Question

The average production cost for major movies is 59 million dollars and the standard deviation is 18 million dollars. Assume the production cost distribution is normal. Suppose that 40 randomly selected major movies are researched. Answer the following questions. Round all answers to 4 decimal places where possible.

1. What is the distribution of XX? XX ~ N(_____,_____)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(______,______)
   1. For a single randomly selected movie, find the probability that this movie's production cost is between 58 and 61 million dollars.______
3. For the group of 40 movies, find the probability that the average production cost is between 58 and 61 million dollars.______
4. For part d), is the assumption of normal necessary? No/Yes

Answer 1

Solution :

Given that ,

mean = = 59

standard deviation = =18

a) The distribution is approximately normal.

X ~ N ( 59,18)

b) The sampling distribution is approximately normal.

~ N (

n = 40

= = 59

= / n = 18 / 40 = 2.8460

P( 58< x < 61 ) = P[( 58 -59 )/18 ) < (x - ) /  < (61-59 ) /18 ) ]

= P( -0.06 < z < 0.11 )

= P(z < 0.11) - P(z <-0.06 )

Using standard normal table

= 0.5438

Probability = 0.5438

c) P( 58 <   < 61 )  

= P[(58 -59) /2.8460< ( - ) / < (61 -59 ) / 2.8460 )]

= P( -0.35 < Z < 0.70)

= P(Z <0.70 ) - P(Z < -0.35)

= 0.758 - 0.3632 = 0.3948

probability =0.3948

d) Yes.