## Introduction

Inferential statistics are used under the circumstances that the behavior in a population, which is too large, needs to be understood. Samples are taken from the populations and the behavior of the sample is assumed to bear similar characteristics to the larger population (Black 273). The most important postulation is that the sample average should fall within the specified significance level. In sampling and sampling procedures, it is a known fact from the central tendency theorem that the value of the sample average will have a normal distribution.

Samples are selected from the larger populations to measure and learn more from that population. The point of measurement is the sample mean, which is assumed to be the same as the population means. The sampling process that would lead to more knowledge on the entire population is hypothesis testing. In other words, hypothesis testing is a method of investigating assertions or thoughts concerning a particular group (Black 273).

## The case involving hypothesis-testing dilemma

The case involves an organization determining if its sales representatives are selling on average 2.7 units per day. Within the organization’s total population of 1000 sales representatives, a sample of 100 sales executives was selected to determine if the assertion is true. The company sales manager wanted to determine if its estimate is true by picking 100 sales representatives to test the assertion.

## The testing procedures

The first step involves making out the stated assertions. For instance, in this case, the parameter is the average sales units per day. The company claims that the mean quantity of sales the employees make per day is 2.7 units. In a company with over 1000 sales representatives, it is not easy to substantiate this claim unless the appropriate measurement procedure is carried out.

The second step in hypothesis testing is to decide on the criterion upon which the claim can be substantiated (Black 273). As in this case, 2.7 is assumed to be the mean of the population and the average sample sales of the selected sales representatives should be equal or close to the population mean. In other words, this step involves choosing the level or extent of deviation that should be acceptable to decide whether the sample mean measures the population means.

The third step involves performing the random sampling procedure to come up with the sample that represents the population. For instance, in the case of the company only 100 sales representatives were selected. The alternative hypothesis is established. The mean sales of the selected sample are then measured.

The final step is to compare the mean of the selected sample to the expected or that of the population to help in drawing the inferences. The inference will be drawn depending on the discrepancy between the sample mean and that of the population. In the circumstances, that the difference is small there is a correlation between the sample mean and that of the population thus, the hypothesis is accepted. In the situation that the differences between the means are large, the claim is rejected. The large differences mean that the measured sample characteristics are different from that of the population.

Applying the steps and using the case, the hypothesis will be stated as follows:

- H0: average sales per day are 2.7 units
- H1: average sales per day are more than 2.7 units per day

Given that 100 sales representative have a mean sales of 2.9 and a standard deviation of 0.6, the p-value will be P(X > 2.9) = P[Z>(2.9 – 2.7)/(0.6/√100] = P[Z> 3.33] = 0.0004.

The null hypothesis is rejected since the p-value is too small to be found within the significance level. In other words, the alternative hypothesis is accepted that the average value of sales units per day is more than the 2.7 estimates.

## Reviewed literature on hypothesis testing of a single population

Hypothesis testing is one of the statistical tools normally applied in business decision-making processes. The tool is applied to examine assertions that are normally being made by most managers. In real business scenarios, outcomes are often predicted particularly where output is involved. Therefore, there is a need for a particular tool that would be applied to test the hypothetical predictions to accurately determine the expected outcomes. Organizations’ managers often apply hypothesis testing in circumstances where there is an assertion on the predicted outcome. In most cases, two variables are involved particularly in linear relationships. The aim of the hypothesis testing in a single population is to determine the level of accuracy in some of the propositions being made (Groebner 205).

Hypothesis testing is also applied in almost all decisions being made. In other words, hypothesis testing is applied in all studies whose outcomes are utilized in decision-making. All the research, whether scientific or academic, hypothesis testing is used to examine the initial propositions being made. As such, hypothesis testing is part of the larger studies that are made by businesses to ascertain particular outcome relations that either involve two or more variables depending on the study design.

## Ethical issues in violation of assumptions in statistics

Like any other statistical tool, hypothesis testing is conducted under various assumptions upon which the final decision of the outcome is based. The most common assumptions made in hypothesis testing include the equality of the means of the sample and the population. The other assumption is that the means follow the normal distribution.

The proof of whether the assertion being made is true involves testing the stated parameter, which in this case is the mean. While testing the parameters, violating these assumptions is likely. The result of violating the assumptions is the types one or two errors characterized by the p-values leading to the rejection or acceptance of the null hypothesis (Groebner 307). In other words, both types one and two errors are measured by the levels of the p-values. In the circumstances that the type one error is made, the p-value does not fall within the significance level hence the rejection of the null hypothesis. On the other hand, type two errors are made when the p-value falls within the significance level.

The most important implication for the violation of the assumption resulting in the rejection or acceptance of the null hypothesis is that wrong decision are likely to be made regarding the assertion being tested. In essence, the outcomes will not be accurate particularly when the assumptions are not highly considered.

## Conclusion

Hypothesis testing is also known as significance testing which is the systematic way of investigating a particular claim concerning a population parameter using the quantified data from the selected sample of the population. The procedure tests the suppositions through the determination of the probability that the sample statistic was selected in case the premise concerning the parameter of the group is real.

## Works Cited

Black, Ken. *Business Statistics: For Contemporary Decision Making*. Hoboken, NJ: Wiley Global Education, 2011. Print.

Groebner, David. *Business Statistics: A Decision-making Approach*. London, UK: Pearson Education, 2011. Print.