QUARTER 4 HYPOTHESIS TESTING/ TYPE 1 AND II ERROR/ TRADITIONAL AND P-VALUE METHOD
- 2. Identify the Appropriate test- statistics Learning Objectives Formulate the appropriate null and alternative hypotheses Illustrates the null hypothesis, alternative hypothesis, level of significance, rejection region and types of errors in hypothesis testing
- 4. HYPOTHESIS TESTING Hypothesis is a decision-making process for evaluating claims about a Population based on the characteristics of a sample purportedly coming from that population..
- 5. define the population state the hypothesis give the significance level select a sample collect data run tests on the data draw conclusion
- 7. null hypothesis denoted by H0, is a statement that there is no difference between a parameter and a specific value, or that there is no difference between two parameters.
- 8. alternative hypothesis is denoted by H1, is a statement that there is a difference between a parameter and a specific value, or that there is a difference between two
- 11. Two-tailed test/ Non-directional is a test with two rejection regions. In this test, the null hypothesis should be rejected when the test value is in either of the two critical regions Example A nutritionist claims that his/her developed bread is fortified with Vitamin B. A certain combination of fruits provides tge daily requirement for Vitamin C.
- 12. One-tailed test indicates that the null hypothesis should be rejected when the test values is in the critical region on one side of the parameter. Example A musician believes that listening to classical music affects the mood of a person. A mother wants to prove that reading books to children improves their thinking process.
- 13. Hypothesis A statistical hypothesis is an inference about a population parameter. This inference may or may not be true. Example Brand X detergent will wash white clothes sparkling white. Claims of product quality in TV commercials A certain type of gasoline can get your car more kilometers covered to the kiter than before.
- 14. Examine sample instead of population to draw conclusions. The only sure way of finding the truth or falsity of a hypothesis is by examining the entire population. Because this is always not feasible, a sample is instead examined
- 15. Parameter is a numerical value that states something about the entire population being studied. Example
- 16. Ho = parameter = specific value H1 = parameter specific value This is a two-tailed test. NON - DIRECTIONAL
- 17. Ho = parameter = specific value H1 = parameter < specific value This is a left-tailed test. DIRECTIONAL
- 18. Ho = parameter = specific value H1 = parameter > specific value This is a right-tailed test. DIRECTIONAL
- 21. Advantages Disadvantages Flexibility in Data Analysis Increased sample sizes Objective and Open- minded approach Reduced Precision Comprehensive understanding Potential Post-hoc interpretation Greater Sensitivity Limited theoretical guidance NONDIRECTIONAL
- 22. Advantages Disadvantages Hypothesis generation Potential missed interpretation Replication and generalizability NONDIRECTIONAL Note: H1 determines the direction of the test, not the null hypothesis.
- 23. Advantages Disadvantages Specific prediction Risk of Type I error Testable and refutable Narrow focus Efficiency and resource application Limited generalizability Practical applications Biased interpretation Enhanced communication Reduced flexibility DIRECTIONAL
- 24. level of significance is a pre-determined error which the researcher is willing to risk in rejecting the null hypothesis when it is true.
- 25. rejection region/ critical region is the range of values of the test value that indicates that there is a significance difference and that the null hypothesis should be rejected.
- 26. Acceptance region is the range of values of the test value that indicates that there is no significance difference and that the null hypothesis should be accepted
- 27. Type I error If the null hypothesis is true and accepted, or if it is false and rejected, the decision is correct. If the null hypothesis is true and rejected, the decision is incorrect.
- 28. Type II error If the null hypothesis is false and accepted, the decision is incorrect Example
- 29. What mistakes do people make?
- 30. On a moonlit night, a young man declares that there are two moons. Read and identify errors. Bryan thinks that he is a six-footer. His actual height is 156 cm. Mark says, “I am virtuous!” In the next moment, he finds himself in jail.
- 31. On a beachfront, a signage reads, “No littering of plastic wrappers, empty bottles, and cans.” A few yards away, environmentalists are picking up the rubbish left behind by picnic lovers. Read and identify errors. Thousands of years ago, Ptolemy declared that the earth is flat.
- 32. Common phrases of hypothesis testing
- 33. EXAMPLES
- 34. BOTTLED FRUIT JUICE CONTENT The owner of a factory that sells a particular bottled fruit juice claims that the average capacity of a bottle of their product is 250 mL.
- 35. BOTTLED FRUIT JUICE CONTENT The owner of a factory that sells a particular bottled fruit juice claims that the average capacity of a bottle of their product is 250 mL. H0: ‘The bottled drinks contain 250mL per bottle.’ Ha: ‘The bottled drinks do not contain 250mL per bottle.’
- 36. WORKING STUDENTS A university claims that working students earn an average of Php 20 per hour.
- 37. Retirement The mean number of years Americans work before retiring is 34.
- 38. Elections At most 60% of Filipino votes in presidential elections.
- 40. Reject H0 Do not reject H0 H0 is TRUE. Type I error correct decision H0 is FALSE. correct decision Type II error POSSIBLE OUTCOMES IN HYPOTHESIS TESTING The null hypothesis may or may not be true. The decision to reject or not to reject is based on the data obtained from the sample of the population. table 1
- 41. type I error Type I error occurs if one rejects the null hypothesis when it is true.
- 42. type II error Type II error occurs if one does not reject the null hypothesis when it is false.
- 44. The level of significance is the maximum probability of committing a type I error. This probability is symbolized by (greek letter alpha). That is, P(type I error ) = . The probability of type II error is symbolized by ß (greek letter beta). That is, P (type II error) = ß. Although, in most hypothesis testing situations, ß cannot be computed LEVEL OF SIGNIFICANCE
- 45. Critical value determines the critical and non-critical regions. the critical region or the rejection region is the range of the test value that indicates that there is a significant difference and the null hypothesis should be rejected. The non-critical region is the range of values of the test value that indicates that difference was probably due to the chance and that the null hypothesis should not be rejected. CRITICAL VALUE
- 46. CRITICAL VALUE If your computed statistic is found in the rejection region, then you reject Ho. If it is found outside the rejection region, you accept Ho.
- 48. when to use t-test or z-test?
- 49. Z- TEST T- TEST
- 51. z-Test
- 52. t-Test
- 54. 1 STEP Describe the population parameter of interest.
- 55. 2 STEP Formulate the hypotheses: null and alternative hypothesis. State the null hypothesis in such a way that a Type I error can be calculated.
- 56. 3 STEP Check the assumptions. 1. Is the sample size large enough to apply Central Limit Theorem? 2. Do small samples come from normally distributed populations? Are the samples selected randomly?
- 57. 4 STEP Choose a significance level for a. Is the test two-tailed or one-tailed? b. Get the critical values from the test statistical table. c. establish the critical regions.
- 58. 5 STEP Compute the test statistic.
- 59. 6 STEP State the decisio rule.
- 60. 7 STEP Compare the test statistic and the critical value. . Based on the decision rule, decide whether to reject or not to reject . Interpret the result.
- 61. Example Bryan administered a Mathematics achievement test to a random sample of 50 graduating pupils. In this sample, and s=10. The population parameters are Is the performance of the sample above average? and
- 65. 1 STEP Describe the population parameter of interest.
- 66. 2 STEP Formulate the hypotheses: null and alternative hypothesis. State the null hypothesis in such a way that a Type I error can be calculated.
- 67. 3 STEP Check the assumptions. 1. Is the sample size large enough to apply Central Limit Theorem? 2. Do small samples come from normally distributed populations? Are the samples selected randomly?
- 68. 4 STEP Choose a significance level size for Make small when the consequences of rejecting a true null hypothesis severe . Is the test one-tailed or two-tailed?
- 69. 5 STEP Select the appropriate test statistic. a. compute the test statistic b. compute the p-value
- 70. 6 STEP State the decision rule for rejecting the null hypothesis. Reject if the computed p-value is
- 71. 7 STEP Compare the computed probability and . Based on the decision rule, decide whether to reject or not to reject . Interpret the result.
- 72. Example Bryan administered a Mathematics achievement test to a random sample of 50 graduating pupils. In this sample, and s=10. The population parameters are Is the performance of the sample above average? and
- 76. examples: Direction: For each of the following, draw the normal curvem locate the z-value, and indicate if the z-value is in the rejection or in acceptance zone. 1. z=2, 95% confidence, two-tailed. 2. z=2.68, 95% confidence, two-taied.
- 77. 1. z=2, 95% confidence, two-tailed. 2. z=2.68, 95% confidence, two-taied.
- 81. 1. z=2, 95% confidence, two-tailed.
- 82. 1. z=2, 95% confidence, two-tailed. 2. z=2.68, 95% confidence, two-taied.
- 83. Examples A manufacturer claims that the average lifetime of his lightbulbs is 3 years or 36 months. The standard deviation is 8 months. Fifty bulbs are selected, and the average lifetime is found to be 32 months. Should the manufacturer’s statement be rejected at = 0.1?
- 84. STEP 3: COMPUTE THE VALUE OF THE TEST STATISTIC.
- 85. STEP 4: MAKE THE DECISION RULE.
- 86. STEP 5: MAKE THE DECISION.
- 87. sample 1. The father of a senior high school student is listing down the expenses he will incur when he sends his daughter to the university. At the university where he wants his daughter to study, he hears that the average tuition fee is atleast P20 000 per semester. He wants to do hypothesis testing. Suppose from a simple random sample of 16 students, a sample mean pf P 19750 was obtained. Furthermore, the tuition in the university is said to be normally distributed with an assumed population variance equal to P 160 000 and level of significance at
- 91. sample 1. The father of a senior high school student is listing down the expenses he will incur when he sends his daughter to the university. At the university where he wants his daughter to study, he hears that the average tuition fee is atleast P20 000 per semester. He knows the variable of interest, which is the tuition fee, is measured for atleast in the interval scale. He assumes that the variable of interest follows normal distribution but both population mean and variance are unknown. The father asks at random 25 students at the university about their tuition fee per semester. He is able to get an average of P20050 with a standard deviation of P 500. Suppose the level of significance is
- 95. Example A high-end computer manufacturer sets the retail cost of their computers based in the manufacturing cost, which is 28000. However, the company thinks there are hidden costs and that the average cost to manufacture the computers is actually much more. The company randomly selects 40 computers from its facilities and finds that the mean cost to produce a computer is 32000 with a standard deviation of 800. Run a hypothesis test to see if this thought is true.
Editor's Notes
- #1: We make decisions every day. Some of these are important while others are not. In decision-making, we usually follow certain processes: weigh alternatives, collect evidence, and make a decision. After a decision is made, an appropriate interpretation is made (or an action is undertaken). We follow these basic processes in testing hypothesis in Statistics.
- #2: Hypothesis is a decision-making process for evaluating claims about a Population. In this process, the researcher must define the population under study, state the particular hypotheses to be investigated, give the significance level, select a sample from the population. Collect a data, perform the requires test, and reach a conclusion. Every hypothesis begins with the statement of the hypothesis. The only sure way of finding the truth or falsity of a hypothesis is by examining the entire population.
- #13: Every hypothesis testing begins with the statement of a hypothesis. A statistical hypothesis is an inference about a population parameter. This inference may or may not be true.
- #14: Every hypothesis testing begins with the statement of a hypothesis. A statistical hypothesis is an inference about a population parameter. This inference may or may not be true.