Assume that human body temperature are
normally distributed with a mean 98.21 F
and a standard deviation of 0.61F
a)A hospital uses 100.6 as the lowest
temperature considered to be a fever. What
percentage of normal and healthy persons
would be considered to have a fever. Does
this percentage suggest that a cut off of 100.6 F is appropriate?
b) Physicians want to select a minimum temperature for requiring further medical tests. What should that temperature, be if we only 5.0 of healthy people to exceed it.(Such a result is a false positive, meaning that
the test result is positive, but the subject is not really sick.
ISK
a) To determine the percentage of normal and healthy persons who would be considered to have a fever, we can use the standard normal distribution table.
First, we need to calculate the z-score for the cutoff temperature of 100.6°F using the formula:
z = (x - μ) / σ
where x is the cutoff temperature, μ is the mean, and σ is the standard deviation. Substituting the values, we get:
z = (100.6 - 98.21) / 0.61
z = 3.86
Using the standard normal distribution table, we can find the proportion of values greater than 3.86. However, since the table usually provides the proportion to the left of the z-score, we need to subtract the proportion from 1 to get the proportion greater than 3.86.
From the standard normal distribution table, the proportion to the left of 3.86 is approximately 0.9999. Subtracting this from 1, we find that the proportion greater than 3.86 is approximately 0.0001.
To convert this proportion to a percentage, we multiply by 100:
Percentage = 0.0001 * 100 = 0.01%
So, only 0.01% of normal and healthy persons would be considered to have a fever based on a cutoff of 100.6°F. This suggests that using a cutoff of 100.6°F may not be appropriate for identifying fever in normal and healthy individuals, as it would classify a very small percentage of them as having a fever.
b) To determine the minimum temperature for requiring further medical tests while only allowing 5.0% of healthy people to exceed it, we need to find the corresponding z-score.
We can use the standard normal distribution table again, but this time we need to find the z-score that corresponds to a cumulative proportion of 0.95 (since we want the top 5% to exceed the minimum temperature). From the standard normal distribution table, the z-score that corresponds to a cumulative proportion of 0.95 is approximately 1.645.
Now, we can use the z-score formula to find the minimum temperature:
z = (x - μ) / σ
Substituting the values, we have:
1.645 = (x - 98.21) / 0.61
Solving for x:
1.645 * 0.61 = x - 98.21
0.004 = x - 98.21
x = 98.21 + 0.004
x ≈ 98.214
Therefore, the minimum temperature for requiring further medical tests, while only allowing 5.0% of healthy people to exceed it, should be approximately 98.214°F.
To answer these questions, we need to use the concept of Z-scores, which tells us the number of standard deviations a data point is away from the mean in a normally distributed data set. Here's how we can approach both parts of the question:
a) To find the percentage of normal and healthy persons that would be considered to have a fever, we need to find the proportion of the population with a temperature greater than or equal to 100.6°F.
First, we convert 100.6°F to a Z-score using the formula:
Z = (x - μ) / σ
Where:
x = temperature (100.6°F)
μ = mean (98.21°F)
σ = standard deviation (0.61°F)
Z = (100.6 - 98.21) / 0.61 ≈ 3.8
To find the percentage using the Z-score, we can refer to a Z-table or use a statistical calculator. Looking up the Z-score of 3.8, we find that the area to the right is approximately 0.9999.
This means that approximately 99.99% of normal and healthy persons would be considered to have a fever based on the cutoff of 100.6°F. Given that this percentage is very close to 100%, it suggests that the chosen cutoff of 100.6°F might be too strict, as it would classify almost all normal and healthy persons as having a fever.
b) To select a minimum temperature for requiring further medical tests while only allowing 5% of healthy people to exceed it (resulting in a false positive), we need to find the Z-score that corresponds to the upper 5% of the distribution.
Using the Z-table, we can reverse lookup the Z-score that corresponds to an area of 0.05 (since we want the upper 5%). This Z-score is approximately 1.645.
Using the Z-score formula again, we can solve for x:
Z = (x - μ) / σ
1.645 = (x - 98.21) / 0.61
Rearranging the equation, we get:
x - 98.21 = 0.61 * 1.645
x - 98.21 = 1.0014
x ≈ 99.2114
Therefore, a minimum temperature of approximately 99.2°F should be set as the cutoff for requiring further medical tests, as this would ensure that only 5% of healthy people exceed it, minimizing false positives.
a) Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
Do you consider it appropriate?
b) It would help if you proofread your questions before you posted them. Only “5.0 of healthy people”? Five people out of how many?