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.