Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C.

Question

A single thermometer is randomly selected and tested. Let Z represent the reading of this thermometer at freezing. What reading separates the highest 0.74% from the rest? That is, if P(Z>c) = 0.0074, find c.


c= ______ °C

Answer 1

To find the value of \( c \) such that \( P(Z > c) = 0.0074 \), we first need to determine the corresponding percentile for the standard normal distribution.

Since \( P(Z > c) = 0.0074 \), it follows that \( P(Z \leq c) = 1 - 0.0074 = 0.9926 \). We want to find the z-score that corresponds to the cumulative probability of 0.9926.

We can use a z-table or a calculator that provides the inverse of the standard normal distribution.

Using the inverse normal distribution function:

\[ c = Z_{0.9926} \]

Using a z-table or an online calculator, we can find that:

\[ Z_{0.9926} \approx 2.41 \]

Thus, the reading \( c \) that separates the highest 0.74% from the rest is:

\[ c \approx 2.41 °C \]

So, the answer is:

\[ c = 2.41 °C \]