Question

What could be done to lower the conditional probabilities of making the type 1 error and the type 2 error in their examples? Can both be lowered at the same time? How?

just give an example of how to do it

Answer 1

Since probability of type 1 and type 2 error are interlinked to each other.if we minimize one of them then other will increase in same proportion.

But we can make some effort to minimize both at some extent.

The probability of a type 1 error (rejecting a true null hypothesis) can be minimized by picking a smaller level of significance α before doing a test (requiring a smaller p-value for rejecting H0).

Once the level of significance is set, the probability of a type 2 error (failing to reject a false null hypothesis) can be minimized either by picking a larger sample size or by choosing a "threshold" alternative value of the parameter in question that is further from the null value. This threshold alternative value is the value you assume about the parameter when computing the probability of a type 2 error.

To be "honest" from intellectual, practical, and perhaps moral perspectives, however, the threshold value should be picked based on the minimal "important" difference from the null value that you'd like to be able to correctly detect (if it's true). Therefore, the best thing to do is to increase the sample size.

Explanation:

The level of significance α of a hypothesis test is the same as the probability of a type 1 error. Therefore, by setting it lower, it reduces the probability of a type 1 error. "Setting it lower" means you need stronger evidence against the null hypothesis H0 (via a lower p-value) before you will reject the null. Therefore, if the null hypothesis is true, you will be less likely to reject it by chance.

Reducing α to reduce the probability of a type 1 error is necessary when the consequences of making a type 1 error are severe (perhaps people will die or a lot of money will be needlessly spent).

Once a level of significance α has been decided on. To reduce the probability of a type 2 error (because the consequences could be severe as well), you can either increase the sample size or choose an alternative value of the parameter in question that is further from the null value.

By increasing the sample size, you 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.

By choosing a threshold value of the parameter (under which to compute the probability of a type 2 error) that is further from the null value, you 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 (in the rejection region).

For example, suppose we are testing the null hypothesis H0:μ=10 versus the alternative hypothesis Ha:μ>10 and suppose we decide on a small value of α that leads to rejecting the null if ¯x>15 (this is the rejection region). Then an alternative value of μ=16 will lead to a smaller than 50% chance of incorrectly failing to reject H0 when μ=16 is assumed to be true, while an alternative value of μ=14 will lead to a greater than 50% chance of incorrectly failing to reject H0 when μ=14 is assumed to be true. In the former case the sampling distribution of ¯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 ¯x is centered on 14 and the area under it to the left of 15 will be greater than 50%.