Errors in Statistical Hypotheses Testing

ERRORS IN STATISTICAL HYPOTHESES TESTING

A statistical decision in a hypotheses testing problem is either of rejecting or not rejecting H₀ based on a given random sample. Statistical decisions are governed by certain rules, developed applying a statistical theory, which are known as decision rules. The decision rule leading to rejection of H₀ is called as rejection rule.

The null hypothesis may be either true or false, in reality. Under this circumstance, there will arise four possible situations in each hypotheses testing or decision making problem as displayed in Table 1.5.

It must be recognized that the final decision of rejecting H₀ or not rejecting H₀ may be incorrect. The error committed by rejecting H₀, when H₀ is really true, is called type I error. The error committed by not rejecting H₀, when H₀ is false, is called type II error.

Example 1.3

A soft drink manufacturing company makes a new kind of soft drink. Daily sales of the new soft drink, in a city, is assumed to be distributed with mean sales of ₹40,000 and standard deviation of ₹2,500 per day. The Advertising Manager of the company considers placing advertisements in local TV Channels. He does this on 10 random days and tests to see whether or not sales has increased. Formulate suitable null and alternative hypotheses. What would be type I and type II errors?

Solution:

The Advertising Manager is testing whether or not sales increased more than ₹40,000.

Let μ be the average amount of sales, if the advertisement does appear.

The null and alternative hypotheses can be framed based on the given information as follows:

Null hypothesis: H_o: μ = 40000

i.e., The mean sales due to the advertisement is not significantly different from ₹40,000.

Alternative hypothesis: H₁: μ > 40000

i.e., Increase in the mean sales due to the advertisement is significant.

(i) If type I error occurs, then it will be concluded as the advertisement has improved sales. But, really it is not.

(ii) If type II error occurs, then it will be concluded that the advertisement has not improved the sales. But, really, the advertisement has improved the sales.

The following may be the penalties due to the occurrence of these errors:

If type I error occurs, then the company may spend towards advertisement. It may increase the expenditure of the company. On the other hand, if type II error occurs, then the company will not spend towards advertisement. It may not improve the sales of the company.