Temperatures in June in the Paradise City are distributed nearly normally with mean 75 degrees and standard deviation 4 degrees F. Which of the following temperatures would be considered unusual?
72.2
76.2
84.6
73.8
69.8

To determine which of the given temperatures would be considered unusual, we can use the concept of standard deviations in a normal distribution.

For a normally distributed data set:

- **Mean (μ)** = 75 degrees F
- **Standard deviation (σ)** = 4 degrees F

A temperature is considered unusual if it falls more than 2 standard deviations away from the mean (either above or below).

### Step 1: Calculate the Z-score
The Z-score is calculated using the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- \( X \) is the temperature
- \( \mu \) is the mean
- \( \sigma \) is the standard deviation

### Step 2: Compute Z-scores for each temperature
1. **For 72.2 degrees**:
\[
Z = \frac{72.2 - 75}{4} = \frac{-2.8}{4} = -0.7
\]

2. **For 76.2 degrees**:
\[
Z = \frac{76.2 - 75}{4} = \frac{1.2}{4} = 0.3
\]

3. **For 84.6 degrees**:
\[
Z = \frac{84.6 - 75}{4} = \frac{9.6}{4} = 2.4
\]

4. **For 73.8 degrees**:
\[
Z = \frac{73.8 - 75}{4} = \frac{-1.2}{4} = -0.3
\]

5. **For 69.8 degrees**:
\[
Z = \frac{69.8 - 75}{4} = \frac{-5.2}{4} = -1.3
\]

### Step 3: Determine which temperatures are unusual
- A Z-score greater than 2 or less than -2 indicates that the temperature is unusual.
- In this case, the only Z-score greater than 2 or less than -2 is for **84.6 degrees** (Z = 2.4).

### Conclusion:
**84.6 degrees** would be considered an unusual temperature.

Would you like any more details or have any questions? Here are some related questions:

1. How does the concept of Z-scores help in understanding normal distributions?
2. What is the probability of observing a value more than 2 standard deviations away from the mean in a normal distribution?
3. Can you explain why 84.6 degrees is considered unusual based on the Z-score?
4. What are the implications of a temperature being unusual in a dataset?
5. How would the Z-scores change if the standard deviation was larger or smaller?

**Tip:** Understanding Z-scores can help you quickly determine how far away a data point is from the mean, making it easier to identify outliers in a dataset.

Learn how to determine unusual temperatures in a normal distribution with a mean of 75 degrees and a standard deviation of 4 degrees F. Analyze temperatures such as 72.2, 76.2, 84.6, 73.8, and 69.8 using Z-scores. Understand the concept of normal distribution and Z-scores in statistical analysis. Suitable for high school students.

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