The mean human body temperature is 98.20°F with a standard deviation of 0.62°F. Sally's temperature corresponds to a z-score of -1.5. What is her temperature? Round your answer to the nearest hundredth.
Mathematics · College · Thu Feb 04 2021
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Answer: To find Sally's temperature, we need to use the formula for converting a z-score to a raw score in the context of a normal distribution:
\[ \text{Raw score (X)} = (\text{z-score} \times \text{Standard deviation}) + \text{Mean} \]
In this case, the mean (\( \mu \)) is 98.20°F and the standard deviation (\( \sigma \)) is 0.62°F. Sally's z-score is -1.5. Plugging these values into the formula gives us:
\[ X = (-1.5 \times 0.62) + 98.20 \] \[ X = -0.93 + 98.20 \] \[ X = 97.27 \]
Therefore, Sally's temperature is 97.27°F when rounded to the nearest hundredth.
Extra: The concept of a z-score is fundamental in statistics. A z-score, also known as a standard score, represents how many standard deviations an element is from the mean. If the z-score is negative, it means the score is below the mean. If it's positive, the score is above the mean.
The formula used above transforms a z-score into the actual value (also known as the raw score) on the original scale of measurement. This process is vital in statistics when you're dealing with variable data and you need to compare different scores that are normally distributed.
In the context of the human body temperature, knowing how far a temperature is from the average can help determine if someone might be experiencing a fever or hypothermia. Sally's temperature being -1.5 standard deviations away from the mean indicates that she is below the average body temperature.