked Solution & Example Answer:The distances travelled to work, D km, by the employees at a large company are normally distributed with $D \sim N(30, 8^2)$ - Edexcel - A-Level Maths Statistics - Question 7 - 2010 - Paper 2

Step 1

Find the probability that a randomly selected employee has a journey to work of more than 20 km.

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Answer

To find the required probability, we need to standardize the value of 20 km using the mean and standard deviation:

$Z = \frac{X - \mu}{\sigma} = \frac{20 - 30}{8} = -1.25$

Now, we look for the probability:

$P(D > 20) = P(Z > -1.25) = 1 - P(Z \leq -1.25)$

Using a standard normal distribution table: $P(Z \leq -1.25) \approx 0.8944$

Thus: $P(D > 20) \approx 0.1056$

Step 2

Find the upper quartile, $Q_3$, of D.

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The upper quartile, $Q_3$ , is found at the 75th percentile of the normal distribution. Using the properties of the normal distribution:

$Q_3 = \mu + z \cdot \sigma$

For $z$ corresponding to 0.75, approximately 0.674: $Q_3 = 30 + 0.674 \cdot 8 = 35.39$

Thus, $Q_3 \approx 35.4$ .

Step 3

Write down the lower quartile, $Q_1$, of D.

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The lower quartile, $Q_1$ , is at the 25th percentile: Using the property: $Q_1 = \mu + z \cdot \sigma$ For $z$ corresponding to 0.25, approximately -0.674: $Q_1 = 30 - 0.674 \cdot 8 = 24.61$

Thus, $Q_1 \approx 24.6$ .

Step 4

Find the value of h and the value of k.

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Answer

To find $h$ and $k$ :

Calculate $h$ : $h = Q_1 - 1.5(Q_3 - Q_1)$ $h = 24.6 - 1.5(35.4 - 24.6) = 24.6 - 16.2 = 8.4$
Calculate $k$ : $k = Q_3 + 1.5(Q_3 - Q_1)$ $k = 35.4 + 1.5(35.4 - 24.6) = 35.4 + 16.2 = 51.6$

Thus, $h \approx 8.4$ and $k \approx 51.6$ .

Step 5

Find the probability that the distance travelled to work by this employee is an outlier.

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Answer

To find this probability:

Calculate the probabilities: $P(D < h) \implies P(D < 8.4)$ $= P\left(Z < \frac{8.4 - 30}{8}\right) = P(Z < -2.675) \approx 0.0035$
Calculate: $P(D > k) \implies P(D > 51.6)$ $= P\left(Z > \frac{51.6 - 30}{8}\right) = P(Z > 2.675) \approx 0.0035$
Therefore: $P(D < h \text{ or } D > k) \approx 0.0035 + 0.0035 \approx 0.007$

The distances travelled to work, D km, by the employees at a large company are normally distributed with $D \sim N(30, 8^2)$ - Edexcel - A-Level Maths Statistics - Question 7 - 2010 - Paper 2

Worked Solution & Example Answer:The distances travelled to work, D km, by the employees at a large company are normally distributed with $D \sim N(30, 8^2)$ - Edexcel - A-Level Maths Statistics - Question 7 - 2010 - Paper 2

Find the probability that a randomly selected employee has a journey to work of more than 20 km.

Find the upper quartile, $Q_3$, of D.

Write down the lower quartile, $Q_1$, of D.

Find the value of h and the value of k.

Find the probability that the distance travelled to work by this employee is an outlier.

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