Outcomes and the Type I and Type II Errors Quiz

PDF page 391: When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H0 and the decision to reject or not. The outcomes are summarized in the following table:

PDF page 392: Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities. α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true: rejecting a good null.

PDF page 392: as "rejecting a good null". As we will see later, it is this type of error that we will guard against by setting the probability of making such an error. The goal is to NOT take an action that is an error.

PDF page 392: called "accepting a false null". In this situation you have allowed the status quo to remain in force when it should be overturned. As we will see, the null hypothesis has the advantage in competition with the alternative.

PDF page 392: Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities. α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true: rejecting a good null.

PDF page 392: β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null

PDF page 392: hypothesis is false. (1 − β) is called the Power of the Test. α and β should be as small as possible because they are probabilities of errors. Statistics allows us to set the probability that we are making a Type I error. The probability of making a Type I error is α. Recall that the confidence intervals in the last unit were set by choosing a value called Zα (or tα) and the alpha value determined the confidence level of the estimate because it was the probability of the interval failing to capture the true mean (or proportion parameter p). This alpha and that one are the same. The easiest way to see the relationship between the alpha error and the level of confidence is with the following figure.

PDF page 392: β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null

PDF page 392: hypothesis is false. (1 − β) is called the Power of the Test. α and β should be as small as possible because they are probabilities of errors. Statistics allows us to set the probability that we are making a Type I error. The probability of making a Type I error is α. Recall that the confidence intervals in the last unit were set by choosing a value called Zα (or tα) and the alpha value determined the confidence level of the estimate because it was the probability of the interval failing to capture the true mean (or proportion parameter p). This alpha and that one are the same. The easiest way to see the relationship between the alpha error and the level of confidence is with the following figure.

PDF page 392: hypothesis is false. (1 − β) is called the Power of the Test. α and β should be as small as possible because they are probabilities of errors. Statistics allows us to set the probability that we are making a Type I error. The probability of making a Type I error is α. Recall that the confidence intervals in the last unit were set by choosing a value called Zα (or tα) and the alpha value determined the confidence level of the estimate because it was the probability of the interval failing to capture the true mean (or proportion parameter p). This alpha and that one are the same. The easiest way to see the relationship between the alpha error and the level of confidence is with the following figure.

PDF page 392: In the center of Figure 9.2 is a normally distributed sampling distribution marked H0. This is a sampling distribution of – X and by the Central Limit Theorem it is normally distributed. The distribution in the center is marked H0 and represents the distribution for the null hypotheses H0: μ = 100. This is the value that is being tested. The formal statements of the null and alternative hypotheses are listed below the figure. The distributions on either side of the H0 distribution represent distributions that would be true if H0 is false, under the alternative hypothesis listed as Ha. We do not know which is true, and will never know. There are, in fact, an infinite number of distributions from which the data could have been drawn if Ha is true, but only two of them are on Figure 9.2 representing all of the others. To test a hypothesis we take a sample from the population and determine if it could have come from the hypothesized distribution with an acceptable level of significance. This level of significance is the alpha error and is marked on Figure 9.2 as the shaded areas in each tail of the H0 distribution. (Each area is actually α/2 because the distribution is symmetrical and the alternative hypothesis allows for the possibility for the value to be either greater than or less than the hypothesized value--called a two-tailed test). – If the sample mean marked as X is in the tail of the distribution of H0, we conclude that the probability that it could have 1 come from the H0 distribution is less than alpha. We consequently state, "the null hypothesis cannot be accepted with (α) – level of significance". The truth may be that this X did come from the H0 distribution, but from out in the tail. If this is 1 so then we have falsely rejected a true null hypothesis and have made a Type I error. What statistics has done is provide an

PDF page 392: In the center of Figure 9.2 is a normally distributed sampling distribution marked H0. This is a sampling distribution of – X and by the Central Limit Theorem it is normally distributed. The distribution in the center is marked H0 and represents the distribution for the null hypotheses H0: μ = 100. This is the value that is being tested. The formal statements of the null and alternative hypotheses are listed below the figure. The distributions on either side of the H0 distribution represent distributions that would be true if H0 is false, under the alternative hypothesis listed as Ha. We do not know which is true, and will never know. There are, in fact, an infinite number of distributions from which the data could have been drawn if Ha is true, but only two of them are on Figure 9.2 representing all of the others. To test a hypothesis we take a sample from the population and determine if it could have come from the hypothesized distribution with an acceptable level of significance. This level of significance is the alpha error and is marked on Figure 9.2 as the shaded areas in each tail of the H0 distribution. (Each area is actually α/2 because the distribution is symmetrical and the alternative hypothesis allows for the possibility for the value to be either greater than or less than the hypothesized value--called a two-tailed test). – If the sample mean marked as X is in the tail of the distribution of H0, we conclude that the probability that it could have 1 come from the H0 distribution is less than alpha. We consequently state, "the null hypothesis cannot be accepted with (α) – level of significance". The truth may be that this X did come from the H0 distribution, but from out in the tail. If this is 1 so then we have falsely rejected a true null hypothesis and have made a Type I error. What statistics has done is provide an