Answer to Question #341844 in Statistics and Probability for Natalia

Question

Question.1:

An adult has on average 5.4 liters of blood. Assume the variable is normally distributed and has a standard deviation of 0.4. Find the percentage of people who have less than 5.7 litres of blood in their system.

Question.2 :

The average annual salary for all U.S teachers is $ 48,675. Assume that the distribution is normal and the standard deviation is $ 4995. Find the probability that a randomly selected teacher earns:

(i) Between $36000 and $42000 a year.

(ii) More than 40,000 a year.

Question.3:

The average daily jail population in the U.S is 706,242. If the distribution is normal and the standard deviation is 52,145, find the probability that on a randomly selected day the jail population is:

(i) Greater than 750,000

(ii) More than 600,000 but not more than 700,000

Answer 1

Question1:

$$P(X<5.7)=P(Z<\dfrac{5.7-5.4}{0.4})$$

$$=P(Z<0.75)=0.7734$$

Question2:

(i):

$$P(36000<X<42000)$$

$$=P(Z<\dfrac{42000-48675}{4995})$$

$$-P(Z\le\dfrac{36000-48675}{4995})$$

$$=P(Z<-1.3363)-P(Z\le-2.5375)$$

$$\approx0.0851$$

(ii):

$$P(X>40000)=1-P(Z\le\dfrac{40000-48675}{4995})$$

$$\approx1-P(Z\le-1.7367)\approx0.9588$$

Question3:

(i):

$$P(X>750000)=1-P(Z\le\dfrac{750000-706242}{52145})$$

$$\approx1-P(Z\le0.83916)\approx0.2007$$

(ii):

$$P(600000<X<700000)$$

$$=P(Z<\dfrac{700000-706242}{52145})$$

$$-P(Z\le\dfrac{600000-706242}{52145})$$

$$=P(Z<-0.1197)-P(Z\le-2.0374)$$

$$\approx0.4316$$