Statistics and Probability Quarter 4 – Module 1: Testing Hypothesis Statistics and Probability – Grade 11 Alternative Alternativ e Delivery Mode Quarter 4 – Module 1: Testing Hypothesis First Edition, 2020 Republic Act 8293, Section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Undersecretar y: Diosdado M. San Antonio Development Developm ent Team of the Module Writer: Josephine B. Ramos Editors: Gilberto M. Delfina, Delfina, Josephine P. De Castro, and Pelagia Pelagia L. Manalang Manalang Reviewers: Josephine V. Cabulong, Nenita N. De Leon, and Tesalonica C. Abesamis Illustrator: Jeewel C. Cabriga Layout Artist: Edna E. Eclavea Management Team: Wilfredo E. Cabral, Regional Director Job S. Zape Jr., CLMD Chief Elaine T. Balaogan, Regional ADM Coordinator Coordinator Fe M. Ong-ongowan, Regional Librarian Aniano M. Ogayon, Ogayon, Schools Division Division Superintendent Superintendent Maylani L. Galicia, Assistant Schools Division Superintendent Randy D. Punzalan, Assistant Schools Division Superintendent Imelda C. Raymundo, CID Chief Generosa F. Zubieta, EPS In-charge of LRMS Pelagia L. Manalang, EPS Printed in the Philippines by ________________________ ________________________ Department Departm ent of Education – Region IV-A CALABARZON Office Address: Telefax: Gate 2 Karangalan Karangalan Village, Barangay San Isidro Cainta, Rizal 1800 02-8682-5773/8684-4914/8647-7487 E-mail Address: region4a@deped.gov.ph Statistics and Probability Quarter 4 – Module 1: Testing Hypothesis Introductory Introduct ory Message For the facilitator: Welcome to the Statistics and Probability for Senior High School Alternative Delivery Mode (ADM) Module on Testing Hypothesis! Hypothesis! This module was collaborative collaboratively ly designed, designed, developed, developed, and reviewed by educators both from public and private institutions to assist you, the teacher or the facilitator, in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances. In addition to the material in the main text, you will also see this box in the body of the module: Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners. As a facilitator, you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module. ii For the learner: Welcome to the Statistics and Probability for Senior High School Alternative Delivery Mode (ADM) Module on Testing Hypothesis! Hypothesis! The hand is one o ne of the most symbolical parts of the human body. It is often used to depict skill, action, and purpose. Through our hands, we may learn, create, and accomplish. Hence, the hand in this learning resource signifies that as a learner, you are capable and empowered to successfully achieve the relevant competencies and skills at your own pace and time. Your academic success lies in your own hands! This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. This module module has the foll following owing parts and corresponding corresponding ic icons: ons: What I Need to Know What I Know What’s In What’s New What Is It What’s More This will give give you an idea on the skills or competencies you are expected to learn in the module. This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module. This is a brief drill or review to help you link the current lesson with the previous one. In this portion, the new lesson will be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity, or a situation. This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills. This comprises comprises activities for independent independent practice to solidify your understanding and skills of the topic. You may check iii the answers to the exercises using the Answer Key at the end of the module. What I Have Learned This includes questions or blank sentences/paragraphs to be filled in to process what you learned from the lesson. What I Can Do This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns. concerns. Assessment This is a task which aims to evaluate your level of mastery in achieving the learning competency. Additional Activities In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also aims for retention r etention of learned concepts. Answer Key This contains answers to all activities in the module. At the end of this module, you will also find: This is a list of all sources used in developing this module. The following following are s some ome reminders in usi using ng this module: module: 1. Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises. 2. Don’t forget to answer What I Know before moving on to the other activities included in the module. 3. Read the instruction carefully before doing each task. 4. Observe honesty and integrity in doing the tasks and checking your answers. 