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 17.34% from the rest? That is, if P(Z>c) = 0.1734, 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 17.34% from the rest? That is, if P(Z>c) = 0.1734, find c.
c=_____ °C
Answers
GPT-4o mini
To find the value of \( c \) such that \( P(Z > c) = 0.1734 \), we first need to determine the corresponding z-score for the cumulative probability. This can be done more easily by using the complement:
\[
P(Z \leq c) = 1 - P(Z > c) = 1 - 0.1734 = 0.8266
\]
Now we need to find the z-score that corresponds to a cumulative probability of 0.8266.
Using a standard normal distribution table or a calculator, we look for the z-score that yields an area of 0.8266 to its left. The z-score that corresponds to this cumulative probability is approximately:
\[
z \approx 0.95
\]
Now, we can conclude that:
\[
c \approx 0.95 \, °C
\]
Thus, the reading that separates the highest 17.34% from the rest is:
\[
c = 0.95 \, °C
\]
\[
P(Z \leq c) = 1 - P(Z > c) = 1 - 0.1734 = 0.8266
\]
Now we need to find the z-score that corresponds to a cumulative probability of 0.8266.
Using a standard normal distribution table or a calculator, we look for the z-score that yields an area of 0.8266 to its left. The z-score that corresponds to this cumulative probability is approximately:
\[
z \approx 0.95
\]
Now, we can conclude that:
\[
c \approx 0.95 \, °C
\]
Thus, the reading that separates the highest 17.34% from the rest is:
\[
c = 0.95 \, °C
\]