Question

The lengths of salamanders have a normal distribution with mean 15cm, and standard deviation 2cm.


What is the probability a salamander has length less than 13cm?


(provide your answer as a decimal)

Answer 1

Answer: 0.1587

Step-by-step explanation:

So we were given;

x = 13

Mean = 15

Standard deviation = 2

We will be using the Z score formula

Z = (x - mean) / standard deviation

So z = ( 13 - 15 ) / 2

z = - 2/2

z = -1

So we check for -1 in the z table

z = 0.1587

Answer 2

Using the normal distribution, it is found that there is a 0.1587 probability that a salamander has length less than 13cm.

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In a normal distribution, the z-score is used, which in a set with mean $\mu$ and standard deviation $\sigma$, is given by:  

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

• It measures how many standard deviations the measure is from the mean.
• The p-value of a z-score, found from the z-table, is it's percentile.

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• Mean of 15 means that $\mu = 15$
• Standard deviation of 2 means that $\sigma = 2$
• The probability of a length less than 13cm is the p-value of Z when X = 13, thus:

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

$Z = \frac{13 - 15}{2}$

$Z = -1$

$Z = -1$ has a p-value of 0.1587.

Thus, 0.1587 probability a salamander has length less than 13cm.

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