One of the problems to be solved by statistical inference is that of testing a hypothesis. That is, given a certain claim about a population, usually about one of its parameters, we want to know whether the experimental results from a sample contradict that claim or not.The objective of statistical hypothesis testing is, therefore, to provide a methodology that allows us to verify whether the sample data provide evidence that supports or does not support a formulated (statistical) hypothesis.The central idea of hypothesis testing is to assume that the hypothesis in question is true and verify whether the observed sample is “plausible” under these conditions.An ExampleExample: An industry uses, as one of the components of the machines it produces, an imported screw that must satisfy certain requirements. One of these is tensile strength. These screws are manufactured by several countries, and the technical specifications vary from country to country. For example, the catalog of country A states that the average tensile strength of its screws is 145 kg, with a standard deviation of 12 kg. For country B, the mean is 155 kg and the standard deviation is 20 kg. Could we discover where the screw is from based on its strength?A batch of these screws, of unknown origin, will be auctioned at a very attractive price. In order for the industry to know whether or not to make an offer, it needs to know which country produced these screws. The auctioneer’s notice states that, shortly before the auction, the average strength ⎯ x of a sample of 25 screws from the batch will be disclosed. What decision rule should the industry use to say whether the screws are from country A or B?The simplest way to solve this problem would be to name as the producing country the one for which the item is closest to the population mean.Thus, we would take the midpoint between the two countries:\[\text{Midpoint} = \frac{145 + 155}{2} = 150 \]Suppose that the average strength of the batch were 148; according to our decision rule, we would say that the screws are from origin A. Could we be wrong in this conclusion? Or, in other words, is it possible that in origin B there is a screw with strength 148? Yes, it is possible. So, to better understand the adopted decision rule, it is interesting to study the types of errors we can make and their respective probabilities.The Types of ErrorType 1 ErrorSaying that the screws are from A when in reality they are from B. Being B even though the sample mean is less than 150 kg.Type 2 ErrorSaying that the screws are from B when in reality they are from A. Being A even though the sample mean is greater than 150 kg.The Null HypothesisTo make things even easier, let us define two hypotheses, also numbered:\(H_0\) (also called the null hypothesis): the screws are from origin B. With mean 155 and standard deviation 20.\(H_1\): the screws are from origin A. With mean 145 and standard deviation 12.When \(H_0\) is true, that is, the screws are from B, we know that X will have an approximately normal distribution, with mean 155 and standard deviation equal to 20/√25 = 4. Therefore, for screws with strength less than 150 kg:\[ P(\text{error 1})=P(\text{H0 is true}) \]\[ P(\text{error 1})=P(X<150) \]\[ P(\text{error 1})=P(Z<150−1554) \]\[ P(\text{error 1})=P(Z<−1.25)=0.1056=10.56% \]Similarly, when \(H_1\) is the true alternative, we will have:\[ P(\text{error 2})=P(\text{H1 is true}) \]\[ P(\text{error 2})=P(X>150) \]\[ P(\text{error 2})=P(Z>150−1452.4) \]\[ P(\text{error 2})=P(Z>2.08)=0.01876=1.88% \]These probabilities are better presented in the table:Actual origin of the screwStrength < 150Strength > 150ANo errorType II error1.88%BType I error10.56%No errorWe can note that, if the screws are really from B (second row) and the sample has a mean greater than 150 (second column), we will say they are from B, and we will not make any error. On the other hand, if the mean ⎯ x is less than 150 (first column), we should say they are from A, and we will be making an error whose probability in this case is 10.56%. Similarly, we will have an interpretation for the case in which the screws are really from A (first row).For each decision rule adopted, that is, if we choose any value instead of 150, only the probabilities of type I and type II errors will change.Therefore, there must be a point at which the type I error is equal to the type II error, that is, a decision rule in which the probability of making an error against A is the same as making an error against B. Show that this point is 148.75 kg, and in this case α = β = 5.94%.From the above, we see that, once a value of x is chosen, we can find the probabilities of making each type of error. But we can also proceed in the opposite way: fix one of the errors, say α, and find the decision rule that will correspond to the probability of type I error equal to α.For example, let us set α at 5%, and see what the corresponding decision rule is. We have:\[ P(\text{error 1})=5% \]\[ P(\text{error 1})=P(Z<−1.645) \]\[ −1.645=x−1554 \]\[ x=148.42 \]So, the decision rule will be:“If the mean of the studied batch is less than 148.42, we say that the batch is from A; otherwise, we say it is from B.”