Question

According to the article "A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich" published in the Journal of the American Medical Association, the body temperatures of adults are normally distributed with a mean of 98.242 and a standard deviation of 0.713.

1) What is the probability that a randomly selected person's body temperature is between 97.9 and 98.51? Round your answer to 4 decimal places.
2) What is the probability that the average body temperature of 4 randomly selected people is between 97.9 and 98.51? Round your answer to 4 decimal places.
3) Why did the probability increase?

A)The probability increased since the sample size decreased resulting in a wider distribution which increased the chances of being in an interval containing the mean.
B)The probability increased since the sample size increased resulting in a narrower distribution which increased the chances of being in an interval containing the mean.

Answer 1

1) The probability that a randomly selected person's body temperature is between 97.9 and 98.51 can be found using the z-score formula to standardize the temperatures and using a standard normal distribution table or calculator. 2) The probability that the average body temperature of 4 randomly selected people is between 97.9 and 98.51 can be found using the standard error of the mean and the central limit theorem. 3) The probability increased because the sample size decreased, resulting in a wider distribution.

Step-by-step explanation:

1) To find the probability that a randomly selected person's body temperature is between 97.9 and 98.51, we can standardize the temperatures using z-scores. The z-score formula is z = (x - mean) / standard deviation. So, for 97.9, the z-score is (97.9 - 98.242) / 0.713 = -0.479. And for 98.51, the z-score is (98.51 - 98.242) / 0.713 = 0.37. Now, we can use a standard normal distribution table or a calculator to find the probabilities associated with these z-scores. The probability is the difference between the two probabilities: P(97.9 < x < 98.51) = P(z < 0.37) - P(z < -0.479).

2) To find the probability that the average body temperature of 4 randomly selected people is between 97.9 and 98.51, we can use the central limit theorem. The distribution of sample means tends to be normal regardless of the shape of the population. We can find the standard error of the mean using the formula standard deviation / sqrt(sample size). In this case, the standard error of the mean is 0.713 / sqrt(4) = 0.3565. Now we can use the z-score formula to standardize the range of temperatures as in part 1, and find the probabilities associated with those z-scores using a standard normal distribution table or calculator. The probability is the difference between the two probabilities as in part 1.

3) The probability increased since the sample size decreased resulting in a wider distribution which increased the chances of being in an interval containing the mean. A)The probability increased since the sample size decreased resulting in a wider distribution which increased the chances of being in an interval containing the mean.