5. Finish the task at hand before proceeding to the next. 6. Return this module to your teacher/facilitator once you are through with it. References If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it! iv What I Need to Know Hypothesis testing can allow us to measure data in samples to learn more about the data in populations that are often too large or inaccessible. We can measure a sample mean to learn more about the mean in a population. Here, we can either accept or reject our assumption using hypothesis testing. This ADM module in hypothesis testing will help you study the different concepts and steps in hypothesis testing as well as its application in real-life situations. After going through this module, you are expected to: 1. define and illustrate the null hypothesis, alternative hypothesis, level of significance, rejection region, and types of errors in hypothesis testing; 2. identify the rejection and non-rejection regions and the critical values; and 3. different differentiate iate Type I and Type II errors in claims and decisions. Are you ready now to study hypothesis testing using your ADM module? Good luck and may you find it helpful. What I Know Directions: Choose the best answer to the given questions or statements. Write the letter of your answer on a separate sheet of paper. 1. It is a proposed explanation, assertion, assertion, or assumption about a population parameter or about the distribution of a random variable. A. Decision C. Probability B. Statistics D. Hypothesis Hypothesis 2. What is the statistical method used in making decisions using experimental data? experimental A. Simple analysis B. Analytical testing C. Hypothesis testing D. Experimental testing 3. It is also the probability of committing an incorrect decision about the null hypothesis. 1 A. Level of error B. Level of hypothesis C. Level of acceptanc acceptance e D. Level of significance 4. Which of the following describes an alternative hypothesis using two- tailed test? A. = 100 B. ≠ 100 C. > 100 D. < 100 5. In a one-tailed test, in which critical value listed below will the computed z of 2.313 fall in the acceptance region? A. 1.383 C. 2.228 B. 1.533 D. 2.365 6. Which of the following would be an appropriate null hypothesis? A. B. C. D. The The The The mean of mean of mean of mean of a a a a sample is e equal qual to 75. population is equal to 75. sample is not equal to 75. population is greater than than 75. 7. When is a Type I error committed? A. We reject a null hypothesis that is false. B. We reject a null hypothesis that is true. C. We fail to reject a null hypothesis that is true. D. We fail to reject a null hypothesis that is false. 8. When is a Type II error committed? committed? A. We reject a null hypothesis that is true. B. We reject a null hypothesis that is false. C. We fail to reject a null hypothesis that is true. D. We fail to reject a null hypothesis that is false. 9. Which of the following is a Type I error? C. is true; fail to reject . A. is true; reject . B. is false; reject . D. is false; fail to reject . 10. Which of the following describes an alternative hypothesis in a left-tailed test? A. > 100 B. < 100 C. = 100 D. ≠ 100 11. Which of the following must be used as the level of significance if we want a higher possibility of correct decision? A. 1% B. 5% C. 10% D. 25% 12. Which of the following would be an appropriate alternative hypothesis for one-tailed test? A. < 100 B. = 100 C. ≥ 100 D. ≤ 100 2 13. Using a left-tailed test, which of the following value of z falls in the rejection region where the critical value is – 1.725? A. – 1.700 B. – 1.715 C. – 1.724 D. – 1.728 14. If the computed z-value is 2.015 and a nd the critical value is 1.833, which of the following statements could be true? A. It lies in the rejection region, must be rejected. B. It lies in the rejection region, we failed to reject . C. It lies in the non-rejection region, must be rejected. D. It lies in the non-rejection region, we failed to reject . 15. If the computed z-value is – 1.290 and the critical value is – 2.571, which of the following stateme statements nts could be true? A. It lies in the rejection region, must be rejected. rejected. B. It lies in the rejection region, we failed to reject . C. It lies in the non-rejection region, must be rejected. D. It lies in the non-rejection region, we failed to reject . Lesson 1 Testing Hypothesis Have you at a certain time asked yourself how you could possibly decide to put a business in place and gain your expected profit? Or wonder if a judge in a trial could have given a wrong decision in determining who’s guilty? Or think if your classmates’ average weight s differ significantly among your age? Or imagine how a newly discovered medicine is being tested for human treatment? This lesson will help you make sound decisions in dealing with these situations. 3 What’s In Where Am I Now? Directions: Identify the region where each of the given values falls. Region B Region A Region C -3 1. 2. 3. 4. 5. -2.5 -2 = 1.95 = 0.15 = 1.45 = 2.4 = 2.73 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 ______________________________ ______________________________ ______________________________ ______________________________ ______________________________ Answer the following questions. 1. Are you familiar with the shape of the curve used in Activity 1? 2. What is the name of that curve? 3. In what type of distribution is this kind of curve used? 4. How were you able to locate in which region the given value falls? 5. What mathematical concepts did you apply in locating the region? Notes to the Teacher Check the student’s level of readiness for the next topic. If she/he did not answer most of the items and the guide questions, you may provide another review activity about normal curve. 