In: Advanced Math
(answer all three parts} Normal (i.e. average)
internal temperature for humans is approximately 98 degrees,
If a given population has a standard deviation of 1.2 degrees, what
is the maximum temperature for the lowest 35% of the
population? What is the minimum temperature for the highest
25% of the population.
2. If you take a 50 question true-false final exam
(two-points each question) and you never paid attention in the
class (or knew anything about the topic on your own), what is
probability you will receive a grade of 56 or less, in which case
you will fail the class? What is the probability you will get
at least 30 questions right, which will give you a passing
grade in the class?
Solution:---
X : Internal temperature of humans
X is normally distributed with mean 98 degrees and standard deviation 1.2 degrees.
Let X35 be the maximum temperature for the lowest 35% of the population i.e
P(X<X35) = 0.35
Let Z35 be the z-score of X35
Z35 = (X35 - mean)/Standard deviation = (X35-98)/1.2
X35 = 98+1.2Z35
P(Z<Z35) = P(X<X35) = 0.35
From standard normal tables ,
P(Z<--0.38) = 0.3520; P(Z<-.39) = 0.3483
Therefore, average of -.38 and -.39 i.e -.385 is a good approximation
P(Z<-.385) = (0.3520+0.3483)/2=0.35015
Z35 = -.385
X35 = 98+1.2Z35 = 98 + 1.2 * (-.385) =98-0.462 = 97.538
Maximum temperature for the lowest 35% of the population = 97.538
----------------
Let X25 minimum temperature for the highest 25% of the population i.e P(X>X25) =0.25
P(X>X25) =1-P(X<X25) = 0.25; P(X<X25) =0.75
Z25 be the Z-score for X25
Z25 = (X25 - mean)/Standard deviation = (X25-98)/1.2
X25 = 98+1.2Z25
P(Z<Z25) = P(X<X25) = 0.75
From standard normla tables,
P(Z<0.67) = 0.7486 ; P(Z<0.68) = 7517
Therefore, average of .67 and .68 i.e .675 is a good approximation
P(Z<.675) = (0.7486+0.7517)/2=0.75015
Z25 = 0.675
X25 = 98+1.2Z25 =98+1.2 * 0.675 = 98+0.81=98.81
minimum temperature for the highest 25% of the population = 98.81
2
This is a binomial distribution question with
n = 50
p = 0.5
q = 1 - p = 0.5
This binomial distribution can be approximated as Normal
distribution since
np > 5 and nq > 5
Since we know that
The z-score at x = 28.5 is,
z = 0.99
This implies that
The z-score at x = 29.5 is,
z = 1.2728
This implies that