Answer :
Final answer:
The probability that the mean body temperature of a random sample of 16 people is between 98.5 and 99 is approximately 0.746, assuming body temperatures are normally distributed. This uses the principles of the Central Limit Theorem and Z-scores.
Explanation:
The subject here is statistics, a branch of mathematics. We can find the probability using the concept of the Central Limit Theorem and the Normal Distribution. Given the population mean (μ) is 98.6, the population variance (σ²) is 0.36, and the sample size (n) is 16. The Central Limit Theorem tells us that the sampling distribution of the sample means will be approximately normally distributed with mean μ and variance σ²/n.
In this case, the mean of the distribution of sample means is 98.6 and the standard deviation (square root of variance) is the square root of (0.36/16) = 0.15.
To find the probability that the mean body temperature of the randomly selected 16 people is between 98.5 and 99, we convert these temperatures to z-scores using the formula z = (x - μ) / σ.
The z-scores corresponding to 98.5 and 99 are (98.5-98.6)/0.15 = -0.67 and (99-98.6)/0.15 = 2.67. We need to find the probability that the z-score is between -0.67 and 2.67. We can look up these z-scores on a standard normal (Z) table or use statistical software to find that the probability is approximately 0.746.
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