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Assume that the readings on the thermometers are normally distributed with a mean of 0 degrees and standard deviation of 1.00degreesC. A thermometer is randomly selected and tested. Draw a sketch and find the temperature reading corresponding to Upper P 96​, the 96 th percentile. This is the temperature reading separating the bottom 96 % from the top 4 %.


To solve this problem, we need to find the temperature reading corresponding to the 96th percentile (\(P_{96}\)) from a normal distribution with a mean of \(0^\circ C\) and a standard deviation of \(1^\circ C\). This means that \(P_{96}\) represents the temperature where 96% of the values lie below it and 4% above it.

### Step 1: Use Z-scores
For a normal distribution:
- Mean (\(\mu\)) = 0
- Standard deviation (\(\sigma\)) = 1

The 96th percentile corresponds to a Z-score that separates the bottom 96% from the top 4%. To find this Z-score, we can use a standard normal distribution table or a calculator.

### Step 2: Find the Z-score for \(P_{96}\)
Using a Z-score table or calculator, the Z-score corresponding to the 96th percentile is approximately **1.75**.

### Step 3: Calculate the temperature for \(P_{96}\)
Since the mean (\(\mu\)) is 0 and the standard deviation (\(\sigma\)) is 1, the temperature corresponding to \(P_{96}\) is simply equal to the Z-score because \(X = \mu + Z \times \sigma\).

Thus, the temperature for \(P_{96}\) is approximately:

\[
X = 0 + 1.75 \times 1 = 1.75^\circ C
\]

### Step 4: Select the Correct Graph
The correct graph should show a shaded region to the left that covers 96% of the area under the curve, with a cutoff around the temperature \(1.75^\circ C\), which is located slightly to the right of the mean. Based on the image, **Graph C** represents this distribution correctly.

Thus:
- The graph representing \(P_{96}\) is **Graph C**.
- The temperature for \(P_{96}\) is approximately **1.75°C**.

Would you like further details or have any questions?

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Here are 5 related questions to consider:
1. How would the graph change if the standard deviation increased?
2. What does the Z-score tell us about the distribution of data?
3. How would the calculation change if we were looking for the 90th percentile instead of the 96th?
4. Why is the normal distribution symmetric, and what implications does this have for percentiles?
5. How do outliers affect the mean and standard deviation of a dataset?

**Tip:** When working with normal distributions, always double-check if the problem specifies whether it's a standard normal distribution (mean 0, standard deviation 1) or if you need to transform the values accordingly!

Assume that the readings on the thermometers are normally distributed with a mean of 0 degrees and standard deviation of 1.00 degrees Celsius. A thermometer is randomly selected and tested. Find the temperature reading corresponding to P96​, the 96th percentile. This is the temperature reading separating the bottom 96% from the top 4%.

Learn how to find the temperature corresponding to the 96th percentile (P96) in a normal distribution with a mean of 0°C and standard deviation of 1°C. The solution involves using the Z-score for percentile calculation.

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