Topic: Hypotheses Testing

Question:

State the null hypothesis and alternative hypothesis in (a) words and in (b) symbols for each of the following:

1. A librarian of a school claims that all their grade 8 students read an average of 10 storybooks a month with a standard deviation of 2 books. A random sample of grade 8 students read an average 12 books a month and a standard deviation of 1 book. The confidence statement is 95%.

2. According to a factory employer, the mean working time of workers in the factory is 6 hours with a standard deviation of 0.5 hours. A researcher interviewed 50% of the employees and found out that their mean working time is 8 hours with a standard deviation of 1 hour. The a level is 0.05.

3. A random sample of 200 students got a mean score of 62 with a standard deviation score of 5 in a knowledge test in mathematics. In the standardization of the test, ? =50 with ? = 10.

References: See attached photos. answer the following based from the given references provided by my prof. thank you. God Bless.

I. INTRODUCTORY CONCEPT! PANIMULANG KDNSEP'TO In the earlier lessons, you learned about the mean and the standard deviation as descriptions of a set of numerical data. You have also learned the importance of random sampling as well as the properties of the normal curve and how to compute the proportions of areas under it. All this concept including the probability distribution is very important in analyzing statistical data especially in hypothesis testing. In this lesson, you will explore and understand the hypothesis testing. Before we discuss the process of conducting hypothesis testing, it is necessary to study first the different concepts and terms that associated to hypothesis testing. II. LEARNING SKILLS FROM MELCs Illustrates: (a)Null hypothesis, (b)Alternative hypothesis, (c)Level of significance (deejection region, and (e)Types of errors of hypothesis testing. III. DISCUSSION Types of Hypotheses A manufacturer of emergency equipment asserts that a respirator that it makes delivers pure air for ?5 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 ?'5 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 u as to test an assertionor a hypothesisabout it, namely, whe'mer it is as large as 75 or not. The agency is not necessarily interested in the actual value of p, just whether it is as claimed. Their sampling is done to perform a test of hypotheses, the subject of this chapter. A hwothesis 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. Denition The nulf hypothesis, denoted HD, 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 Hg, 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. DEfll'llIlOl'l 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 result of a hypotheses testing procedure is a choice of one of the following two possible conclusions: 'I . Reject Ho (and therefore accept Ha), or 2. Fail to reject Ho (and therefore fail to accept H.). The null hypothesis typically represents the status quo, or what has historically been true. In the example _I l_ of the respirators, we would believe the claim of the manufacturer unless there is reason not to do so, so the null hypotheses is Hn: u=75.Hn: p=75. The alternative hypothesis in the example is the contradictory statement H.: p5.Ha: p5_ The null hypothesis will always be an assertion containing an equal sign but depending on the situation the alternative hypotesis can have any one of three forms: with the symbol \"," or with the symbol \"s" The following two examples illustrate the latter two cases. EXAMPLE 1 A publisher of college textbooks claims that the average price of all hardbound college textbooks is P12150. 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 publishers claim unless there is compelling evidence to the contrary. Thus, the null hypothesis is Ha: p=127.50. Since the student group thinks that the average textbook price is greater than the publisher's gure, the alternative hypothesis in this situation is H.: u>12?.50. Answer. i-Iu: p=1 2150 and Hal u:12?.50 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 altemative 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 Ho: p=8.D. Since 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 H.: peso. Answer. Ho: u=B.EI and H.: peat) EXAMPLE 3 The owner of a factory that sells: a narticular bottled frIIit i . . claims that the. averaae nanacitv of a EXAMPLE 3 The owner of a factory that sells a particular bottled fruitjuice claims that the average capacity of a bottle of their product is 250 ml. Is the claim true? To test the claim, the members of a consumer group did the following: 1. Get a sample of 100 such bottles. 2. Calculate the capacity of each bottle. 3. Compare the sample mean and the claim. The observed mean capacity i of the 100 bottles is 243 ml. The sample standard deviation is 'ID ml. In the example, the owner's statement (called claim) is a general statement. The claim is that the capacity of all their bottled products is 250 ml per bottle. So, the population mean p is 250 ml, clearly a sample mean. There is a difference of 7" ml. Thus, the two hypotheses would be: He : The bottled drinks contain 250 ml per bottle. (this is the claim) Ha :The bottled drinks do not contain 250 ml per bottle. (This is the opposite of the claim In svmhnls 'I. if (as in the respirator example} Ha has the form H.: mm, we reject Ha if x is far to the left of paper, that J L is, to the left of some number C, so the rejection region has the form of an interval (00, C]. 2. if (as in the textbook example) H. has the form H.: p>pn, we reject Hn if)? is far to the right of |..lo, that is, to the right of some number C, so the rejection region has the form of an interval [C, 00). 3. if (as in the baked good example) H. has the form H.: j..l#|..la, we reject Ho if i is far away from pa in either direction, that is, either to the left of some number C or to the right of some other number C', so the rejection region has the form of the union of two intervals (00, C] U[C',oo). The key issue in our line of reasoning is the question of how to determine the number C or numbers C and 0', called the critical value or critical values of the statistic, that determine the rejection region. Denition The critical value or critical values of a test of hypotheses are the number or numbers that determine the rejection region. Suppose the rejection region is a single interval, so we need to select a single number 0. Here is the procedure for doing so. We select a small probability, denoted a, say 1%, which we take as our definition of "rare event" an event is \"rare\" if its probability of occurrence is less than u. The probability that} takes a value in an interval is the area under its density curve and above that interval, so as shown in figure below (drawn under the assumption that He is true, so that the curve centers at pa) the cn'tical value C is the value of} that cuts off a tail area do in the probability density curve of x . when the rejection region is in two pieces, that is, composed of two intervals, the total area above both must be a, so the area above each one is 9', as also shown in gure below. may..." mung.- C M P\" C H.-.-.-r H;. . Nu . Rm: H. The numberu is the roralarea ofa tail or a pair of rails. Under the normal curve, the rejection region refers to the region where the value of the test statistic lies for which we will reject the null hypothesis. This region is also called critical region. So, if your computed statistic is found in the rejection region, then you reject Hn. If it is found outside the rejection region, you accept Ho. Also note the line that separates the rejection region from the non-rejection region (10). This line passes through the condence coefcients, which are also called critical values. The critical values can be obtained from the critical values table of the test statistics. For example, if the test statistic is a z, the critical values can be obtained from the z-table. So, for 95% confidence level, the critical values for a non-directional test are -1.96 and + 1.96. when the confidence level is 99%, for a non-directional test, the critical values are - 2.58 and +2.58. The Signicance Level The signicance level, also denoted as alpha or o, is the probability of rejecting the null hypothesis The Significance Level The significance level, also denoted as alpha or a, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference. When the confidence level (1-a) is 95% the significance level is 5% or 0.05. The significance level determines how far out from the null hypothesis value we'll draw that line on the graph. To graph a significance level of 0.05, we need to shade the 5% of the distribution that is furthest away from the null hypothesis. Two-Tailed Critical Region for a Significance Level of 0.05 L T. df =24 330.6 Density 0.029 0.025 17 229 291 172 2GE Energy Cost In the graph above, the two shaded areas are equidistant from the null hypothesis value and each area has a probability of 0.025, for a total of 0.05. In statistics, we call these shaded areas the critical region for a two-tailed test. If the population mean is 260, we would expect to obtain a sample mean that falls in the critical region 5% of the time. The critical region defines how far away our sample statistic must be from the null hypothesis value before we can say it is unusual enough to reject the null hypothesis. EXAMPLE 4 In the context of Example 2, suppose that it is known that the population is normally distributed with standard deviation o = 0.15 gram, and suppose that the test hypotheses Hop=8.0 versus Hap=8.0 will be performed with a sample of size 5. Construct the rejection region for the test for the choice a=0.10. Explain the decision procedure and interpret it. Solution: If Ho is true, then the sample mean x is normally distributed with mean and standard deviation. MX = #= 8.0, ox = 0 / vn = 0.15 = 0.067 V5 Since Ha contains the = symbol the rejection region will be in two pieces, each one corresponding to a tail of area ?,= .,, = 0.05.. From the z-table, Zo.os=1.645, so C and C' are 1.645 standard deviations of x to the right and left of its mean 8.0: C = 8.0 - (1.645)(0.067) = 7.89 and C' = 8.0+ (1.645)(0.067) = 8.11 Rejection Region for the Choice a=0.10 Ha : u #8.0 * = 0.05 2 = 0.05 7.89 8.0 8.11 Reject Ho 7.89 8.0 8.11 Reject Ho The decision procedure is taking a sample of size 5 and compute the sample mean x . If x is either 7.89 grams or less or 8.11 grams or more then reject the hypothesis that the average amount of fat in all servings of the product is 8.0 grams in favor of the alternative that it is different from 8.0 grams. Otherwise, do not reject the hypothesis that the average amount is 8.0 grams. The reasoning is that if the true average amount of fat per serving were 8.0 grams, then there would be less than a 10% chance that a sample of size 5 would produce a mean of either 7.89 grams or less or 8.11 grams or more. Hence if that happened it would be more likely that the value 8.0 is incorrect (always assuming that the population standard deviation is 0.15 aram). mg tray Definition If Ha has the form =Ho the test is called a two-tailed test. If H. has the form up. the test is called a left-tailed test. If Ha has the form u>Ho the test is called a right-tailed test. Each of the last two forms is also called a one-tailed test. LL When the alternative hypothesis utilizes the = symbol, the test is said to be non-directional. When the alternative hypothesis utilizes the > or the symbol 200 000.00 Two Types of Errors The format of the testing procedure in general terms is to take a sample and use the information it contains to come to a decision about the two hypotheses. As stated, before our decision will always be either reject the null hypothesis Ho in favor of the alternative H. presented, or do not reject the null hypothesis Ho in favor of the alternative H. presented. There are four possible outcomes of hypothesis testing procedure, as shown in the following table: True State of Nature Ho is true Ho is false Do not reject Ho | Correct decision Type II error Our Decision Reject Ho Type I error Correct decision As the table shows, there are two ways to be right and two ways to be wrong. Typically to reject H. when it is actually true is a more serious error than to fail to reject it when it is false, so the former error is labeled "Type I" and the latter error "Type II." Type I and Type II errors Type I and Type II errors True situation The truth The null hypothesis is The null hypothesis is Guilty true Innocent Reject the null Type | error Guilty hypothesis (Rejecting a true null Correct decision Correct decision hypothesis) Our verdict Our decision Fail to reject the Type II error Failing to reject a Innocent null hypothesis Correct decision Correct decision Error false null hypothesis) EXAMPLE 6 Maria insists that she is 30 years old when, in fact, she is 32 years old. What error is Mary committing? Solution: Mary is rejecting the truth. She is committing a Type | error. EXAMPLE 7 Stephen says that he is not bald. His hairline is just receding. Is he committing an error? If so, What type of error? L IV. EVALUATION: Answer the following A. Determine whether the test is two-tailed or one-tailed. If it is one-tailed, is it left-tailed or right-tailed? Then, sketch the graphical representation of the test. 1. A nutritionist claims that her developed bread is fortified with vitamin B. 2. A musician believes that listening to classical music affects mood 3. A storekeeper thinks that time of day influences sale of ice cream. A mother wants to prove that reading books to children improves their thinking processes