4 What’s New Keep Me Connected! Directions: Analyze the situation below and answer the questions that follow. fo llow. According to a survey, the average daily usage of social media worldwide of global internet users amounts to 142 minutes per day. Sofia conducts her own survey among her friends to find out if their time spent on social media is significantly higher than the global survey. Before her survey, she formulated the following claims: Claim A: The average daily usage of social media of her friends is the same as the global average usage. Claim B: The average daily usage of social media of her friends is higher than the global average usage. The table table shows Sofia’s friends and their respective time spent on social media. Friend’s Name Allen Bryan Ellen Jake Mindie Shamsi Candice Dory Mitch Mila Minutes per Day Spent on Social Media 132 148 165 157 120 144 136 160 185 173 Answer the following questions: 1. What statistical data is/are needed to prove which of Sofia’s claims is accepted or rejected? 5 2. What is the average daily usage of social media of her friends? Compare it with the previous average usage. 3. Which of the two claims could probably be true? Why? 4. If Sofia computed the average daily internet usage of her friends to be higher than the global survey, do you think it would be significantly higher? 5. What is your idea of an average value being significantly higher than the global average value? 6. What do you think is the difference between simple comparison of data and hypothesis testing? What Is It Hypothesis testing is a statistical method applied in making decisions using experimental data. Hypothesis testing is basically assumption that we make about a population. testing an A hypothesis is a proposed explanation, assertion, or assumption about a population parameter parameter or about the distrib distribution ution of a random variable. Here are the examples of questions you can answer with a hypothesi hypothesis s test: Does the mean height of Grade 12 students differ from 66 inches? Do male and female Grade 7 and Grade 12 students differ in height on average? Is the proportion of senior male students’ height significantly higher than that of senior female students? Key Terms and Concepts Used in Test Hypothesis The Null and Alternative Hypothesis The null hypothesis is an initial claim based on previous analyses, which the researcher tries to disprove, reject, or nullify. It shows no significant difference between two parameters. It is denoted by . The alternative hypothesis is contrary to the null hypothesis, which shows that observations are the result of a real effect. It is denoted by . Note: You can think of the null hypothesis as the current value of the population populatio n parameter, which you hope to disprove in favor of your alt lte erna nattiv ive e h ot othe hesi sis s. 6 Take a look at this example. The school record claims that the mean score in Math of the incoming Grade 11 students is 81. The teacher wishes to find out if the claim is true. She tests if there is a significant difference between the batch mean score and the mean score of students in her class. Solution: Let be the population mean score and ̅ be the mean score of students in her class. You may select any of the following statements as your null and alternative hypothesis as shown in Option 1 and Option 2. Option 1: : The mean score of the incoming Grade 11 students is 81 or = 81. : The mean score of the incoming Grade 11 students is not 81 or ≠ 81. Option 2: : The mean score of the incoming Grade 11 students has no significant difference with the mean score of her students or = ̅ . : The mean score of the incoming Grade 11 students has a significant difference with the mean score of her students or ≠ ̅ . Now, it’s your turn! Based on the first claim of Sofia in Activity 2 that “the average daily usage of social media of her friends is the same as the global average usage”, formulate two hypotheses about the global average usage ( ) and the average usage of her friends ( ̅ ) on the blanks provided provided below. : _____________________________________________ : _____________________________________________ You can verify your answer to your teacher and start working on the next activity. Here is another key term you should know! Level of Significance The level of significance denoted by alpha or refers to the degree of significance in which we accept or reject the null hypothesis. 100% accuracy is not possible in accepting or rejecting a hypothesis. The significance significance level α is also the probability of making the wrong decision when the null hypothesis is true. Do you know that the most common levels of significance used are 1%, 5%, or 10%? Some statistics books can provide us table of values for these levels of si ni icance. 7 Take a look at this example. Maria uses 5% level of significance in proving that there is no significant change in the average number of enrollees in the 10 sections for the last two years. It means that the chance that the null hypothesis ( ) would be rejected when it is true is 5% . = 0.05 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 = 0.05 is actually the area under the normal curve curve within within the re ection rree ion. It’s your turn! If Sofia used a 0.10 level of significance, what are the chances that she would have a wrong conclusion if the two values have no significant difference? Here is another key term you should know! Two-Tailed Test vs One-Tailed Test When the alternative hypothesis is two-sided like : ≠ , it is called two-tailed test. When the given statistics hypothesis assumes a less than or greater than value, it is called one-tailed test. Here are some examples. The school registrar believes that the average number of enrollees this school year is not the same as the previous school year. In the above situation, let be the average number of enrollees last year. : = : ≠ If uses tailed test. 2 2 2 8 ≠, use a two- However, if the school registrar believes that the average number of enrollees this school year is less than the previous school year, then you will have: : : = < Use the left-tailed when contains the symbol <. On the other hand, if the school registrar believes that the average number of enrollees this school year is greater than the previous school year, then you will have: have: : : = > Use the right-tailed test when contains the symbol >. Now back to the two claims of Sofia, Sofia , what do you think should be the type of test in her following claims? Claim A: The average daily usage of social media of her friends is the same as the global average usage. Claim B: The average daily usage of social media of her friends is higher than the global average usage. Here is the other concept! 9 Illustration of the Rejection Region The rejection region (or critical region) is the set of all values of the test statistic that causes us to reject the null hypothesis. The non-rejection region (or acceptance region) is the set of all values of the test statistic that causes us to fail to reject the null hypothesis. The critical value is a point (boundary) on the test distribution that is compared to the test statistic to determine if the null hypothesis would be rejected. Non-Rejection Region -3 -2.5 -2 -1.5 Rejection Region -1 -0.5 0 0.5 1 1.5 2 2.5 3 Critical Value Illustrative Example 1: Now, let’s take a look at Sofia’s first claim. She assumed that the average online usage of her friends is the same as the global usage ( ). She computed for the t-value using the formula = 152, = ̅− √ where = 142, ̅ s = 19.855, and n = 10. = This t-test formula was discussed in the last chapter. ̅ √ = = 152 152 142 142 19.855 √10 . = 1.593 10 Use a scientific calculator to verify the computed tvalue. From the table of t-values, determine the critical value. Use df = n-1 = 9, one-tailed test at 5% level of significance. The critical critical t-value iis s 1.833. How did we get that value? Look at this illustratio illustration n! The table of t-values can be found at the last part of this module. Now, you can sketch a t distribution curve and label showing the rejection area (shaded part), the non-rejection region, the critical value, and the computed t-value. This is how your t distribution curve should look like! Rejection Re ion Non-Rejection Region -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1.593 (Computed Value) 1 1.5 2 2.5 3 1.833 (Critical Value) As you can see from your previous illustration, the computed tvalue of 1.593 is at the left of the critical value 1.833. So, in which region do you think the computed value falls? The computed computed value is less than the critical value. value. : The average online usage of her friends is the same as the global usage. : The average online usage of her friends is higher than the global usage. The computed computed t-value is at the non-rejection region. 11 We fail to reject the null hypothesis, . Illustrative Example 2: A medical trial is conducted to test whether or not a certain drug reduces cholesterol level. Upon trial, the computed z-value of 2.715 lies in the rejection area. -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 The computed z-value of 2.715 can be found here! The computed computed value is g greater reater than the critical v value. alue. : The certain drug is effective in reducing cholesterol level by 60%. : The certain drug is not effective in reducing cholesterol level by 60%. The computed computed z-value is at the rejection rejecti on region. We reject the null hypothesis, hypothesis, in favour of . Illustrative Example 3: Sketch the rejection region of the test hypothesis with critical values of ±1.753 and determine if the computed t-value of –1.52 lies in that region. Solution: Draw a t-distribution curve. Since there are two critical values, it is a two tailed test. Locate the critical values and shade the rejection regions. Now, locate the computed t-value of –1.52. You can clearly see that it is not at the rejection region as shown in the following figure. The computed t-value is at the non-rejection region. Therefore, we fail to reject the null hypothesis, . – 1.52 – 1.753 (critical value) 1.753 (critical value) 12 Type I and Type II Errors Rejecting the null hypothesis when it is true is called a Type I error with probability denoted by alpha (). In hypothesis testing, the normal curve that shows the critical region is called the alpha region. Accepting the null hypothesis when it is false is called a Type II error with probability denoted by beta (). In hypothesis testing, the normal curve that shows the acceptance region is called the beta region. The larger larger the value of alpha, alpha, the small smaller er is the val value ue of beta. This is the region of Type I error. α = P [Type I error] = P [ is true, Reject ] Region where is true α This is the region of Type II error. β = P [type II error] = P [ is false, Fail to reject ] Region where is false To summarize summarize the differe difference nce between the Type I and T Type ype II errors, take take a look at the table below. Null Hypothesis True False Fail to Reject Reject Correct Decision Type I Error - Failed to reject when it is true - Rejected when it is true Type II Error Correct Decision - Failed to reject when it is false - Rejected when it is false 13 Now, complete the statements that follow. Analyze the possibilities of Sofia’s conclusion. Identify if it is a Type I Error, Type II Error, or a Correct Decision. Decision. If Sofia finds out that her null hypothesis is … 1. true and she fails to reject it, then she commits a ____________________. ____________________. 2. true and she rejects it, then she commits a _____________________. 3. false and she fails to reject it, then she commits a __________________. 4. false and she rejects it, then she commits a _____________________. Your answers should be: 1) Correct Decision, 2) Type I Error, 3) Type II Error, and 4) Correct Decision. Illustrative Example: Bryan is starting his own food cart business and he is choosing cities where he will run his business. He wants to survey residents and test at 5% level of significance whether or not the demand is high enough to support his business before he applies for the necessary permits to operate in his selected city. He will only choose a city if there is strong evidence that the demand there is high enough. We can state the null hypothesis for his test as: : The demand is high enough. What would be the consequence of a Type I error in this setting? _____ He doesn't doesn't choose a c city ity wher where e demand is actually high high enough. _____ He chooses chooses a city where demand is actually high enoug enough. h. _____ He chooses chooses a city where demand iisn't sn't actually high enough enough. The Type I error is the first statement because he rejected the true null hypothesis. What would be the consequence of a Type II error in this setting? _____ He doesn't doesn't choose a c city ity wher where e demand is actually high high enough. _____ He chooses chooses a city where demand is actually high enoug enough. h. _____ He chooses chooses a city where demand isn't actuall actually y high enough. enough. The Type Type II error is the thi third rd statement because h he e failed to reject the false null hypothesis. What is the probability of Type I error? _____ 0.10 _____ 0.25 _____ 0.05 _____ 0.01 The probability of Type I error iis s 0.05 because it is the level of significance used. 14 What’s More Activity 1.1 Null Vs Alternative Directions: State the null and the alternative hypotheses of the following statements. 1. A medical trial is conducted to test whether or not a new medicine reduces uric acid by 50%. : ____________________________________________________ : ____________________________________________________ 2. We want to test whether the general average of students in Math is different from 80%. : ____________________________________________________ : ____________________________________________________ 3. We want to test whether the mean height of Grade 8 students is 58 inches. : ____________________________________________________ : ____________________________________________________ 4. We want to test if LPIHS students take more than four years to graduate from high school, on the average. : ____________________________________________________ : ____________________________________________________ 5. We want to test if it takes less than 60 minutes to answer the quarterly test in Calculus. : ____________________________________________________ : ____________________________________________________ 6. A medical test is conducted to determine whether or not a new vaccine reduces the complications of dengue fever. : ____________________________________________________ : ____________________________________________________ 7. The enr enrolment olment in high high school this school year increases increases by 10%. : ____________________________________________________ : ____________________________________________________ 8. The intelligenc intelligence e quotient of male grade 11 students students is the same as the female students. : ____________________________________________________ : ____________________________________________________ 15 9. The school want to test if the students in Grade 7 prefer online distance learning as the method of instruction. : ____________________________________________________ : ____________________________________________________ 10. The school librarian wants to find out if there was an increase in the number of students accessing the school library. : ____________________________________________________ : ____________________________________________________ Activity 1.2 The Tale of Tails Directions: Determine if one-tailed test or two-tailed test fits the given alternative hypothesis. 1. The mean height height of Grad Grade e 12 students iis s less than than 66 inches. inches. 2. The standard standard deviation deviation of their height height is not equal to to 5 inches. 3. Male Grade 7 and Grade 12 1 2 students differ in height on average. The proportion of senior male students’ height is significantly higher than 4. that of senior female students. 5. The average grade of Grade 11 students in Statistics is lower than their average grade in Calculus. 6. The newly found vaccine reduces the risks of viral infections of the patience. 7. The enrolment enrolment in elem elementary entary schools is not the s same ame as the enrolment enrolment in the secondary schools. 8. Male adolescents have higher intelligence quotient level than the female adolescents. 9. The average number of o f internet users this year is significantly significantly higher as compared last year. 10. Paracetamol and Ibuprofen have the same rate of time to reduce the headache of the patients. Activity 1.3 Are You In or Out? Directions: Illustrate the rejection region given the critical value and identify if the t-values lie in the non-rejection region or rejection region. 1. critical t-value of 1.318 computed t-value of 1.1 The computed computed t-value is at the __________ region. 16 1.671 computed t-value of 2.45 2. critical t-value of The computed computed t-value is at the __________ region. 1.725 computed t-value of 2.14 3. critical t-value of The computed computed t-value is at the __________ region. ±1.311 computed t-value of 1.134 4. critical t-value of The computed computed t-value is at the __________ region. 1.701 computed t-value of 2.48 5. critical t-value of The computed computed t-value is at the __________ region. 6. critical t-value of 2.12 computed t-value of 2.15 The computed computed t-value is at the __________ region. 17 2.306 computed t-value of 2.110 7. critical t-value of The computed computed t-value is at the __________ region. 2.228 computed t-value of 1.987 8. critical t-value of The computed computed t-value is at the __________ region. ±1.812 computed t-value of 1.915 9. critical t-value of The computed computed t-value is at the __________ region. 1.860 computed t-value of 2.3 10. critical t-value of The computed computed t-value is at the __________ region. Activity 1.4 Type I or Type IIII Directions: Check the box that corresponds to your answer. Situation 1: A quality control expert wants to test the null hypothesis that an imported solar panel is an effective source of energy. 18 1. What would be the consequence of a Type I error in this context? They do not conclude that the They do not conclude that the solar panel is effective when it is solar panel is effective when it is actually effective. not actually effectiv effective. e. They conclude that the solar They conclude that the solar panel panel is effective when it is is effective when it is not actually effective. actually effective. 2. What would be the consequence of They do not conclude that the solar panel is effective when it is not actually effective effective.. They conclude that the solar panel is effective when it is actually effective. a Type II error? They do not conclude conclude that the solar panel is effective when it is actually effective. They conclude that the solar panel is effective when it is not actually effective. Situation 2: A resort owner does a daily water quality test in their swimming pool. If the level of contaminants is too high, then he temporarily closes the pool to perform a water treatment. We can state the hypotheses for his test as: : The water quality is acceptable. : The water quality is not acceptable. 3. What would be the consequence of a Type I error in this setting? The owner owner closes the p pool ool when it needs to be closed. The owner owner does not clos close e the pool when it needs to be closed. The owner owner closes the p pool ool when it does not need to be closed. The owner owner does not clos close e the pool when it does not need to be closed. 4. What would be the consequence of a Type II error in this setting? The owner closes the pool pool when it The owner closes the pool when it needs to be closed. does not need to be closed. The owner does not close the pool The owner does not close the pool when it does not need to be when it needs to be closed. closed. 19 5. In terms of safety, which error has more dangerous consequences in this setting? Type I Error Type II Error What I Have Learned Directions: Complete the following statements. Write the answers in your notebook. 1. __________ _____________________ _______________is ____is a statistic statistical al method that is used in making decisions using experimental data. 2. A ________________________ is a proposed explanation, assertion, or assumption assumpti on about a population parameter or about the distribution of a random variable. 3. The null hypothesis is an initial claim which the researcher researcher tries to _____________________ __________ ______________________ _________________. ______. 4. The alternative hypothesis hypothesis is _____________________ __________ _______________________ _________________. _____. contrary to the 5. The level level of significance significance is denoted denoted by____ by_______________ ___________________. ________. 6. The significance level α is also the probability of making the wrong decision when ____________________________________. 7. When the alternative hypothesis ______________________ __________ ___________________. _______. is two-sided, it is called 8. When the given statistics hypothesis assumes a less than or greater than value, it is called ______________________________. 9. The rejection region (or critical region) is the set of all values of the test statistic that cause us to ____________________ _______________________________ ____________ _ 10. Rejecting the null hypothesis when it is true results to what type of error? 20 What I Can Do Directions: Cite five (5) situations in your community where you can apply hypothesis testing. Then, just choose one situation and: 1. create a problem statement; 2. formulate the null and alternative hypothesis; 3. select the level of significance and sketch the rejection region; and 4. state the possible Type I and Type II errors. Assessment Directions: Choose the best answer to the given questions or statements. Write the letter of your answer on a separate sheet of paper. 1. It is the statistical method used in making decisions using experimental data. A. observation B. simple analysis C. analytical testing D. hypothesis testing 2. What term is being used to describe a proposed explanation, assertion, or assumption about a population parameter or about the distribution of a random variable? A. statistic B. decision C. hypothesis D. probabilit probability y 3. It is also referred to as a probability of committing an incorrect decision about the null hypothesi hypothesis. s. A. level of error B. level of hypothesis C. level of acceptanc acceptance e D. level of significance 4. Which of the following would be an appropriate null hypothesis? A. B. C. D. The mean of a sample is equal to 80. The mean of a population is equal to 80. The mean of a population is not equal to 80. The mean of a population is greater th than an 80. 21 5. Which of the following describes a null hypothesis using two-tailed test? A. : = 6. B. : ≠ C. : ≥ D. : ≤ Which of the following describes an alternative hypothesis using twotailed test? A. : < 50 C. : ≠ 50 . : > 50 D. : = 50 7. Which of the following must be used as the significance level if we want a lower possibility of correct decision? A. 1% B. 2% C. 5% D. 10% 8. Which of the following would be an appropriate alternative hypothesis for one-tailed test? A. : = 85 B. : ≥ 85 C. : ≥ 85 D. : < 85 9. In a one-tailed test, in which critical values below will the computed z of 2.312 falls in the non-rejection region? A. 1.383 B. 1.533 C. 2.228 D. 2.354 10. When is a Type I error committed? A. B. C. D. We We We We reject a null hypothesis that is true. reject a null hypothesis that is false. fail to reject a null hypothesis that is true. fail to reject a null hypothesis that is false. 11. When is a Type II error committed? A. B. C. D. We We We We reject a null hypothesis that is true. reject a null hypothesis that is false. fail to reject a null hypothesis that is true. fail to reject a null hypothesis that is false. 12. Which of the following is a Type I error? A. is true; reject . B. is false; reject . C. is true; fail to reject . D. is false; fail to reject . 13. If the computed z-value is 1.286 and a nd the critical value is 1.383, which of the following statements could be true? A. It lies in the rejection region, must be rejected. B. It lies in the rejection region, hence we fail to reject . C. It lies in the non-rejection region, must be rejected. D. It lies in the non-rejection region, hence we fail to reject . 14. Using a left-tailed test, which of the following value of z will not fall in the rejection region whereB. the– critical is ––1.638? A. – 1.637 1.639 value C. 1.641 22 D. – 1.706 15. If the computed z-value is 1.915 and a nd the critical value is 1.812, which of the following statements could be true? A. It lies in the rejection region, must be rejected. B. It lies in the rejection region, hence we fail to reject . C. It lies in the non-rejection region, must be rejected. D. It lies in the non-rejection region, hence we fail to reject . dditional Activities A medical trial is conducted to test whether or not a certain drug can treat a certain allergy. Upon trial, the t-value is computed as 1.311. Sketch and complete the table below to discuss the findings of the medical trial. -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 : 1 1.5 2 2.5 The computed computed t-value is at the ___________ __________ _ region. : 3 Decision: Justify your your decision by writing an exp explanation lanation in 5-10 5-10 sentenc sentences. es. 23 Answer Key . s ul ucl ac ni t set yl r et r a uq e ht r e ws na ot set uni m06) naht er o mr o( na ht ssel se kat tI : . s ul ucl ac ni t set yl r et r a uq e ht r e ws na ot set uni m06 se kat tI : .) el bi sso pt o nfi sr ae y 4 naht ssel( l oo hcs hgi h morf et a udar g ot sr ae y r uof na ht er o mf o e g ar ev a na se kat st ne d ut S: .l oo hcs hgi h morf et a udar g ot sr aey r uof f o eg ar ev a na se kat st ne d ut S: . se hc ni 85) naht r e hgi h r o naht ssel r o( t o n si st ne d ut s f o t hgi e h nae me hT : . se hc ni 85 si st ne d ut s f o t hgi e h nae me hT: . %08 morf t ner effi d si ht a M ni st ne d ut s f o e g ar ev a l ar e ne g e hT : . %08 ot l a uqe si ht a M ni st ne d ut s f o eg ar ev a l ar e neg e hT: . di c a ci r u ec uder t o nnac e ni ci de m we n e hT : . di c a ci r u ec uder nac e ni ci de m we n e hT: .5 .4 .3 .2 .1 1. 1 yti vi t c A D. 51 A. 41 D. 31 C. A. B. B. C. 5 4 3 2 1 nI s’ t a h W 24 A. 21 A. 11 B. 01 A.9 D. 8 B.7 B.6 D. 5 B.4 D. 3 C.2 D. 1 wo n KI t a h W r orr EII e py T . 5 . des ol c e b ot s dee n ti ne h wl oo p e ht es ol c t o n se o d r e n wo e hT . 4 . des ol c e b ot dee n t o n se o d ti ne h wl oo p e ht ses ol c r e n wo e hT . 3 . evi t ceff e yll a ut c a t o n si ti ne h w evi t ceff e si l e na p r al os e ht t a ht e d ul c noc ye hT . 2 . evi t ceff e yll a ut c a si ti ne h w evi t ceff e si l e na p r al os e ht t a ht e d ul c noc t o n o d ye hT . 1 nI s’ t a h W t set t set t set t set t set t set t set t set t set t set deli at- o wt. deli at - e no deli at - e no deli at - o wt deli at - e no deli at - e no deli at - e no deli at - o wt deli at - o wt deli at- e no noi t cej er. noi t cej er ec nat pecc a ec nat pecc a noi t cej er noi t cej er ec nat pecc a 01 .9 .8 .7 .6 .5 .4 .3 .2 .1 01 .9 .8 .7 .6 .5 .4 noi t cej er . 3 noi t cej er . 2 ec nat pecc a . 1 3. 1 yti vi t c A 2. 1 yti vi t c A . yr ar bil l oo hcs e ht g ni ssecc a st ne d ut s f o r e b mun e ht ni es aer c ni o n s a w er e hT : . yr ar bil l oo hcs e ht g ni ssecc a st ne d ut s f o r e b mun e ht ni es aer c ni na s a w er e hT: . . s noi t c urt s ni f o do ht e m e ht s a g ni nr ael ec nat si d e nil no r ef er p t o n di d st ne d ut s 7 e dar G: . s noi t c urt s ni f o do ht e me ht s a g ni nr ael ec nat si d e nil no r ef er p st ne d ut s 7 e dar G: . st ne d ut s el a mef e ht na ht ) r e wol r o( r e hgi h si st ne d ut s 11 e dar g el a mf o t nei t o uq ec ne gill et ni e hT : . st ne d ut s el a mef e ht s a e mas e ht si st ne d ut s 11 e dar g el a mf o t nei t o uq ec negill et ni e hT: . %01 y b es aer c ni t o n di d r aey l oo hcs si ht l oo hcs hgi h ni t ne m l or ne e hT : . %01 y b ses aer c ni r aey l oo hcs si ht l oo hcs hgi h ni t ne m l or ne e hT: . r evef e ug ne df o s noi t acil p moc e ht ec uder t o nnac e ni cc av we n e hT : . r evef e ug ne df o s noi t acil p moc e ht sec uder e ni cc av we n e hT: 01 .9 .8 .7 .6 ) … t no C( 1. 1 yti vi t c A 25 A. 51 A. 41 D. 31 A. 21 D. 11 A. 01 D. 9 D. 8 D. 7 C.6 A.5 B.4 D. 3 C.2 D. 1 t ne m ss ess A r orr e I e py T. 01 si se ht o py h ll un e ht t cej er . 9 t set deli at - e no . 8 t set deli at - o wt . 7 e urt si si se ht o py h ll un e ht . 6 ) ( a hpl a . 5 si se ht o py h ll un . 4 yfill un r o , t cej er , ev or psi d . 3 si se ht o py h . 2 g ni t set si se ht o py H . 1 de nr ae L eva HI t a h W 26 27 References Textbooks Albacea, Zita, Mark John Ayaay, Imelda Demesa, and Isidro David. Teaching Guide for Senior High School: Statistics and Probability . Quezon City: Commission on Higher Education, 2016. Caraan, Avelino. Introduction to Statistics & Probability . Mandaluyong City: Jose Rizal Universi University ty Press, 2011. 2011. Chan Shio, Christian Paul, and Maria Angeli Reyes. Statistics and Probability for Senior High School. Quezon City: C & E Publishing Inc., 2017. De Guzman, Danilo. Statistics and Probability. Quezon City: C & E Publishing Inc., 2017. Jaggia, Sanjiv, and Alison Kelly. Business Statistics: Communicating with Numbers. 2ndEd. New York: McGraw-Hill Education, 2016. Sirug, Winston. Statistics and Probability for Senior High School CORE Subject A Comprehensive Approach K to 12 Curriculum Compliant. Manila: Minshapers Minshapers Co., Inc., 2017. Online Resources Khan Academy. “Consequences of Errors and Significance.” Accessed February 2, 2019. https://www.khanacademy.org/math/apstatistics/tests-significance-ap/error-probabilitiespower/a/consequences-errors-significance Minitab.com. “About the Null and Alternative Hypotheses. ” Accessed February 4, 2019. https://support.minitab.com/enus/minitab/18/help-and-how-to/statistics/basicstatistics/supporting-topics/basics/null-and-alternative-hypotheses/ Minitab. com. “What are Type I and Type II Errors? ” Accessed February 4, 2019. https://support.minitab.c https://support.minitab.com/en-us/mi om/en-us/minitab/18/he nitab/18/help-and-howlp-and-howto/statistics/basic-statistics/supporting-topics/basics/type-i-and-typeii-error/ Zaiontz, Charles. “Null and Alternative Hypothesis.” Accessed February 2, 2018. http://www.real-statistics.com/hypothesis-testing/nullhypothesis/ 28 For inquiries or feedback, please write or call: Department of Education - Bureau of Learning Resources (DepEd-BLR) Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (632) 8634-1072; 8634-1054; 8631-4985 Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph