Question
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.
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
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
Answers
GPT-4o mini
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:
\[
c = 2.41 °C
\]
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:
\[
c = 2.41 °C
\]