Question

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0degrees°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 0degrees°C ​(denoted by negative​ numbers) and some give readings above 0degrees°C ​(denoted by positive​ numbers). Assume that the mean reading is 0degrees°C and the standard deviation of the readings is 1.00degrees°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. A quality control analyst wants to examine thermometers that give readings in the bottom​ 4%. Find the temperature reading that separates the bottom​ 4% from the others. Round to two decimal placesA. -1.48°CB. -1.89°CC. -1.63°CD. -1.75°C

Answer 1

D. -1.75°C

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (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 pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

Assume that the mean reading is 0degrees°C and the standard deviation of the readings is 1.00degrees°C. This means that
`\mu = 0, \sigma = 1`.

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

The bottom 4% if the 4th percentile.

This is the value of X when Z has a pvalue of 0.04. This is
`Z = -1.75`.

`Z = (X - \mu)/(\sigma)`

`-1.75 = (X - 0)/(1)`

`X = -1.75`

The correct answer is:

D. -1.75°C