Empirical Rule The Empirical Rule applies to a normal, bell-shaped curve than is symmetrical about the mean. It states that within one standard deviation of the mean (both left-side and right-side) there is about 68% of… Empirical Rule and z-score Probability
The Finite Population Correction Factor, sometimes just called the FPC factor, is used when the sample size is large relative to the population size. For most situations, the population is so large, typical sample sizes are far too small to worry about the need for the FPC.
The guidance is that we need to use the FPC when the ratio of the sample size n to the population size N is greater than 5%. For example, if the population size is 300 and the sample size is 30, we have a ratio of 10% and thus need to use the FPCF.
The most common formula for calculating the FPC isFinite Population Correction factor
Excel’s Data Analysis ToolPak has three tools for running tests of hypotheses using the t-distribution – t-tests. The output from the tools can be a bit confusing because, unlike other statistical software, these do not… Tail of the Test: Interpreting Excel Data Analysis t-test output
Here is a common problem from intro stats: [7.4.30t] A random sample of 100 observations from a population with a standard deviation of 44 yielded a sample mean of 108. Test the null hypothesis that… Single-sample z-test for the mean [7.4.30t]
Every time I teach the Central Limit Theorem, I get questions from students on why we divide the population standard deviation, sigma, by the square root of the sample size to calculate the standard deviation of the sampling distribution which we call the standard error.
Recall that the equation for the standard error is
where σ is the population standard deviation and n is the sample size.
One can find any number of precise “academic explanations” of why this is true, and I give my students links to those references. But I often get follow-up questions from students who look at those references and then ask for a simpler, clearer explanation that does not involve a lot of algebra.
So, I am going to attempt one here and in the companion video.
Let’s begin with our reliable dice as an example of a population. If we use the standard six-sided dice, and assuming the dice are fair, and the rolls are fair, then each face has an equal chance of coming up. Because there are six faces, 1,2,3,4,5,6, each face has 1/6 or 16.6666666…% chance of coming up.
Let’s also assume that each roll of the dice is a sample of size n=1 since we just have one die. We know from basic probability that we need to look at the long-term when we are looking at empirical probabilities, so let us make 10,000 rolls of our single die. So, we then have 10,000 samples of sample size n = 1.
Back in the dark ages when access to computers was not all that common, I was faced with developing a project schedule for, to me, a complex construction project. I was not that long out of school, so I sought out my boss with the hope he would give me some guidance on how to approach the problem.
He told me to use three-point estimation and to talk to some of the older engineers in the firm to get their ideas on the likely outcomes. So, I did and learned that the three points he was talking about were the worst case, the best case, and the most likely case for what would happen during the project. (Wikipedia, n.d.)
He also directed me to consider using PERT. I did and learned that form of project management scheduling including consideration of the optimistic time estimate (o), the most likely or normal time estimate (m), and the pessimistic time estimate (p). In PERT, instead of using probabilities for each estimate of the time required, the task time is calculated as (o + 4m + p) ÷ 6. (Taylor Jr., 2011)
To model a three-point estimate with a probability distribution you need to use a triangular distribution. Today, three-point estimates are commonly used in business and engineering, so it is somewhat surprising that Excel does not have a built-in function to help. I was recently faced with this dilemma in my quantitative methods course which I am trying to migrate away from expensive software solutions.