Assume that body temperatures of healthy adults are normally distributed with a mean of 98.2°F and a standard deviation of 0.62°F. Suppose we take a sample of eight healthy adults. What is the probability that their mean body temperature will be greater than 98.4?

To find the probability that the mean body temperature of a sample of eight healthy adults will be greater than 98.4°F, we can use the concept of the sampling distribution of the sample mean.

The mean of the sampling distribution of the sample mean is equal to the mean of the population, which is 98.2°F.

The standard deviation of the sampling distribution of the sample mean, also known as the standard error, is equal to the standard deviation of the population divided by the square root of the sample size. In this case, the standard deviation of the population is 0.62°F, and the sample size is 8. Therefore, the standard error is 0.62°F divided by the square root of 8.

To find the probability, we can calculate the z-score, which measures how many standard errors the sample mean is away from the population mean. The formula for the z-score is:

z = (x - μ) / σ

where x is the value we want to find the probability for (98.4°F in this case), μ is the mean of the population (98.2°F), and σ is the standard deviation of the sampling distribution of the sample mean (standard error).

Using this formula, we can calculate the z-score:

z = (98.4 - 98.2) / (0.62 / √8)

Calculating this, we get:

z = 0.2 / (0.62 / 2.83)

z = 0.2 / 0.22

z ≈ 0.909

Once we have the z-score, we can look up the probability associated with that z-score using a standard normal distribution table or a calculator. In this case, we are interested in finding the probability that the mean body temperature is greater than 98.4°F, so we need to find the area under the normal curve to the right of the z-score of 0.909.

Using a standard normal distribution table or a calculator, we find that the probability associated with a z-score of 0.909 is approximately 0.819. Therefore, the probability that the mean body temperature of the sample of eight healthy adults will be greater than 98.4°F is approximately 0.819, or 81.9%.