Question

Assume that the readings on the thermometers are normally distributed with a mean of and standard deviation of 1.00C. Assume ​% of the thermometers are rejected because they have readings that are too high and another ​2.3% 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

To find the cutoff values separating the rejected thermometers from the others, we need to determine the z-scores associated with the given probabilities. Using a standard normal distribution table, the upper cutoff value has a z-score of approximately 2.05, and the lower cutoff value has a z-score of approximately -2.05.

Step-by-step explanation:

In order to find the cutoff values separating the rejected thermometers from the others, we need to determine the z-scores associated with the given probabilities.

Since we know the mean and standard deviation of the thermometer readings (mean = 0 and standard deviation = 1), we can use a standard normal distribution table or a calculator to find the z-scores.

To find the z-score corresponding to the upper cutoff value, we subtract the given percentage from 1 (to find the area under the normal curve to the left of the cutoff value).

This gives us 1 - 0.023 = 0.977. Using the standard normal distribution table, we find that the z-score associated with an area of 0.977 is approximately 2.05.

To find the z-score corresponding to the lower cutoff value, we subtract the given percentage from 1 and then subtract the resulting value from 1 again (since the area corresponds to the left tail).

This gives us 1 - 0.023 - 0.977 = 0.023. Using the standard normal distribution table, we find that the z-score associated with an area of 0.023 is approximately -2.05.