A statistical measurement of Spencer’s inventory is conducted for finding the variables for controlling the quality of output of the production. Quality control in production means managing the variance of output products from the production.The quality control officer or the project manager actively collects production samples to measure the production quality; the project manager collects a sample of production to conclude about the whole production units. The project manager has to use a systematic plan for collecting perfect samples that ideally represent the population parameter.

For measuring the variance of Spencer’s production, a set of random sampling methods is adopted. For having a more representative sample of the population, the random sampling method is the most suitable one. The product type that is to be measured should follow a standard measurement, and all the produced items should have a measurement around the standard version, and literally, it should the ideal production scenario of Spencer.

As the products are being produced in the same process so the quality should have a standard dispersion, if there are any variations in the production, then the random selection of different products would be the best measurement for having an idea of the product quality of different times. For this specific analysis, the minimum sample size would be 388 to support the 95% confidence interval.

The samples were randomly chosen from the population measurement by applying Rand () formula in an Excel sheet, which has generated a great a random number for each observation. Then the data were sorted as per the randomly generated number and finally, a total of 388 samples were precisely selected from the total population, which represents the characteristics of the population data and the sample is assuming a 5% sampling error.

The process that uses sample measurement’s result and concludes about the population parameter is recognized as the generalization process. The generalization process requires calculation of the sample parameter i.e. mean, median, sample variance, and standard deviation; and then the process requires calculating and formulating the confidence interval which provides us with an idea of the range in which population means can fall within.

Using the sampling process is relatively easier, but it is not so easy to conclude about the population parameter correctly. The accuracy of a generalization process depends on several things, sample size and sample selection process are the most important issues among them. The bigger the sample size, the more accurate the generalized result would be; and the better the sample represents the population, the closer the sample result would be to the population parameter.

The next thing for generalizing the sample statistics is the calculation of confidence interval which is a range of value derived from the sample statistics and contains the possibility of having the unknown population parameter within its range. Putting emphasise on the importance of the usage of confidence interval (Cumming, 2014, p.25) showed different possible uses of the confidence interval and error calculation in the modern research idea named meta-analysis.

The confidence level of the confidence interval signifies the probability of having the true population parameter value within the intervals. For Spencer’s inventory a 95% confidence interval signifying that there is a probability of 95% that the population parameter will be within the interval range and only 5% possibility of having a result outside the normal distribution curve. Hoekstra, Morey, Rouder, &Wagenmakers (2014) concluded that a great number of researchers often misinterpret the confidence interval, so it is essential to carefully interpret the confidence interval of Spencer’s inventory.

The confidence interval of Spencer’s inventory should follow a t distribution as it is assumed that we do not have the population data, so the population variation is unknown, and we have a sample size of 388. From the calculation of the collected data of Spenser’s inventory we find the following:

Mean,

So, the confidence interval of Spencer’s inventory (CI), .

So, the confidence interval is, (155.52, 288.41). That means there is a 95% possibility that the mean of Spencer’s inventory will be within the range starting from 155.52 to 288.41. From the calculation of the confidence interval, it can be seen that mean of the selected random sample of Spencer’s inventory should be within the given confidence interval. But as statisticians, we need to make sure that the confidence interval is containing the population mean. For testing the viability of calculated confidence interval, a further experiment on the statistic is necessary. The confidence interval derived from the selected random samples under some assumption

- The sample is selected randomly or each observation having equal probability of being selected in the sample.
- The level of confidence is 95%, population mean unknown, population variations unknown
- Sample size is large enough to represent the population characteristics properly
- Population mean follows a standard normal distribution

To conclude about the population mean, a further test is required to check the integrity of the data received from confidence interval calculation. In order to prove the data integrity, I would like to hypothesize that the population Mean is equal to as the measurement is considered to be the standard while measuring the dispersion; so, the expectation of having an equal measurement would match the expectation.

Null Hypothesis, [Sample mean and population mean are equal]

Alternative Hypothesis, [Sample mean and population mean are not equal]

Level of Significance or 5% level of significance is allowed for the test.

The sample will follow a “t” statistics as the population parameter is unknown, and the t statistic is,

The test result suggests that is less than the table value of “t”.

## Test Decision: Do not reject the null hypothesis

In other words, it can be concluded that there is a 95% confidence that the population mean is between 155.52 and 288.41, and there is little or no significant difference between the sample mean and the population mean. The hypothesis testing especially linear mixed effect models have been serving as the best model for testing the statistical parameter, except from the F minus model (Barr, Levy, Scheepers, & Tily, 2013).

Based on the calculation of the sample data it can be said that there is a 95% possibility that population mean will remain within the calculated range of confidence interval, and sample mean and the population mean are the same; which means there is little or no difference between the sample mean and the population mean.

## References

Barr, D. J., Levy, R., Scheepers, C., &Tily, H. J. (2013). *Random effects structure for confirmatory hypothesis testing: Keep it maximal*. *Journal of memory and language*, *68*(3), 255-278. Web.

Cumming, G. (2014). *The new statistics: Why and how*. *Psychological science*, *25*(1), 7- 29. Web.

Hoekstra, R., Morey, R. D., Rouder, J. N., &Wagenmakers, E. J. (2014). *Robust misinterpretation of confidence intervals*. *Psychonomic bulletin & review*, *21*(5), 1157-1164. Web.