Probability

Question

If the reading of a thermometer is normally distributed with a mean 0.0C and a standard distribution of 1.0C, find the probability that a reading is:

a) between 0.0 and 8C

b) Less than -8 C

Answer 1

This thermometer has a standard normal distribution, so we are being asked for the probability that a random variable with a standard normal distribution is between 0 standard deviations and 8 standard deviations above the mean, that is, we are looking for 

 

P(0 < z < 8)

 

We can always turn this kind of statement into 

 

P(z < 8) - P(z < 0)

 

For a standard normal distribution, P(z < 8) is 1 to at least 15 significant figures, while P(z < 0) = 0.5 by the symmetry property of normal distributions.  Therefore,

 

P(0 < z < 8) =

 

P(z < 8) - P(z < 0) =

 

1 - 0.5 =

 

0.5.

 

For part b, again by the symmetry property, since P(z < 8) = P(z > -8), then

 

1 = P(z < -8) + P(z > -8)

 

1 = P(z < -8) + P(z < 8)

 

1 = P(z < -8) + 1

 

0 = P(z < -8)

 

to 15 significant figures.