The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of $0 ^ { \circ } \mathrm { C }$ 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 $0 ^ { \circ } \mathrm { C }$ (denoted by negative numbers) and some give readings above $0 ^ { \circ } \mathrm { C }$ (denoted by positive
numbers). Assume that the mean reading is $0 ^ { \circ } \mathrm { C }$ and the standard deviation of the readings is $1.00 ^ { \circ } \mathrm { C }$ Also assume that the
frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and
tested. Find the temperature reading corresponding to the given information.
-If 6.3% of the thermometers are rejected because they have readings that are too high and another 6.3% are rejected because they have readings that are too low, find the two readings that are cutoff values separating the
Rejected thermometers from the others.
A) -1.46° , 1.46°
B) -1.53° , 1.53°
C) -1.45° , 1.45°
D) -1.39° , 1.39°

Question

Quizplus · Accepted Answer

Since the mean reading is $0 ^ { \circ } \mathrm { C }$ and the standard deviation is $1.00 ^ { \circ } \mathrm { C }$, we can find the cutoff values using the normal distribution table.

Let x be the cutoff value for rejected thermometers with readings that are too high. Then we have:

$$P(X > x) = 0.063$$
where X is the random variable representing the readings of the thermometers.

Using the normal distribution table, we find the z-score corresponding to a probability of 0.063 to be approximately 1.5. Therefore:

$$z = \frac{x - \mu}{\sigma} = 1.5$$
$$x = \mu + 1.5\sigma = 1.5 ^ { \circ } \mathrm { C }$$

Similarly, let y be the cutoff value for rejected thermometers with readings that are too low. Then we have:

$$P(X < y) = 0.063$$

Using the normal distribution table, we find the z-score corresponding to a probability of 0.063 to be approximately -1.5. Therefore:

$$z = \frac{y - \mu}{\sigma} = -1.5$$
$$y = \mu - 1.5\sigma = -1.5 ^ { \circ } \mathrm { C }$$

Therefore, the two cutoff values are -1.53° and 1.53°, so the answer is B.

The Precision Scientific Instrument Company Manufactures Thermometers That Are Supposed 0∘C0 ^ { \circ } \mathrm { C }0∘C

The Precision Scientific Instrument Company Manufactures Thermometers That Are Supposed $0 ^ { \circ } \mathrm { C }$