Question

Assume that the readings on the thermometers are normally distributed with a mean of 0 degrees and standard deviation of 1.00degreesC. Assume 2.8​% of the thermometers are rejected because they have readings that are too high and another 2.8​% 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:

1.91° and -1.91°.

Step-by-step explanation:

2.8% of the thermometers are rejected on either end of the curve.  The bottom end, where the readings are too far below the mean, will have an area from this point to the left tail of the curve of 0.028.

The top end, where the readings are too far above the mean, will have an area from this point to the left tail of the curve of 1-0.028 = 0.972.

We look in a z table for these values.  We look within the cells of the table; the closest value to 0.028 is 0.0281, which corresponds with a z score of -1.91.  The closest value to 0.972 is 0.9719, which corresponds with a z score of 1.91.

We substitute these values into the z score formula, along with our values for the mean (0) and the standard deviation (1):

$-1.91=\frac{X-0}{1}$

Simplifying the right hand side, X-0 = X; X/1 = X.  This means X = -1.91.

For the second value,

$1.91=\frac{X-0}{1}$

Simplifying the right hand side, X-0 = X; X/1 = X.  This means X = 1.91.

This means the two values are 1.91° and -1.91°.