An Appreciation of the Role of Statistical Hypotheses in Decision Making

 Analyze, Criticize, and Explain the following statement: "What is Significant to a Manager may not be Statistically Significant. What is not Significant to a Manager may be Statistically Significant"

preamble
One of the problems faced by a manager in an enterprise is in the context of acceptance or rejection of a decision based on acceptance or rejection of a hypothesis. The difficulty in formulating a hypothesis gets further complicated when it involves a one sided (one tail) test. For a moment, let us forget the nuances of one sided or two sided hypothesis and focus on the role of statistical hypothesis in decision making in the light of the above statement. Are there clear implications to a manager if certain situations apparently looking significant may not be statistically significant and likewise, situations not looking significant to a manager may be statistically significant? In what way this can affect the decision? What are the consequences? This article attempts to throw some light on the role of statistical significance.

Situation 1
An advertising agency is interested in understanding the behavior of a certain class of adult population watching programs on television. The agency feels that in view of excess violence and crimes shown in some programs, this class of people have started watching more children's programs. The client who has hired the agency to do an assignment on the viewing pattern of the audience feels that it is worth the money and efforts only if more than 20% of the adults watch children's program. The Advertising agency using a pilot study conjectures that more than 20% of adults watch children programs. A comprehensive survey is then conducted to find out the merits of the conjecture. The survey utilized a random sample of 200 respondents in this class of adults. The sample proportion (p) of adults watching the children program is found to be 0.24. The agency feels happy about the sample results and does recommend to its client to go ahead with their plan.

Non statistically speaking, this is a case where the sample result is significant to the agency on the face of it. Let us see whether the recommendation is statistically valid.

 Structuring the Null and alternative Hypotheses: Null Hypothesis: alternative hypothesis: H0: π ≤ 0.20 H1: π > 0.20

 Test Statistic to be used: The sample size is large and therefore the appropriate test statistic to use is z test. The formula is: where N = 200, p= 0.24, p = 0.20, and . Therefore, z = 1.326.

Interpretation:
The critical value for the one tailed test at the 0.05 level is 1.65. The calculated value of Z is less than the this critical value of Z. The conclusion is we have no strong evidence to reject the null hypothesis. The inference is that the data are consistent with hypothesis that "20% or less of the adult population is watching children's programs." The sample proportion of 0.24 may have arisen due to chance. So, what is significant to a manger is not statistically significant here. Looking at the sample proportion, one should not get carried away and conclude that more than 20% of adults watch children program unless the statistical test substantiates this claim. Decisions, which involve a large financial outlay, should not be taken merely because it is significant to a manager on the face of it. On the other hand, we should not accept the null hypothesis and conclude that 20% or less of the adult population is watching children's programs. The data hint that this hypothesis is not true even though they are not strong enough to rule it out.

Situation 2:
The manufacturer of detergents has a decision to make- whether or not to adjust the process based on sample results. The detergent of one type must weigh 500 grams. Customers have started complaining that the cake weighs significantly less than 500 grams. A random sample of 25 cakes of this detergent was taken from the current lot kept for shipment and it was found that the average weight was 498 grams with a standard deviation of 4 grams. The product specification is 500 grams plus or minus two grams. To the manufacturer, the weight reduction is with in the specification and not significant.

Let us see what the statistical analysis says.

 Structuring the Null and alternative Hypotheses: Null Hypothesis: alternative hypothesis: H0: μ ≥ 500 H1: μ < 500

Test Statistic to be used:
The appropriate test statistic is the t-test: = -2.5 where M= 498, m =500, N =25,  s = 4

Interpretation:
The table value of t for 24 degrees of freedom at 5% level of significance is 1.71.
Therefore, assuming the null hypothesis is true, the probability of a t less than or equal to -1.71 is 0.05. The probability of obtaining a t less than or equal to -2.5(0.0196) is lower than 0.05 and therefore the null hypothesis can be rejected in favor of the alternative hypothesis. The inference is that the mean weight of the cake is less than 500 grams. The customers are right in saying that, on average, the weight is less than the expected weight of 500 grams. In other words, to the manufacturer the weight reduction is not significant because it is within the specification. But the truth of the matter is that the average cake weighs less than 500 grams and the difference is not due to chance. The manufacturer must initiate some quality control studies to reduce the variation and ensure that the average weight is at least 500 grams.

Concluding Remarks:

A careful analysis of the given statement in the light of the above examples bring into sharp focus the role of statistical hypothesis in decision making. The word "Statistical Significance" is key to decision making; is key to minimizing the risk; is key to accept or reject a proposed decision based on acceptance or rejection of a hypothesis. Statistical thinking is essential and this article in some measure highlights that hypothesis testing is not a statistical technique to delight the statisticians. It has a very meaningful and important role indecision making. Looking just at sample mean or sample proportion which are of course point estimators, one should not jump to conclusions. The statistical validation is indeed very crucial to the manager in taking the right decision. Therefore, the statement:

"What is Significant to a Manager may not be Statistically Significant. What is not Significant to a Manager may be Statistically Significant."

is profoundly true.

(An article by P.K. Viswanathan, Adjunct Professor and Management Consultant, Chennai, India)