Questions: 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. Find P72, the 72-percentile. This is the temperature reading separating the bottom 72% from the top 28%. P72= °C (Round answer to three decimal places)

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. Find P72, the 72-percentile. This is the temperature reading separating the bottom 72% from the top 28%.
P72= °C
(Round answer to three decimal places)

Transcript text: Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of $0^{\circ} \mathrm{C}$ and a standard deviation of $1.00^{\circ} \mathrm{C}$. A single thermometer is randomly selected and tested. Find $P_{72}$, the 72 -percentile. This is the temperature reading separating the bottom $72 \%$ from the top $28 \%$. $P_{72}=$ $\square$ ${ }^{\circ} \mathrm{C}$ (Round answer to three decimal places) Question Help: $\square$ Video Message instructor Add Work Calculator Submit Question

Solution

Solution Steps

Step 1: Determine the Z-Score for the 72nd Percentile

To find the 72nd percentile ($P_{72}$) of a normally distributed dataset, we first calculate the z-score corresponding to this percentile. The z-score for the 72nd percentile is given by:

\[ z_{0.72} \approx 0.5828 \]

Step 2: Convert the Z-Score to the Original Scale

Next, we convert the z-score back to the original temperature scale using the mean ($\mu$) and standard deviation ($\sigma$) of the distribution. The formula to convert the z-score to the original value is:

\[ P_{72} = \mu + z_{0.72} \cdot \sigma \]

Substituting the values:

\[ P_{72} = 0 + 0.5828 \cdot 1 = 0.5828 \]

Step 3: Round the Result

Finally, we round the result to three decimal places:

\[ P_{72} \approx 0.583 \, ^\circ C \]