Assume that the readings on the thermometers are normally distributed with a mean of 0 degrees and standard deviation of 1.00degreesC. Assume 3.3% of the thermometers are rejected because they have readings that are too high and another 3.3% are rejected because they have readings that are too low. Draw a sketch and find the two readings that are cutoff values separating the rejected thermometers from the others.

The variable of interest is the readings on thermometers. This variable is normally distributed with mean μ= 0 degrees C and standard deviation σ= 1.00 degrees C.

The objective is to find the readings that are in the top 3.3% of the distribution and the lowest 3.3% of the distribution.

Symbolically:

The lower value P(X≤a)=0.033

Top value P(X≥b)=0.033

(see attachment)

Lower value:

The accumulated probability until "a" is 0.03, since the variable has a normal distribution, to reach the value of temperature that has the lowest 3.3%, you have to work under the standard normal distribution.

First we look the Z value corresponding to 0.033 of probability:

Z= -1.838

Now you reverste the standardization using the formula Z= (a-μ)/δ

a= (Z*δ)+μ

a= (-1.838*1)+0

a= -1.838

Top value:

P(X≥b)=0.033

This value has 0.033 of the distribution above it then 1 - 0.033= 0.967

is below it.

You can rewrite the expression as:

P(X≤b)=0.967

Now you have to look the value of Z that corresponds to 0.967 of accumulated probability:

b= (Z*δ)+μ

b= (1.838*1)+0

b= 1.838

The cutoff values that separates rejected thermometers from the others are -1.838 and 1.838 degrees C.

I hope it helps!