Question

(a) The height of trees a normally distributed with mean 160 cm and standard deviation is 8 cm.
Find the probability data randomly selected tree has a height greeter than 170 cm
Find the probability that tree has a height greeter than 180 cm.
Give intuitive reason why your answers are relative.
(b) Find median and mode.

Answer 1

Solution a)

$\mu = 160$

$\sigma = 8$

Let tree height be X

$p(X > 170)$

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

$= \frac{170-160}{8} = 1.25$

At Z = 1.25 the probability p(X>170) = 0.1056

Answer: 0.1056

$p(X>180)$

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

$Z=\frac{180−160}{8} = 2.5$

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At Z = 2.5 the probability p(X>180) = 0.0062

Answer: 0.0062

These solutions are relative since the calculated probabilities are based on the unique distribution of the data defined by the mean (160) and standard deviation (8)

Solution b)

For a normal curve

$mean = median = mode$

Answer: Median = Mode = 160