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Significance Tests / Hypothesis Testing
Power is computed for a one-sample test of a hypothesized µ valueof 0 when the standard deviation is assumed known, i.e., power iscomputed for the standard z-test. The user has animation controls tochange the values of delta (the difference between the null and thealternative means), the population standard deviation, the samplesize, and the level of significance.
When the critical region ( i.e. null hypothesis rejection region ) is only one side of the normal distribution curve - either left tail or right tail it is called one-tailed null hypothesis.
A one tailed test with the rejection region in one tail.
It is also called directional hypothesis testing because the alternate hypothesis rejects the null hypothesis by comparing the test statistic value to be either greater than or less than the reference value specified in the null hypothesis.
This however, does not mean that there is 4 % probability that null hypothesis is true. It means that assuming null hypothesis is true, the sample data exhibits the statistical test with probability 0.04.
# to reject or not to reject the null hypothesis
Similarly, the acceptable level of Type II error is denoted as β. β is probability of not rejecting null hypothesis when it is false; so (1-β) is correctly rejecting the null hypothesis when it is false and hence it is called the power of the statistical test.
When the critical region ( i.e. null hypothesis rejection region ) is on both sides of the normal distribution curve - left and right tail it is called two-tailed null hypothesis testing.
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the null hypothesis is rejected when it is true b.
Probability Value is the probability of the statistic used to test the null hypothesis. p-value along with probability of occurrence of type I and type II errors determines whether to reject or not reject the null hypothesis.
the null hypothesis is not rejected when it is false c.
Is the acceptable level of Type I error denoted as α. Say a 5 percent of tolerance is allowed for Type I error beyond which the null hypothesis is rejected. That is out of 20 samples, we're willing to accept one rejection of null hypothesis even if it is true. α can also be 1 percent if consequences of Type I error are costly.
Does the test statistic (c) fall in the critical region (d)?
Depending on which error costs more the level of tolerance called significance is set while testing the null hypothesis using a particular statistic.
One can never prove the truth of a statistical (null) hypothesis.
Type II error occurs when the study fails to reject the null hypothesis and concludes that both the medicines are effective in treating the patients when actually one of them is not.
failing to reject the null hypothesis when it is false.
It is also called non-directional hypothesis testing because the alternate hypothesis rejects the null hypothesis if the test statistic value is not equal to the reference value specified in the null hypothesis - regardless of whether it is greater than or less than the specified value.
failing to reject the null hypothesis when it is true.
Type I error occurs when the study rejects the null hypothesis and concludes that the two medicines are different when actually they have the same effect.
rejecting the null hypothesis when it is true.
To see the effect of two medicines the null hypothesis is the default statement that there is no difference between the medicines. To validate this statement null hypothesis testing is done on a sample data of the population. There are two datasets representing two samples for the effect of two medicines and the test statistic means is used.
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