With a large and infinite population size we cannot evidently embark on measuring every single member of the population dataset. For this very reason we have to donate a small sample set from the entire population and then represent it via the variable n which would be expressed as:
The above sample set provides us with (X Bar) which will be the arithmetic mean of the selected dataset. To make the selection for the population completely unbiased we will employ random sampling based on the receipts that are generated by customer purchases. Every day for a month one customer will be selected at random (using ballot method) from each branch and their data be tabulated for analysis ( Mara & Cribbie, 2018) . After 30 days of successful balloting the sample size would be large enough to provide statistical inference and concurrently we will employ the following statistical methods:
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Arithmetic Mean & Standard deviation
The Sigma symbol denotes summation and is used frequently to denote the sum of all values. The arithmetic mean is then followed by the standard deviation of the population dataset which can be attributed to providing the net average of the deviation that we may find in the data set while the expression ca be denoted as follows:
The standard deviation gives precedence to unbiased population variance which can be represented as the square of the Standard deviation. The population variance hence becomes:
The population variance gives us the total variance on how each element in the data set differs from each other and provides a key snapshot into the accuracy of the sample procured.
The Use of Z Scores
In a normal distribution Z scores are used to calculate the probability of a sample data set and how far it is from the sample mean. Z scores are expressed in terms of standard deviations and can be notated using the following:
Now if we are to apply the same Z score parameters on our data set we would know exactly by how much each value is deviating from the sample mean. To take an example of such an instance we can isolate the population variance, which are 930 in our sample data set, and using the preceding formula we can find that the number procured is 0.086 standard deviations away from the sample mean (Shahzad et al, 2018).
Similarly, we can also calculate a Z score for overall statistical inference which amounts to 0.57 standard deviations away from the sample mean. The exercise of providing Z scores in a normally distributed data set helps us identify outliners in a set of observations and therefore enables the governing body to make adjustments to policy as well (Sengupta, 2018).
References
Mara, C. A., & Cribbie, R. A. (2018). Equivalence of population variances: Synchronizing the objective and analysis. The Journal of Experimental Education, 86(3), 442-457.
Shahzad, U., Hanif, M., Koyuncu, N., & Sanaullah, A. (2018). On the estimation of population variance using auxiliary attribute in absence and presence of non-response. Electronic Journal of Applied Statistical Analysis, 11(2), 608-621.
Sengupta, S. (2018). Admissible unbiased estimation of finite population variance under a randomized response model. Communications in Statistics-Theory and Methods, 47(20), 5077-5082.
