Question

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 78 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean μ = 78 tons and standard deviation σ = 1.2 ton.

(a) What is the probability that one car chosen at random will have less than 77.5 tons of coal? (Round your answer to four decimal places.)


(b) What is the probability that 21 cars chosen at random will have a mean load weight x of less than 77.5 tons of coal? (Round your answer to four decimal places.)


(c) Suppose the weight of coal in one car was less than 77.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment?

Yes No    

Suppose the weight of coal in 21 cars selected at random had an average x of less than 77.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Yes, the probability that this deviation is random is very small. Yes, the probability that this deviation is random is very large.     No, the probability that this deviation is random is very small. No, the probability that this deviation is random is very large.

Answer 1

(a)

= 78

= 1.2

To find P(X<77.5):

Z = (77.5 - 78)/1.2

= - 0.4167

Table of Area Under Standard Normal Curve gives area = 0.1628

So,

P(X<77.5) = 0.5 - 0.1628 = 0.3372

So,

Answer is:

0.3372

(b)

= 78

= 1.2

n = 21

SE = /

= 1.2/

= 0.2619

To find P(<77.5):

Z = (77.5 - 78)/0.2619

= - 1.9094

Table of Area Under Standard Normal Curve gives area = 0.4719

So,

P(<77.5) = 0.5 - 0.4719 = 0.0281

So,

Answer is:

0.0281

(c)

Absolute value of Z Score = 0.4167 < 2. So, the value is not more than 2 standard deviation away from mean. So, it is not unusual value.

So,

Correct option:

No

(d)

Absolute value of Z Score = 1.9094 < 2. So, the value is not more than 2 standard deviation away from mean. So, it is not unusual value.

So,

Correct option:

No, the probability that this deviation is random is very large.