Review the Learning Activity titled “Introduction to Hypothesis Testing.” Describe the difference...

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Review the Learning Activity titled “Introduction to Hypothesis Testing.” Describe the difference between a Level 1 and Level 2 Error, including where the error is actually determined. Then, refer to the table called “Key Takeaway” at the end of the Activity. Do any of the steps identified include resolving whether a Type 1 or Type 2 error has occurred? If so, which step is involved, and what specifically should you do? If not, suggest a revision to the Key Takeaway that would include such a determination.A manufacturer of emergency equipment asserts that a respirator that it makes delivers pure air for 75 minutes on average. A government regulatory agency is charged with testing such claims, in this case, to verify that the average time is not less than 75 minutes. To do so, it would select a random sample of respirators, compute the mean time that they deliver pure air, and compare that mean to the asserted time 75 minutes. In the sampling that we have studied so far, the goal has been to estimate a population parameter. But the sampling done by the government agency has a somewhat different objective, not so much to estimate the population mean μ as to test an assertion—or a hypothesis—about it, namely, whether it is as large as 75 or not. The agency is not necessarily interested in the actual value of μ, just whether it is as claimed. Their sampling is done to perform a test of hypotheses, the subject of this objective. Types of Hypotheses A hypothesis about the value of a population parameter is an assertion about its value. As in the introductory example, we will be concerned with testing the truth of two competing hypotheses, only one of which can be true. The null hypothesis, denoted H0, is the statement about the population parameter that is assumed to be true unless there is convincing evidence to the contrary. The alternative hypothesis, denoted Ha, is a statement about the population parameter that is contradictory to the null hypothesis and is accepted as true only if there is convincing evidence in favor of it. Hypothesis testing is a statistical procedure in which a choice is made between a null hypothesis and an alternative hypothesis based on information in a sample. The end result of a hypotheses testing procedure is a choice of one of the following two possible conclusions: 1. Reject Ho (and therefore accept Ha). 2. Fail to reject Ho (and therefore fail to accept Ha). The null hypothesis typically represents the status quo, or what has historically been true. In the example of the respirators, we would believe the claim of the manufacturer unless there is reason not to do so, so the null hypotheses is  The alternative hypothesis in the example is the contradictory statement  The null hypothesis will always be an assertion containing an equals sign, but depending on the situation, the alternative hypothesis can have any one of three forms: with the symbol “,” or with the symbol “≠”. Many statisticians believe that it is only appropriate to state the outcomes of hypothesis testing in terms of the null hypotheses: (1) reject the null hypothesis or (2) fail to reject the null hypothesis. To say "accept" either the null hypothesis or the alternative hypothesis is viewed by some researchers as incorrect. They believe that any single hypothesis is providing some evidence for the case of either the null or alternative hypothesis, but not enough evidence to completely establish either truth. Because the language is trickier when you "fail to reject the null hypothesis," we will use the convention of "accepting" the null hypothesis in this context, but be sure to consider your audience when discussing this topic with others. Example 1 A publisher of college textbooks claims that the average price of all hardbound college textbooks is $127.50. A student group believes that the actual mean is higher and wishes to test their belief. State the relevant null and alternative hypotheses. Solution The default option is to accept the publisher’s claim unless there is compelling evidence to the contrary. Thus, the null hypothesis is  Because the student group thinks that the average textbook price is greater than the publisher’s figure, the alternative hypothesis in this situation is  Example 2 The recipe for a bakery item is designed to result in a product that contains 8 grams of fat per serving. The quality control department samples the product periodically to ensure that the production process is working as designed. State the relevant null and alternative hypotheses. Solution The default option is to assume that the product contains the amount of fat it was formulated to contain unless there is compelling evidence to the contrary. Thus, the null hypothesis is  Because to contain either more fat than desired or to contain less fat than desired are both an indication of a faulty production process, the alternative hypothesis in this situation is that the mean is different from 8.0, so  In Example 1, the textbook example, it might seem more natural that the publisher’s claim be that the average price is at most $127.50, not exactly $127.50. If the claim were made this way, then the null hypothesis would be , and the value $127.50 given in the example would be the one that is least favorable to the publisher’s claim, the null hypothesis. It is always true that if the null hypothesis is retained for its least favorable value, then it is retained for every other value. Thus, in order to make the null and alternative hypotheses easy for the student to distinguish, in every example and problem in this competency, we will always present one of the two competing claims about the value of a parameter with an equality. The claim expressed with an equality is the null hypothesis. This is the same as always stating the null hypothesis in the least favorable light. So, in the introductory example about the respirators, we stated the manufacturer’s claim as “the average is 75 minutes” instead of the perhaps more natural “the average is at least 75 minutes,” essentially reducing the presentation of the null hypothesis to its worst case. The first step in hypothesis testing is to identify the null and alternative hypotheses. The Logic of Hypothesis Testing Although we will study hypothesis testing in situations other than for a single population mean (for example, for a population proportion instead of a mean or in comparing the means of two different populations), in this section, the discussion will always be given in terms of a single population mean μ. The null hypothesis always has the form  for a specific number (in the respirator example, ; in the textbook example, ; and in the baked goods example, ). Because the null hypothesis is accepted unless there is strong evidence to the contrary, the test procedure is based on the initial assumption that H0 is true. This point is so important that we will repeat it here:  The test procedure is based on the initial assumption that H0 is true. The criterion for judging between H0 and Ha based on the sample data is: if the value of  would be highly unlikely to occur if H0 were true, but favors the truth of Ha, then we reject H0  in favor of Ha. Otherwise, we do not reject H0. Supposing for now that  follows a normal distribution, when the null hypothesis is true, the density function for the sample mean  must be as in the figure below: a bell curve centered at  Thus, if H0 is true, then  is likely to take a value near and is unlikely to take values far away. Our decision procedure therefore reduces simply to: 1. If Ha has the form , then reject H0 if  is far to the left of  2. If Ha has the form , then reject H0 if  is far to the right of  3. If Ha has the form  then reject Ho if  is far away from  in either direction. Figure 5.1 (The Density Curve If  Is True) Think of the respirator example, for which the null hypothesis is , the claim that the average time air is delivered for all respirators is 75 minutes. If the sample mean is 75 or greater, then we certainly would not reject H0 (because there is no issue with an emergency respirator delivering air even longer than claimed). If the sample mean is slightly less than 75, we would logically attribute the difference to sampling error and not reject H0 either. Values of the sample mean that are smaller and smaller are less and less likely to come from a population for which the population mean is 75. Thus, if the sample mean is far less than 75, say around 60 minutes or less, then we would certainly reject H0, because we know that it is highly unlikely that the average of a sample would be so low if the population mean were 75. This is the rare event criterion for rejection: what we actually observed () would be so rare an event if μ = 75 were true that we regard it as much more likely that the alternative hypothesis μ

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Level 1 Error
The first kind of error that involves the rejection of a null hypothesis that is actually true. This type of error is called a Level 1 error.
Level 1 errors are equivalent to false positives. 
Level 2 errors can be controlled. The value of alpha, which is related to the level of significance that is selected has a direct bearing on type I errors. Alpha is the maximum probability that we have a type I error. For a 95% confidence level, the value of alpha is 0.05. This implies that there is a 5% probability that a true null hypothesis will be rejected.

Review the Learning Activity titled “Introduction to Hypothesis Testing.” Describe the difference between a Level 1 and Level 2 Error, including where the error is actually determined. Then, refer to...

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