The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm. (a) Find the probability that a randomly chosen adult female is taller than 150 cm. Sarah is a young girl. She visits her doctor and is told that she is at the 60th percentile for height. (b) Assuming that Sarah remains at the 60th percentile, estimate her height as an adult. The heights of an adult male population are normally distributed with standard deviation 9.0 cm. Given that 90% of adult males are taller than the mean height of adult females, (c) find the mean height of an adult male.

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The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm - Edexcel - A-Level Maths Statistics - Question 7 - 2012 - Paper 2

Question 7

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The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm. (a) Find the probability that a randomly chosen a... show full transcript

Worked Solution & Example Answer:The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm - Edexcel - A-Level Maths Statistics - Question 7 - 2012 - Paper 2

Step 1

Find the probability that a randomly chosen adult female is taller than 150 cm.

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Answer

To find this probability, we start by calculating the z-score for 150 cm using the formula:

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

where:

$X = 150$ cm
$\mu = 162$ cm (mean)
$\sigma = 7.5$ cm (standard deviation)

Calculating:
$z = \frac{150 - 162}{7.5} = -1.6$

Next, we can look up this z-score in the standard normal distribution table to find the corresponding probability. The area to the left of z = -1.6 is approximately 0.0548. Therefore, the probability that a randomly chosen adult female is taller than 150 cm is:

$P(X > 150) = 1 - P(Z < -1.6) = 1 - 0.0548 = 0.9452$

Step 2

Assuming that Sarah remains at the 60th percentile, estimate her height as an adult.

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Answer

The 60th percentile corresponds to a z-score of approximately 0.25 (from z-tables). We can use the inverse of the z-score formula to estimate Sarah's height as an adult:

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

Assuming she maintains this percentile:

$\mu = 162$ cm (mean height of adult females)
$\sigma = 7.5$ cm (standard deviation)

Calculating: $X = 162 + 0.25 \cdot 7.5 = 162 + 1.875 = 163.875$

Thus, Sarah is estimated to be approximately 163.9 cm tall as an adult.

Step 3

Find the mean height of an adult male.

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Answer

Given that 90% of adult males are taller than the mean height of adult females, we can infer that the mean height of adult males is at the 10th percentile of their height distribution. Therefore, we find the z-score corresponding to the 10th percentile, which is approximately -1.28. Using the z-score formula again, we can set up the following:

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

Substituting the known values:

$\sigma = 9.0$ cm (standard deviation for adult males)
For the adult female mean height: $\mu = 162$ cm

Calculating: $X = 162 + (-1.28) \cdot 9.0 = 162 - 11.52 = 150.48$

Thus, the mean height of an adult male is approximately 150.5 cm.

The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm - Edexcel - A-Level Maths Statistics - Question 7 - 2012 - Paper 2

Worked Solution & Example Answer:The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm - Edexcel - A-Level Maths Statistics - Question 7 - 2012 - Paper 2

Find the probability that a randomly chosen adult female is taller than 150 cm.

Assuming that Sarah remains at the 60th percentile, estimate her height as an adult.

Find the mean height of an adult male.

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