Recall the body temperature data reported for healthy men and women, aged 18-40 years, who were volunteers in Shigella vaccine trials at the University of Maryland Center for Vaccine Development, Baltimore. For these adults, the mean body temperature was found to be 98.249 ^oF with a standard deviation of 0.733 ^oF. 

In Investigation 2.5, we decided that these data provided convincing evidence that μ, the mean body tempreature of healhty adults in the U.S., differed from the previous benchmark of 98.6 degrees, with a 95% confidence interval of (98.12, 98.38) degrees.

Research Question: Based on these data, can we predict the temperate of a healthy adult in the U.S.?

Goals: In this lab, you will

• Explore the distinction bewteen a confidence interval and a prediction interval
• Learn how to calculate and interpret a prediction interval for a quantitative variable

(b) Suppose we assume that μ = 98.249 and σ = 0.733, between what two values do we estimate the middle 95% of body temperatures falls? (Hint: Check both "X" boxes. You can use trial and error or recomember the empirical rule...

It is very important to keep in mind that the confidence interval only makes statements about the population mean, not individual body temperatures. Instead, we want a prediction interval. A prediction interval is a type of confidence interval, but it applies to an individual observation rather than a population mean. The Empirical Rule gave us a rough estimate of an interval for individual body temperatures, but that assumed we knew the population mean and population standard deviation. To calculate a prediction interval, we need to incorporate both the uncertainty in our estimate of μ and the variability in the individual temperatures around the population mean.

 

To formally construct a prediction interval, we will add these two sources of uncertainty together.

The square root of this simplifies to

se(individual) = 

You should have found a standard error of 0.0643.  When the sample size n is large, this will be similar to the value of s.  Otherwise, it will be larger to reflect the uncertainty in our estimate of μ as well. This impacts the width of the interval.

Whne the population has a normal distribuiton, we can use this standard error and a t critical value to construct the interval

Use the t-probability calculator applet to find the critical value for 95% confidence. (Hint: What are you using for the degrees of freedom?)

A prediction interval is a type of confidence interval, but applies to an individual observation rather than a population mean. It is much more difficult to predict an individual's body temperature (e.g., the next person to walk in the room) than it is to predict the population mean. This results in a much wider, but perhaps more relevant, confidence interval.

Keep in mind that this prediction interval procedure is valid only if the population distribution is normally distributed. We often don't have access to the population so we could look at the sample or based on the context of the variable involved, we may consider the population model reasonable but we should admit this caveat when we share our results.

If you do have access to the individual data values, in JMP you can choose Analyze > Distribution, and then use the hot spot to select Prediction Interval.  The results will match the formula used here.