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 the probability, we will first calculate the z-score.

The formula for the z-score is: $z = rac{X - ext{mean}}{ ext{standard deviation}}$

Where:

$X = 150$ cm
mean = 162 cm
standard deviation = 7.5 cm

Plugging in the values: $z = rac{150 - 162}{7.5} = rac{-12}{7.5} \\ = -1.6$

Next, we use the z-score to find the corresponding probability from the z-table:

P(Z > -1.6) = 1 - P(Z < -1.6) = 1 - 0.0548 = 0.9452.

Therefore, the probability that a randomly chosen adult female is taller than 150 cm is approximately 0.9452 or 94.52%.

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 0.253 (you can find this value in standard normal distribution tables).

Using the z-score formula: $X = ext{mean} + z imes ext{standard deviation}$

For the adult female population:

mean = 162 cm
standard deviation = 7.5 cm

Substituting the values: $X = 162 + 0.253 imes 7.5 \\ = 162 + 1.8975 \\ ≈ 163.90$

Therefore, her estimated height as an adult would be approximately 163.9 cm.

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, this implies that the mean adult male height corresponds to the 10th percentile of the adult male distribution.

Using the z-score for the 10th percentile, which is approximately -1.28:

The formula is: $X = ext{mean} + z imes ext{standard deviation}$

Let the mean height of adult males be $M$ . We have:

z = -1.28
standard deviation = 9.0 cm

Setting up the equation: $M - 1.28 imes 9.0 = 162$

Solving for $M$ : $M = 162 + (1.28 imes 9.0) \\ = 162 + 11.52 \\ = 173.52$

Therefore, the mean height of an adult male is approximately 173.52 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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