Question

The lengths of lumber a machine cuts are normally distributed with a mean of 89 inches and a standard deviation of 0.3 inches.​

(a) What is the probability that a randomly selected board cut by the machine has a length greater than 89.11 ​inches?​

​The probability is _____?

​(Round to four decimal places as​ needed.)

(b) A sample of 42 boards is randomly selected. What is the probability that their mean length is greater than 89.11 ​inches?

​The probability is _____?

​(Round to four decimal places as​ needed.)

Answer 1

Solution :

Given that ,

mean = = 89

standard deviation = = 0.3

P(x >89.11 ) = 1 - P(x< 89.11)

= 1 - P[(x -) / < (89.11 -89) /0.3 ]

= 1 - P(z <0.37 )

Using z table

= 1 - 0.6443

= 0.3557

probability= 0.3557

(B)

n = 42

= 89

= / n = 0.3 / 42 =0.0463

P( >89.11 ) = 1 - P( <89.11 )

= 1 - P[( - ) / < (89.11 -89) /0.0463 ]

= 1 - P(z < 2.38)

Using z table

= 1 - 0.9913

= 0.0087

probability= 0.0087