Respuesta :
Answer:
Sally's temperature is 97.27 °F.
Step-by-step explanation:
All the information given in the question tells us that the human body temperatures are normally distributed with a population's mean = 98.20°F and a standard deviation = 0.62°F.
The question gives us Sally's temperature in a z-score. We have to remember that the standard normal distribution is a particular case of a normal distribution where the mean = 0 and the standard deviation = 1.
Using the standard normal distribution, we can determine every probability associated with a normal distribution "transforming" the raw scores, coming from normally distributed data, into z-scores.
A z-score gives us the distance from the population's mean and is in standard deviation units. So, a z = 1.5 tells us that the value is 1.5 standard deviations above the mean. Conversely, a z = -1.5 tells us that the raw score is also 1.5 standard deviation from the mean, but in the opposite direction, that is, below the mean.
The formula for a z-score is as follows:
[tex] \\ z = \frac{x - \mu}{\sigma}[/tex] (1)
Where
[tex] \\ x\;is\;the\;raw\;score[/tex].
[tex] \\ \mu\;is\;the\;population\;mean[/tex].
[tex] \\ \sigma\;is\;the\;population\;standard\;deviation[/tex].
Then to find x (or the raw score, that is, Sally's temperature), we need to solve the formula (1) for it to finally solve the question.
Then
[tex] \\ \mu = 98.20^\circF[/tex] °F
[tex] \\ \sigma = 0.62^\circF[/tex] °F
[tex] \\ z = -1.5[/tex]
Thus (with no units)
[tex] \\ -1.5 = \frac{x - 98.20}{0.62}[/tex]
[tex] \\ (-1.5*0.62) = x - 98.20[/tex]
[tex] \\ (-1.5*0.62) + 98.20 = x[/tex]
[tex] \\ x = (-1.5*0.62) + 98.20[/tex]
[tex] \\ -0.93 + 98.20[/tex]
[tex] \\ x = 97.27[/tex]°F
Thus, Sally's temperature is [tex] \\ x = 97.27[/tex]°F (rounding the answer to the nearest hundredth).
Answer:
Sally's temperature is 97.27° F .
Step-by-step explanation:
We are given that Human body temperatures have a mean of 98.20° F and a standard deviation of 0.62° F.
Let X = Human body temperatures
So, X ~ N([tex]\mu = 98.20,\sigma^{2}=0.62^{2}[/tex])
The z score probability distribution is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)
Let Sally's temperature be x ; we are given that Sally's temperature can be described by z = -1.5; i.e.;
-1.5 = [tex]\frac{x-98.20}{0.62}[/tex]
x = 98.20 - (1.5 * 0.62)
x = 98.20 - 0.93 = 97.27° F
Therefore, Sally's temperature is 97.27° F .
