Question

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.

Answer 1

Answer:

Step-by-step explanation:

Hello!

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!