Question

Suppose we wish to test the hypothesis H0 :μ=45vs. H1 :μ>45.

What will be the result if we conclude that the mean is 45 when the actual mean is 50? Choose one of the following.

1. We have made a Type I error.
2. We have made a Type II error.
3. We have made the correct decision.

Answer 1

Before we go on to solve the problem let us know a bit about hypothesis.

Null Hypothesis

In the formulation of testing problem the roles of H₁ and H₂ are not symmetric.In order to decide which of these two hypothesis should be taken as null hypothesis H₀ the difference between the roles and the implication of these two terms should be clearly understood. In testing hypothesis a statistician should be completely impartial and should have no brief for any party or company nor should he/she allow his personal views to influence the decision.

The neutral or non-committal attitude of the statistician before the sample values are taken is key to the choice of NULL HYPOTHESIS.

Keeping in mind the potential losses due to the wrong decision, the decision maker is somewhat conservative in holding the null hypothesis as true unless there is a strong evidence that is false to him the consequences of wrongly rejecting a null hypothesis seems to be more serious than those of wrongly accepting it.

Hence we denote by H₀ that hypothesis the false rejection of which is regarded as more serious and call it the null hypothesis.

Alternative Hypothesis

It is desirable to state what is called an alternative hypothesis in respect of every statistical hypothesis being tested because the acceptance or rejection of null hypothesis is meaningful only when it is being tested against a rival hypothesis which should rather be explicitly mentioned. Alternative hypothesis is denoted by H_A

Example: Let us suppose that two different company manufacture sleeping drugs for inducing sleep. Drug A is manufactured by first company and drug B is manufactured by second company. Each company claims that its drug is better to that of the other and it is desired to test which is superior drug A or drug B? We formulate the statistical hypothesis to see which drug is better. Let X be a random variable which denotes the additional hours of sleep gained by the individual when drug A is given and let the random variable Y denote the additional hours of sleep gained when drug B is used. Hence our null hypothesis and alternative hypothesis is given by,

H₀: There is no difference between the effects of two drugs

H_A: There is difference between the effects of two drugs i.e. (drug A has more effect than drug B or drug B has more effect than drug A)

Now while performing a test one may arrive at a correct decision or may commit one of the two wrong decisions:

(i) Type-I-Error

Rejecting H₀ the null hypothesis, when it is really true.

(ii) Type-II-Error

Accepting H₀ the null hypothesis, when it is really false.

Coming back to our problem

Given,

Suppose we wish to test the hypothesis H₀ :μ=45 vs. H₁ :μ>45.

Now we need to determine the result if we conclude that the mean is 45 when the actual mean is 50.

Clearly here we have accepted the null hypothesis when the actual mean is 50.

Hence we have accepted the null hypothesis when the null hypothesis is really false.

Hence the result is a Type-II-Error.