- Decision ruleWith this rule, the probability of type II error will be:\[ Z=\frac{148.42>145}{2.4} \]\[ Z=1.425=7.93% \]This second type of procedure is widely used, because usually the decision we must make is not only between two possible populations. The screws could be produced by other countries besides those mentioned and, therefore, with other characteristics regarding average strength. Suppose, further, that the industry is interested in making a proposal only if the screw is from origin B. What decision rule should it adopt?The hypothesis that interests us now is:\[ H_0= \text{the screws are from origin B (μ = 155 and σ = 20)} \]\[ H_1=\text{the screws are not from B (μ and σ unknown)} \]Here we cannot specify the parameters under the alternative hypothesis H 1, because if they are not from B, the screws may come from several other countries, each with its own specifications. Some countries may have more sophisticated production techniques and, therefore, produce screws with an average strength greater than 155. Others, as in the example given, with lower strength. The specification of the alternative hypothesis depends greatly on the degree of information one has about the problem. For example, let us assume that the industry of country B in this case is the most developed, and that no other country can produce an average strength greater than its own. Then, our alternative hypothesis would be more explicit:\[ H_1= \text{the screws are not from B ( μ < 155 and any σ)} \]Actual origin of the screwStrength < 148.42Strength > 148.42BType I error5%No errornot BNo errorType II error?%How could the industry say that the screws are not from B, that is, that they do not have mean 155, with a 5% error?\[ H_0= \text{the screws are from B (μ = 155 and σ = 20)} \]\[ H_1= \text{the screws are not from B (μ and σ unknown)} \]\[ P(\text{error 1})=5% \]\[ P(\text{error 1})=P(X<2.5%)orP(X>97.25%) \]\[ P(\text{error 1})=P(Z<−1.96)orP(Z>1.96) \]\[ −1.96=X1−1554=>X1=147.16 \]\[ 1.96=X2−1554=>X2=162.84 \]Therefore, for any value below 147.16 and above 162.84, we can reject the null hypothesis that the screws are from B.The types of errorExercises1) To decide whether the inhabitants of an island are descendants of civilization A or B, we will proceed as follows:(i) we select a sample of 100 adult residents of the island and determine their average height;(ii) if this average height is greater than 176, we will say they are descendants of B; otherwise, they are descendants of A.The height parameters of the two civilizations are: A: μ = 175 and σ = 10; B: μ = 177 and σ = 10. Let us define: Type I error — saying that the inhabitants of the island are descendants of B when, in reality, they are from A. Let us define: Type II error — saying that they are from A when, in reality, they are from B.(a) What is the probability of a Type I error? And of a Type II error?(b) What should the decision rule be if we want to set the probability of a Type I error at 5%? What is the probability of a Type II error in this case?2) In the situations below, choose as the null hypothesis, H0, the one that for you leads to the more important Type I error. Describe the two errors in each case.(a) The job of a radar operator is to detect enemy aircraft. When something strange appears on the screen, they must decide between the hypotheses:1. an attack is beginning;2. everything is fine, just slight interference.(b) In a jury, an individual is being tried for a crime. The hypotheses before the jury are:1. the accused is innocent;2. the accused is guilty.(c) A researcher believes they have discovered a vaccine against the common cold. They will conduct laboratory research to verify the truth of the claim. According to the result, they will or will not launch the vaccine on the market. The hypotheses they can test are:1. the vaccine is effective;2. the vaccine is not effective.3) The variable X, maintenance cost of a loom, can be considered to have a normal distribution with mean μ and standard deviation 20 units. The possible values of μ may be 200 or 210. To verify which of the two values is more likely, a sample of 25 looms will be used. Define:(a) A hypothesis to be tested.(b) A decision rule and find the probabilities of Type I and Type II errors.4) The association of owners of metallurgical industries is very concerned about time lost due to workplace accidents, whose average in recent times has been around 60 man-hours per year and a standard deviation of 20 man-hours. An accident prevention program was attempted, after which a sample of nine industries was taken and the number of man-hours lost due to accidents was measured, which was 50 hours. Would you say, at the 5% level, that there is evidence of improvement?Answer Key1) a) % Type I error = % Type II error = 15.9%; b) Decision rule of 176.64. Type II error of 35.9%;4) It is not possible to reject the null hypothesis.
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