A body temperature of 96.5 degree F given that human body temperatures have a mean of 98.20 degree F and a standard deviation of 0.62 degree.
A) -1.7: not usual
B) 2.7: unusual
C) -2.7: not unusual
D) -2.7: unusual
The correct answer and explanation is :
To determine if a body temperature of 96.5°F is unusual, we can use the concept of z-scores. A z-score represents how many standard deviations a data point is from the mean. The formula for calculating the z-score is:
[
z = \frac{X – \mu}{\sigma}
]
Where:
- ( X ) is the data point (in this case, 96.5°F),
- ( \mu ) is the mean (in this case, 98.2°F),
- ( \sigma ) is the standard deviation (in this case, 0.62°F).
Step 1: Calculate the z-score
Substitute the given values into the formula:
[
z = \frac{96.5 – 98.2}{0.62} = \frac{-1.7}{0.62} \approx -2.74
]
Step 2: Interpret the z-score
A z-score of approximately -2.7 means that the body temperature of 96.5°F is 2.7 standard deviations below the mean.
Step 3: Assess whether it is usual or unusual
In statistics, data points that are more than 2 standard deviations away from the mean (either above or below) are generally considered unusual. Specifically:
- A z-score between -2 and +2 is considered “usual” or “within normal range.”
- A z-score outside this range (i.e., below -2 or above +2) is considered “unusual” or “extreme.”
Since the z-score here is -2.7, which is beyond the -2 threshold, this body temperature of 96.5°F is considered unusual.
Step 4: Conclusion
The correct answer is D) -2.7: unusual.
Explanation:
A body temperature of 96.5°F is significantly lower than the average body temperature of 98.2°F. This result indicates that the temperature falls outside the usual range of normal human body temperatures, suggesting a potential health concern, such as hypothermia or another condition that could lower body temperature. When temperatures fall more than 2 standard deviations away from the mean, they are typically regarded as unusual, and further investigation might be warranted.