SOLUTION: An orange juice producer buys oranges from a large orange grove that has one variety of oranges. The amount of juice squeezed from these oranges is approximately normally distri

Question 1203058: An orange juice producer buys oranges from a large orange grove that has one variety of oranges. The amount of juice squeezed from these oranges is approximately normally distributed, with a mean of 4.10 ounces and a standard deviation of 0.20 ounces. Suppose that you select a sample of 16 oranges.
a. What is the probability that the sample mean amount of juice will be at least 4.01 ounces?

b. The probability is 78% that the sample mean amount of juice will be contained between what two values symmetrically distributed around the population mean? There is a 78% probability that the sample mean amount of juice will be contained between ______ounce(s) and _____ ounce(s) (Round to two decimal places as needed. Use ascending order.)

c. The probability is 73% that the sample mean amount of juice will be greater than what value?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
mean = 4.1 ounces.
standard deviation = .2 ounces.
sample size is 16 oranges.
standard error = standard deviation / square root of sample size = .2 / sqrt(16) = .05.

use the z-score formula of z = (x-m)/s
z is the z-score.
x is the sample mjean.
m is the population mean.
s is the standard error.

a. What is the probability that the sample mean amount of juice will be at least 4.01 ounces?

when x = 4.01, the formula becomes z = (4.01 - 4.1) / .05 = -1.8.
area to the right of that z-score is equal to .9641.

that's the probability that the mean of the sample will be at least 4.01 ounces.

b. The probability is 78% that the sample mean amount of juice will be contained between what two values symmetrically distributed around the population mean? There is a 78% probability that the sample mean amount of juice will be contained between ______ounce(s) and _____ ounce(s) (Round to two decimal places as needed. Use ascending order.)

if the area between the two z-scores is .78, then the area outside of that range will be 1 - .78 = .22 / 2 = .11.

you will need 2 z-scores.
the high z-score will have an area to the left of it of 1 - .11 = .89
the low z-score will have an area to the left of it equal to .11.
using a z-score calculator, i get:
z-score with area of .89 to the left of it = 1.2265.
z-score with area of .11 to the left of it = -1.2265.
area in between will be .89 - .11 = .78.
use the z-score formula to find the raw score.
z = (x-m)/s
z is the z-score
x is the raw score = the mean of the smple
m is the mean
s is the standard error.
solve for x in this formula to get x = s * z + m.
when z = -1.2265 and m = 4.1 and s = .05, the formula becomes x = .05 * -1.2265 + 4.1 = 4.038675.
when z = .2265 and m = 4.1 and s = .05, the formula becomes x = .05 * 1.2265 + 4.1 = 4.161325.
the area between those two sample means will be equal to.78.

c. The probability is 73% that the sample mean amount of juice will be greater than what value?

z-score will have an area of 1 - .73 = .27 to the left of it.
that z-score will be equal to -.6128.
the sample mean will be equal to x = z * s + m which would be equal to x = -.6128 * .05 + 4.1 = 4.06936.

the calculator at https://davidmlane.com/hyperstat/z_table.html can be used to confirm these results.

here are the results.

for part a.

for part b.

for part c.

RELATED QUESTIONS
there is a direct variation between number of oranges squeezed and the weight of the... (answered by ankor@dixie-net.com)

CalJuice Company has decided to introduce three fruit juices made from blending two or... (answered by Theo)

How much of a(n) 90% orange juice drink must be mixed with 16 gallons of a 30% orange... (answered by richwmiller)

A pure fruit juice is made by mixing the juices of oranges and mangoes in the ratio of... (answered by Cromlix)

There is a direct variation between the number of oranges squeezed and the weight of... (answered by mananth)

Mrs Parkson buys some oranges from a supermarket. She finds that: I. If she buys a dozen (answered by josgarithmetic,MathTherapy)

An orange drink claims that it contains 10% orange juice. How much pure orange juice... (answered by ptaylor)

I have a mixture problem that reads: A woman has made 6 quarts of an orange juice punch... (answered by ankor@dixie-net.com)

Leslie set up a fruit stand to sell some of her crop of oranges. The first person who... (answered by ptaylor)

SOLUTION: An orange juice producer buys oranges from a large orange grove that has one variety of oranges. The amount of juice squeezed from these oranges is approximately normally​ distri

SOLUTION: An orange juice producer buys oranges from a large orange grove that has one variety of oranges. The amount of juice squeezed from these oranges is approximately normally distri