Respuesta :
Answer:
The temperature that separates the bottom 12% from the top 88% is 97.5°F.
Step-by-step explanation:
We are given that human body temperatures are normally distributed with a mean of 98.2°F and a standard deviation of 0.62°F.
Let X = human body temperatures
So, X ~ Normal([tex]\mu= 98.2,\sigma^{2} = 0.62^{2}[/tex])
The z-score probability distribution for the normal distribution is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = mean human body temperature = 98.2°F
[tex]\sigma[/tex] = stnadard deviation = 0.62°F
Now, we have to find the temperature that separates the bottom 12% from the top 88%, that means;
P(X < x) = 0.12 {where x is the required temperature}
P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{x-98.2}{0.62}[/tex] ) = 0.12
P(Z < [tex]\frac{x-98.2}{0.62}[/tex] ) = 0.12
Now, the critical value of x that represents the bottom 12% of the area in the z table is given as -1.1835, that is;
[tex]\frac{x-98.2}{0.62} = -1.1835[/tex]
�� [tex]{x-98.2}= -1.1835\times 0.62[/tex]
[tex]x = 98.2 -0.734[/tex] = 97.5°F
Hence, the temperature that separates the bottom 12% from the top 88% is 97.5°F.
