Suppose we have a set of body temperatures with a mean of 98.6 degrees, and a sample standard deviation of 1.0 degree. Assuming a normal distribution of body temperatures in the larger population, between what two values should 95% of all temperatures lie? Show all work
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.475 each way) to get the Z scores. Insert values above and calculate.
To determine the range between which 95% of all body temperatures lie, we need to find the range that captures 95% of the population within the mean ± a certain number of standard deviations.
We can use the concept of the standard normal distribution (also called the Z-distribution) to solve this problem. Z-scores measure the number of standard deviations a certain value is from the mean.
The first step is to find the Z-score that corresponds to the desired percentage. In this case, we want to capture 95% of the data, so we need to find the Z-score that corresponds to the upper 97.5% (since we want to find the range around the mean). We divide this into two because the normal distribution is symmetric.
Next, we can use the formula: Z = (X - μ) / σ, where X is the observed value, μ is the mean, and σ is the standard deviation.
Since we want to find the range around the mean, we want to solve for X in terms of Z: X = μ + (Z * σ).
Now, let's calculate the Z-score for the upper 97.5%:
Z = invNorm(0.975) (using a standard normal distribution table or calculator)
Z ≈ 1.96 (rounded to two decimal places)
Now, we can calculate the range of body temperatures:
Upper Limit: X = μ + (Z * σ) = 98.6 + (1.96 * 1.0) = 100.556
Lower Limit: X = μ - (Z * σ) = 98.6 - (1.96 * 1.0) = 96.644
Therefore, between 96.644 degrees and 100.556 degrees Fahrenheit, approximately 95% of all body temperatures should lie, assuming a normal distribution of body temperatures in the larger population.