Question

The heights of pecan trees are normally distributed with a mean of 10 feet and a standard deviation of 2 feet. Show all work. 14. (a) What is the probability that a randomly selected pecan tree is between 9 and 12 feet tall? (Round the answer to 4 decimal places) (b) Find the 80th percentile of the pecan tree height distribution. (Round the answer to 2 decimal places) (a) For a sample of 36 pecan trees, state the standard deviation of the sample mean (the "standard error of the mean"). (Round your answer to three decimal places) (b) Suppose a sample of 36 pecan trees is taken. Find the probability that the sample mean heights is between 9.5 and 10 feet. (Round your answer to four decimal places)

Answer 1

Solution :

Given that ,

a ) mean = = 10

standard deviation = = 2

P(9 < x <12 ) = P[(9-10)/2 ) < (x - ) /  < (12-10) / 2) ]

= P(-0.5 < z <1 )

= P(z <1 ) - P(z <-0.5 )

= 0.8413 - 0.3085 = 0.5328

Probability =0.5328

b) 80 %

P(Z < z) = 0.80

z =0.84

Using z-score formula,

x = z * +

x = 0.84 * 2+10

x = 11.68

a ) n = 36

= 10

= / n = 2 / 36 = 0.333

Standard deviation of the sample mean = 0.333

P( 9.5< < 10)

= P[(9.5-10) / 0.333< ( - ) / < (10-10) /0.333 )]

= P(-1.50 < Z <0)

= P(Z < 0) - P(Z < -1.50 )

= 0.50 - 0.0668 = 0.4332

Probability 0.4332