Question

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°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°C (denoted by negative numbers) and some give readings a.bove 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 100°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.
A quality control analyst wants to examine thermometers that give readings in the bottom 4%. Find the reading that separates the bottom 4% from the others.
a) -1.89°
b) -1.63°
c) -1.75°
d) -1.48°

Answer 1

Answer:

c) -1.75°

Step-by-step explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean of 0, standard deviation of 1:

This means that $\mu = 0, \sigma = 1$

Find the reading that separates the bottom 4% from the others.

This is the 4th percentile, which is X when Z has a p-value of 0.04, so X when Z = -1.75.

$Z = \frac{X - \mu}{\sigma}$

$-1.75 = \frac{X - 0}{1}$

$X = -1.75*1$

$X = -1.75$

The correct answer is given by option c.