Standard deviations and probabilities

[Sir]

Homework Statement

Suppose a sample containing 0.600 ppm Selenium is analyzed by a method for which the standard deviation of the population is known to be 0.005 ppm.
1) What is the the probability that a single determination would return a value less than 0.590 ppm?
2) What is the probability that the average for 4 determinations would be less than 0.590 ppm?

Homework Equations

the table at the bottom of http://64.233.167.104/custom?q=cache:bdW_K0aRhzEJ:www.palgrave.com/business/taylor/taylor1/lecturers/lectures/handouts/hChap5.doc+chart+area+beneath+normal+curve+standard+deviations&hl=en&ct=clnk&cd=7&client=pub-8993703457585266" page tells the area under a normal curve for different standard deviations.

The Attempt at a Solution

for 1) we look on the table for 2.0 standard deviations below the mean, and see that the area under the curve is 0.4772. and at infinite standard deviations, the area is necessarily 0.5000.
so to return a value in this range (less than 0.590 ppm) the solution is ( 0.5 - 0.4772 ) * 100%
= 2.28%

for 2) i don't even know where to start. the answer given is 0.0032%. this table doesn't go far enough, but the probability of finding a value less than 0.580 (or 4 standard deviations below) is 0.0032%. I don't know whether that is relevant or not.
the internets and my textbook combined were less than helpful on this as well.
please help!

thanks.

 

[Sir]

all right! I found the equation to govern this.

let z be the number of standard deviations away from the mean
... u be the mean
... x be the value of a determination
... n be the number of determinations
... s be the standard deviation

z = (| u - x | * (n)^(1/2)) / s

 

[Architeuthis Dux]

As a scientist, it is important to understand that probabilities and standard deviations are tools used to analyze and interpret data. In this scenario, the standard deviation of 0.005 ppm tells us about the spread or variability of the data, while the probabilities tell us the likelihood of obtaining certain values within that range.

For 1), you correctly used the table to determine the probability of obtaining a value less than 0.590 ppm. This is known as a one-tailed test, where we are only interested in values below a certain threshold. Your calculation of 2.28% is correct.

For 2), we are now interested in the average of 4 determinations, which is known as a sample mean. To find the probability of obtaining a sample mean less than 0.590 ppm, we need to use a different table or formula. The formula for calculating the probability of a sample mean is:

P(x̄ < a) = P(z < (a-μ)/(σ/√n))

where x̄ is the sample mean, a is the threshold value (0.590 ppm in this case), μ is the population mean (0.600 ppm), σ is the population standard deviation (0.005 ppm), and n is the sample size (4 in this case). Using this formula, we can calculate the probability of obtaining a sample mean less than 0.590 ppm to be 0.0032%.

In conclusion, standard deviations and probabilities provide valuable information about the data and can be used to make informed decisions and draw conclusions. It is important to understand their application and limitations in order to accurately interpret results.