Question

How can a type II error be avoided?

Answer 1

Hint: In this question, we can minimize the error because these errors cannot be avoided. We will minimize the probability of a type II error to occur by increasing our sample size. This means by running an experiment for longer and gathering more data will help us make the correct decision with our test results.

Complete step-by-step answer:
A type $2$ error is a statistics term that is used to refer to a type of error that is made when no conclusive winner is declared between a control and a variation when there actually should be one.
The probability of a type $2$ error can be minimized by first picking a smaller level of significance $\alpha $before doing a test, we can either increase the sample size or choose an alternative value of the parameter in question that is further from the null value.
$1)$By increasing the sample size, we can reduce the variability of the statistic in question, which will reduce its chances of failing to be in the rejection region when its true sampling distribution would indicate that it should be in the rejection region.
$2)$By choosing an alternative value of the parameter that is further from the null value, we reduce the chance that the test statistic will be close to the null value, when its sampling distribution would indicate that it should be far from the null value that is in the rejection region.
For example:
Suppose we are testing the null hypothesis \[{{H}_{0}}:\mu =10\] versus the alternative hypothesis \[{{H}_{a}}:\mu >10~\] and suppose we decide on a small value of $\alpha $ that leads to rejecting the null if \[\bar{x}>15~\](this is the rejection region).
So, to solve this, we will use an alternative value of \[\mu =16\] and that will lead to a smaller than \[50\%\] chance of incorrectly failing to reject ${{H}_{0}}$ when \[\mu =16\] is assumed to be true.
While an alternative value of \[\mu =14\] will lead to a greater than \[50\%\] chance of incorrectly failing to reject ${{H}_{0}}$ when \[\mu =14\] is assumed to be true.
In the former case, the sampling distribution of $\bar{x}$ is centered on $16$ and the area under it to the left of $15$ will be less than \[50\%\], while in the latter case the sampling distribution of $\bar{x}$ is centered on $14$ and the area under it to the left of $15$ will be greater than \[50\%\].

Note: While testing a null hypothesis, it is considered that type $1$ error is more dangerous than type $2$ error because type $1$ error is the decision to reject the null (that is say it is false), when in fact it is true; while a type $2$ error is a decision to accept the null (that is ‘fail to reject it’), when in fact it is false.
Therefore, we have to keep in mind to avoid type $1$ error in a null hypothesis.