Question

The mean is 25.2 and the standard deviation is 0.8

Find the probability that X is between 24.3 and 26.0

sketch a normal curve.

Answer 1

We have normal distribution with mean 25.2 and standard deviation 0.8:

m = 25.2

$\sigma = 0.8$

And we need to know:

$P(24.3 < x < 26)$

$P(24.3 < x < 26) = f_{24.3}^{26} f(x) dx$

$f(x)$ - probability density function

The normal distribution has probability density:

$$f (x) = \frac {1}{\sqrt {2 \pi} \sigma} e ^ {- \frac {(x - m) ^ {2}}{2 \sigma^ {2}}}$$

Therefore:

$P(24.3 < x < 26) = f_{24.3}^{26} \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{(x - m)^2}{2\sigma^2}} dx$

Calculating this integral:

$P(24.3 < x < 26) = 0.6827 = 68.27\%$

A normal curve: