Question

Describe how the mean, standard deviation, and standard error are used in estimating the number of defects in a population.

Answer 1

Answer:

The population means is the mean of the entire population which is difficult to estimate as many times we don't know the true population data. Similar is the case for standard deviation and standard error for a population.

Normally we collect a representative sample from the population through a structured sampling process and calculate the mean, standard deviation, and other parameters of the sample.

Standard deviation is the spread of a variable and standard error the variation of the sample mean.

Normally standard deviation is unknown and to get an estimate of the spread we estimate it from the sample:

s is the sample standard deviation, n is the number of sample.

Now given the question where we are being asked to estimate the number of defects in a population, it is not possible to know the exact number of defects of a population unless we have the data of the entire population. Getting this data is difficult and time-consuming.

So we estimate the number of defects of the population with a certain amount of confidence(Confidence Interval) which is the confidence level of having the true population defects within that range. Normally the confidence level is 95% or 99%. The higher the confidence interval the larger the lower and upper limits of the confidence interval.

If we assume that the population is normally distributed and we want to have a 95% confidence level for estimating the population defect, then the range of defect is given by:

....... formula 1

where x-bar is the sample mean, z is the z score, n is the number of samples, s is the standard deviation the sample.

Calculation of z score: In excel use the command NORM.INV(CI, Mean, Std.Dev) = NORM.INV(0.975,0,1)=1.96

So we are 95% confident that the population defects will be between the range:

Summary :

So in order to estimate the population defect rate we need the sample mean, confidence interval, sample standard error. It will be always a range and the spread of the range will depend on the confidence interval.