Question

1) Suppose you are thinking of contracting with Company A to supply a certain component to your manufacturing facility. As part of your quality control regimen, you purchase a small sample and run some tests. Suppose you find the mass of 10 of these components to be

10.1,12.0,9.7,10.0,9.9,9.2,10.1,11.2,9.4,and 10.0.

What is the population in this scenario? What is the sample? What is population mean, and what is population variance? (you won’t be finding values for pop. mean and var., but will be describing them in terms of the theoretical population.)

Estimate population mean and population variance using sample mean and sample variance,respectively. Give the actual numerical values for your estimates. What do we know about sample mean that makes it a good point estimator for population mean? What do we know about sample variance that makes it a good point estimator for population variance?

Answer 1

mass of all components of company A is the population here.

mass of 10 components purchased is the sample.

Population mean is the average mass of all components of company A.

Population variance is average square deviation from average mass of all components of company A from its mean.

Estimate of population mean is sample mean, which is -

10.1 +12.0 + ... + 10.0 / 10

= 10.16

Estimate of population variance is the sample variance, which is -

(10.16 - 10.1)^2 + ... + (10.16 - 10.0)^2 / 10 - 1

= 0.70

Sample mean is a good estimator of population mean as it does not overestimate or underestimate the population mean. For a relatively large sample size, the sample mean is an unbiased estimate of the population mean. This sample size is considered to be 30 but works even for sample size lower than this. All of this is provided the sampling design is simple random sampling.

The only difference between population variance and sample variance is the denominator term where in population variance, we divide by N and in sample variance, we divide it by N-1. This is because the sample variance, if divided by N always underestimates the true population variance. Hence, dividing by N-1 makes it an unbiased estimator of population variance.