SOLUTION: The volume, L litres, of emulsion paint in a plastic tub may be assumed to be normally distributed with mean 10.25 and variance s2. (a) Assuming that s2 0.04, determine P(L<10)

Algebra ->  Rate-of-work-word-problems -> SOLUTION: The volume, L litres, of emulsion paint in a plastic tub may be assumed to be normally distributed with mean 10.25 and variance s2. (a) Assuming that s2 0.04, determine P(L<10)      Log On


   

Question 1076540: The volume, L litres, of emulsion paint in a plastic tub may be assumed to be normally distributed with mean 10.25 and variance s2.
(a) Assuming that s2 0.04, determine P(L<10)
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the volume is a normal distribution with mean = 10.25 and variance = s^2 (?)

if s^2 = .04, then s = sqrt(.04) = .2

s is the standard deviation.

you have:

m =10.25
s = .2

you want to find the probability of the number of liters being 10 or less.

you calculate the z-factors as follows:

z = (x-m)/s

z is the z-factor
x is the raw score
m is raw mean
s is the standared deviation

you get z = (10 - 10.25) / .2

solve for z to get z = -.25 / .2 = -.125

use a z-score calculator or a z-score table to find the area to the left of a z-score of -.125.

i used the the following calculator.

http://stattrek.com/online-calculator/normal.aspx

i used the following table.

http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

i entered a z-score of -.125.
the mean is and the standard deviation is 1 for a z-score.
the calculator then told me that the probability of getting a z-score to the left of that (less than that) was .45026

the calculator told me that the probability of a z-score being less than -.125 is .45026

with the table, i had to do a little interpolating.

this is because the table only goes to 2 decimal places.

i looked for the probability to the left of -.12 and to the left of -.13.

a z-score of -.12 gave an area to the left as .4522.

a z-score of -.13 gave an area to the left as .4483.

since -.125 is roughly in between, then i calculated that the probability would be .45025

the calculator is more accurate than interpolation because interpolation is straight line while the actual values in between are more on a curve.

here's a picture of what the calculator shows:

$$$

here's a picture of what the correct row on the table shows:

$$$

$$$

in the table, the first column gets you to the row that contains a z-score of -.1.

that's one decimal place accuracy only.

on that same row, the third column gets you the area to the left of a z-score of -.12 and the fourth column gets you to the area to the left of a z-score of -.13.

that's twodecimal place accuracy only.

you then interpolate to get the area to the left of a z-score of -.125