Question

Assume that the readings on the thermometers are normally distributed with a mean of 0 and standard deviation of 1.00 Celsius. A thermometer is randomly selected and tested. find probability of the reading with given values 1.50 and 2.25

Answer 1

so we have three significant figures?

Answer 2

We are looking for the probability below:



$$
\Phi\left(\cfrac{x_2-\mu}{\sigma}\right)-\Phi\left(\cfrac{x_1-\mu}{\sigma}\right)\\
\approx\cfrac{1}{2} \text{erfc}\left (\cfrac{\mu-x_2}{\sqrt 2 \times \sigma}\right )-\cfrac{1}{2} \text{erfc}\left (\cfrac{\mu-x_1}{\sqrt 2 \times \sigma}\right) \\
\approx 0.0546
$$
Where \(\Phi(x)\) is the standard normal and \(x_2=2.25, x_1=1.50,\mu=0\) and -\(\sigma=1\). Erfc is the error function complement that approximates the standard normal.

So there is about 5.46% chance that the reading will be between 1.50 and 2.25 degrees.

Does this make sense?

Here are two links explaining these concepts.

http://en.wikipedia.org/wiki/Standard_normal_distribution#Cumulative_distribution
http://en.wikipedia.org/wiki/Error_function#Applications