Question

If the temperature in degrees Fahrenheit at a certain location is normally distributed with a mean of 68 degrees and a standard deviation of 4 degrees, what is the distribution of the temperature in degrees Celsius at the same location?

Short answer

The distribution of the temperature in degrees Celsius at the same location is Normal distribution with mean 20 and variance 4.938.

Step-by-step solution

Given information

At a certain location the temperature in degrees Fahrenheit at a certain location is normally distributed with mean 68 degrees and standard deviation is 4 degrees.

Calculating the distribution of the temperature.

If X denotes the temperature in degrees Fahrenheit, Y denotes the temperature in degrees Celsius.

Then,\(Y = \frac{{5\left( {X - 32} \right)}}{9}\)

Y is the linear function of X, then Y will also have a normal distribution.

Also,

Mean of the distribution:

\(\begin{array}{c}E\left( Y \right) = \frac{5}{9}\left( {68 - 32} \right)\\ = \frac{5}{9}\left( {36} \right)\end{array}\)

\(\begin{array}{c}E\left( Y \right) = \frac{{180}}{9}\\ = 20\end{array}\)

Variance of the distribution:

\(\begin{array}{c}Var\left( Y \right) = {\left( {\frac{5}{9}} \right)^2}\left( {16} \right)\\ = \frac{{25}}{{81}}\left( {16} \right)\\ = \frac{{400}}{{81}}\\ = 4.938\end{array}\)

Therefore, the distribution of the temperature in degrees Celsius at the same location is the normal distribution with mean 20 and variance 4.938