Question 322948: The diameters of oranges in a certain orchard are normally distributed with a mean of 7.10 inches and a standard deviation of 0.50 inches. Show all work.
(A) What percentage of the oranges in this orchard have diameters less than 6.8 inches?
(B) What percentage of the oranges in this orchard are larger than 7.00 inches?
(C) A random sample of 100 oranges is gathered and the mean diameter obtained was 7.00. If another sample of 100 is taken, what is the probability that its sample mean will be greater than 7.00 inches?
(D) Why is the z-score used in answering (A), (B), and (C)?
(E) Why is the formula for z used in (C) different from that used in (A) and (B)?
(Points :10)
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! The diameters of oranges in a certain orchard are normally distributed with a mean of 7.10 inches and a standard deviation of 0.50 inches. Show all work.
(A) What percentage of the oranges in this orchard have diameters less than 6.8 inches?
---
z(6.8) = (6.8-7.1)/0.5 = -0.6
---
P(x < 6.8) = P(z < -0.6) = 27.43%
=============================================
(B) What percentage of the oranges in this orchard are larger than 7.00 inches?
z(7) = (7-7.1)/0.5 = -0.2
---
P(x > 7) = P(z > -0.2) = 57.93%
----------------------------------------
(C) A random sample of 100 oranges is gathered and the mean diameter obtained was 7.00. If another sample of 100 is taken, what is the probability that its sample mean will be greater than 7.00 inches?
--
z(7) = (7-7.1)/[0.5/sqrt(100)] = -2
P(x-bar > 7) = P(z > -2) = 0.9772
---------------
(D) Why is the z-score used in answering (A), (B), and (C)?
I'll leave that to you. Some texts would say you should
use a t-score on problem C.
-------------------
(E) Why is the formula for z used in (C) different from that used in (A) and (B)?
The Central limit theorem say the standard deviation for a sample of size
n = sigma/sqrt(n), where sigma is the standard deviation used in part A an
==============================
Cheers,
Stan H.
|
|
|
