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 above 0°c (denoted by positive numbers). assume that the mean reading is 0°c and the standard deviation of the readings is 1.00°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 places. a. - 1.75° b. - 1.63° c. - 1.89° d. - 1.48°

Explanation:

Step1: Find the z - score

We want to find the z - score \(z\) such that \(P(Z\lt z)=0.04\). Using a standard normal table (or a calculator with a normal - distribution function, such as the inverse of the cumulative distribution function for the standard normal distribution \(N(0,1)\)).

Step2: Calculate the temperature

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 0\) and \(\sigma=1\). Rearranging the formula for \(x\) gives \(x=\mu + z\sigma\). Since \(\mu = 0\) and \(\sigma = 1\), \(x = z\).
Using a calculator (e.g., in R: qnorm(0.04) or in Excel: NORM.S.INV(0.04)), the z - score corresponding to a left - tail area of \(0.04\) is approximately \(z=- 1.75\)

Answer:

A. \(-1.75^{\circ}\)