Question

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0degrees°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of​ water, some give readings below 0degrees°C ​(denoted by negative​ numbers) and some give readings above 0degrees°C ​(denoted by positive​ numbers). Assume that the mean reading is 0degrees°C and the standard deviation of the readings is 1.00degrees°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. A quality control analyst wants to examine thermometers that give readings in the bottom​ 4%. Find the temperature reading that separates the bottom​ 4% from the others. Round to two decimal places.

Answer 1

Explanation:

Hello!

The variable of study is X: Temperature measured by a thermometer (ºC)

This variable has a distribution approximately normal with mean μ= 0ºC and standard deviation σ= 1.00ºC

To determine the value of X that separates the bottom 4% of the distribution from the top 96% you have to work using the standard normal distribution:

P(X≤x)= 0.04 ⇒ P(Z≤z)=0.04

First you have to use the Z tables to determine the value of Z that accumulates 0.04 of probability. It is the "bottom" 0.04, this means that the value will be in the left tail of the distribution and will be a negative value.

z= -1.75

Now using the formula of the distribution and the parameters of X you have to transform the Z-value into a value of X

z= (X-μ)/σ

z*σ = X-μ

(z*σ)+μ = X

X= (-1.75-0)/1= -1.75ºC

The value that separates the bottom 4% is -1.75ºC

I hope this helps!