Question

Part 1: The mean daily production of a herd of cows is assumed to be normally distributed with a mean of 32 liters, and standard deviation of 9 liters.

A) What is the probability that daily production is between 24.8 and 30.9 liters? Do not round until you get your your final answer.

Part 2: The mean daily production of a herd of cows is assumed to be normally distributed with a mean of 31 liters, and standard deviation of 9.8 liters.

A) What is the probability that daily production is less than 58.7 liters?

Answer= (Round your answer to 4 decimal places.)

B) What is the probability that daily production is more than 12.7 liters?

Answer= (Round your answer to 4 decimal places.)

Part 3 Company XYZ know that replacement times for the DVD players it produces are normally distributed with a mean of 7.7 years and a standard deviation of 1.5 years.

Find the probability that a randomly selected DVD player will have a replacement time less than 3.8 years?
P(X < 3.8 years) =

Enter your answer accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

If the company wants to provide a warranty so that only 3.2% of the DVD players will be replaced before the warranty expires, what is the time length of the warranty?
warranty = years

Answer 1

Solution :

Given that ,

1) mean = = 32

standard deviation = = 9

a) P( 24.8< x <30.9 ) = P[(24.8-32)/9 ) < (x - ) /< (30.9-32) /9 ) ]

= P( -0.80< z < -0.12)

= P(z <-0.12 ) - P(z <-0.80 )

Using standard normal table

= 0.4522- 0.2119 = 0.2403

Probability = 0.2403

2)

mean = =31

standard deviation = =9.8

a) P(x < 58.7 ) = P[(x - ) / < (58.7-31) /9.8 ]

= P(z < 2.83 )

= 0.9977

probability = 0.9977

b)

P(x > 12.7) = 1 - p( x< 12.7 )

=1- p P[(x - ) / < (12.7-31) /9.8 ]

=1- P(z < -1.87 )

= 1 - 0.0307 = 0.9693

probability = 0.9693

3)

mean = = 7.7

standard deviation = =1.5

a) P(x < 3.8 ) = P[(x - ) / < ( 3.8-7.7) /1.5 ]

= P(z < -2.600 )

= 0.0047

probability =0.0047

b) 3.2%

P(Z < z ) = 0.032

z = -1.85

Using z-score formula,

x = z * +

x = -1.85 *1.5 +7.7

x = 4.93

ANSWER = 4.93 Years