Question

In Exercise \(1.67,\) Allen Shoemaker derived a distribution of human body temperatures with a distinct mound shape. \(^{9}\) Suppose we assume that the temperatures of healthy humans are approximately normal with a mean of \(98.6^{\circ}\) and a standard deviation of \(0.8^{\circ} .\) a. If 130 healthy people are selected at random, what is the probability that the average temperature for these people is \(98.25^{\circ}\) or lower? b. Would you consider an average temperature of \(98.25^{\circ}\) to be an unlikely occurrence, given that the true average temperature of healthy people is \(98.6^{\circ} ?\) Explain.

Short answer

Answer: The probability of the average temperature of 130 randomly selected healthy people being 98.25 degrees or lower is approximately 0.105%. This occurrence is considered unlikely.

Step-by-step solution

Identify the given information

In this problem, we are given the following information regarding the human body temperatures: Mean (μ): 98.6 degrees Standard Deviation (σ): 0.8 degrees Sample size (n): 130 Sample mean (x̄): 98.25 degrees We will use these values in our calculations.

Calculate the z-score for the sample mean

To find the z-score for the sample mean, we use the following formula: \(Z = \frac{x̄ - μ}{σ / \sqrt{n}}\) Plugging in the given values, we get: \(Z = \frac{98.25 - 98.6}{0.8/\sqrt{130}}\) Now, let's calculate the z-score by simplifying the equation. \(Z = \frac{-0.35}{0.8/\sqrt{130}} \approx -3.075\)

Look up the probability in a standard normal distribution table

Now, let's look up the z-score of -3.075 in a standard normal distribution table. Keep in mind that we are looking for the probability that the average temperature is 98.25 degrees or lower, which means we are looking for the cumulative probability to the left of this z-score. Looking up -3.075 in a standard normal distribution table, we find that the probability is approximately 0.00105.

Answering part (a) of the question

The probability that the average temperature of 130 randomly selected healthy people is 98.25 degrees or lower is approximately 0.00105, or 0.105%.

Answering part (b) of the question

A probability of 0.105% can be considered quite low for a random event. Given that the true average temperature of healthy people is 98.6 degrees, an average temperature of 98.25 degrees for a random sample of 130 people can be considered an unlikely occurrence.

Key concepts

Understanding the Z-Score

The z-score is a measure we use to understand how a value compares to the average of a set data. It tells us the number of standard deviations a particular data point is from the mean.
You can think of a z-score as a way of saying how unusual or typical a value is within a normal distribution.
For example, in the temperature problem we looked at, the z-score helps us understand how the sample mean of 98.25 compares to the usual human body temperature mean of 98.6. It's calculated using the formula:\[ Z = \frac{x̄ - μ}{σ / \sqrt{n}} \]where:

• \(x̄\) is the sample mean.
• \(μ\) is the population mean.
• \(σ\) is the standard deviation of the population.
• \(n\) is the sample size.

A z-score of \(-3.075\) implies that the sample mean is 3.075 standard deviations below the population mean. That's quite far off from what we'd expect!

Exploring Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. In this standard form, it has a mean of 0 and a standard deviation of 1. It's a common reference in statistics because it lets us look up probabilities for any normal distribution by converting our values to z-scores.
When you have data with a normal distribution, like human body temperatures in our problem, you convert real-world values using the z-score formula and then make use of standard normal distribution tables or software to find probabilities.
These tables show the cumulative probability of observing a value less than your z-score. For instance, a z-score of \(-3.075\) in the standard normal distribution gives us a very low probability of 0.00105. This cumulative probability means there is a 0.105% chance that values are as low or lower.

Decoding Cumulative Probability

Cumulative probability refers to the total probability from the left up to a certain point on a probability distribution curve. This is crucial when we are analyzing the likelihood of certain outcomes happening, especially when they are lower than or equal to a specific value.
In the context of our exercise, cumulative probability helps us find the probability of selecting 130 people whose average body temperature is 98.25 or lower.
From our example, when you find the cumulative probability associated with a z-score of \(-3.075\), it indicates how likely it is to randomly draw a sample with such an average. Since the probability was roughly 0.00105, this results in only about a 0.105% chance, making the particular average seem quite unlikely.
This insight is essential for statisticians and researchers when making decisions based on probability and expected outcomes.