SOLUTION: The Precision Scientific Instruments company manufactures thermometers that are supposed to read 0 𝑜𝐶 at the freezing point of water. Tests on these instruments revealed t

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Question 1182375: The Precision Scientific Instruments company manufactures thermometers that
are supposed to read 0
𝑜𝐶 at the freezing point of water. Tests on these
instruments revealed that at the freezing point of water some thermometers
give readings above zero degrees Celsius while other thermometers give
readings below zero degrees Celsius with an average of zero degrees Celsius
and a standard deviation of the readings ion 0.9
𝑜𝐶. Assume that the readings are
normally distributed,
i. Find the probability of observing one thermometer at random whose
reading is between +𝟎. 𝟓𝒐𝑪 and +𝟏. 𝟔𝒐𝑪 at the freezing point of water.
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

x = temperature in degrees Celsius
mu = 0 = population mean temperature in degrees C
sigma = 0.9 = population standard deviation for the temperatures

Convert x = 0.5 to a corresponding z score
z = (x-mu)/sigma
z = (0.5-0)/0.9
z = 0.56 approximately

Do the same for x = 1.6
z = (x-mu)/sigma
z = (1.6-0)/0.9
z = 1.78 approximately

Your teacher is asking you to find P(0.5 < x < 1.6) which is roughly equivalent to P(0.56 < z < 1.78) based on those earlier z score conversions.

Now use a z table such as this one
https://www.ztable.net/
or one you would find in the back of your textbook.

From that table, we see the following approximations
P(z < 0.56) = 0.71226
P(z < 1.78) = 0.96246
which are found as shown below

the stuff in red pertains to z = 0.56, while the stuff in blue is for z = 1.78

Then we subtract those values to get the answer we're after
P(a < z < b) = P(z < b) - P(z < a)
P(0.56 < z < 1.78) = P(z < 1.78) - P(z < 0.56)
P(0.56 < z < 1.78) = 0.96246 - 0.71226
P(0.56 < z < 1.78) = 0.2502
P(0.5 < x < 1.6) = 0.2502

Answer: Approximately 0.2502
Use a calculator to get more accuracy.

There's roughly a 25.02% chance of getting a reading between 0.5 degrees C and 1.6 degrees C (when the true reading should be 0 degrees C).