Answer to Question #329498 in Statistics and Probability for frank

Question

The value of serum sodium in healthy adults approximately follows a normal distribution with a mean of 141meq/l and standard deviation of 3meq/l

(a) What is the probability that a normal healthy adult will have a serum sodium value above 147meq/l ( 5 marks)

(b) What is the probability that a normal healthy adult will have a serum sodium value below 130 meq/l ( 5 marks)

(c) What is the probability that a normal healthy adult will have a serum sodium value between 132 and 150meq/l (5 marks)

Answer 1

We have a normal distribution, $\mu=141, \sigma=3.$

Let's convert it to the standard normal distribution,

$z=\cfrac{x-\mu}{\sigma}.$

$(a) \ z=\cfrac{147-141}{3}=2;\\ P(X>147)=P(Z>2)=\\ =1-P(Z<2)=\\ =1-0.9772=0.0228 \text{ (from z-table).}$

$(b) \ z=\cfrac{130-141}{3}=-3.67;\\ P(X<130)=P(Z<-3.67)=\\ =0.0001 \text{ (from z-table).}$

$(c) \ z_1=\cfrac{132-141}{3}=-3;\\ z_2=\cfrac{150-141}{3}=3;\\ P(132<X<150)=P(-3<Z<3)=\\ =P(Z<3)-P(Z<-3)=\\ =0.9987-0.0013=0.9974 \text{ (from z-table).}$