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O a Let Us Study Hypothesis testing is a statistical method applied in making decisions using experimental data. Hypothesis testing is basically testing an assumption that we make about a population. A hypothesis is a proposed explanation, assertion, or assumption about a population parameter or about the distribution of a random variable. Here are the examples of questions you can answer with a hypothesis test: 0 Does the mean height of Grade 12 students differ from 66 inches? 0 Do male and female Grade 7 and Grade 12 students differ in height on average? Is the proportion of senior male students' height signicantly 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 signicant difference between two parameters. It is denoted by H0. / The alternative hypothesis is contrary to the null hypothesis, which shows that observations are the result of a real effect. It is denoted by Ha. Note: You can think of the null hypothesis as the current value of the population parameter, which you hope to disprove in favor of your alternative hypothesis. Take a look at this example. The school record claims that the mean score in Math of the incoming Grade 1 1 students is 81. The teacher wishes to nd out if the claim is true. She tests if there is a signicant difference between the batch mean score and the mean score of students in her class. Solution: Let u be the population mean score and a? 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. Ootion 1: Ho: The mean score of the incoming Grade 1 1 students is 81 or u = 81. Ha: The mean score of the incoming Grade 1 1 students is not 81 or u at 81. Option 2: Ho: The mean score of the incoming Grade 11 students has no significant difference with the mean score of her students or u = x. Ha: The mean score of the incoming Grade 11 students has a significant difference with the mean score of her students or u * x. Here is another key term you should know! Level of Significance . The level of significance denoted by alpha or a 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 level a 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%? 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 (H.) would be rejected when it is true is 5%. a = 0.05 1.5 2.5 Here is another key term you should know! Two-Tailed Test vs One-Tailed Test When the alternative hypothesis is two-sided like Ha: u # Mo, 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 Mo be the average number of enrollees last year. Ho: H = Ho Ha: H # HoIf Ha uses at, use a two- tailed test: 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: Ho: '1 = #0 Use the left-tailed when Ha contains the symbol <. ha: h on the other hand if school registrar believes that average number of enrollees this year is greater than previous then you will have: o : p use right-tailed test when j> .110 Ha contains the symbol >. Here is the other concept! 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. J 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. x/ 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. NoneRejection Region Rejection Region Critical Value 1. A medical trial is conducted to test whether or not a certain drug reduces cholesterol level by 60%. Upon trial, the computed zvalue of 2.715 lies in the rejection area. 3 2.5 2 -1.5 41 0.5 O 0.5 1 1.5 2 ' 2. 3 The computed zvalue of 2.715 can be found here! The computed value is greater than the critical value. Ho: The certain drug is effective in reducing cholesterol level by 60%. We reject the null hypothesis, H0 in favour of Ha. The computed zvalue is at the Ha: The certain drug is not effective in reducing cholesterol level by 60%. rejection region. 2. Sketch the rejection region of the test hypothesis with critical values of i1.7 53 and determine if the computed tvalue of 1.52 lies in that region. Since there are two critical values, it is a two tailed test. -35 -3 4.! 4 143 -I 4.! 0 II! 1 15 T 3 u I 35 1.753 1.753 (cn'tical value) (critical value) You can clearly see that computed tvalue of 1.52 is not at the rejection region as shown in the gure. The computed tvalue is at the nonrejection region. Therefore, we fail to reject the null hypothesis, H0. 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 ((1). 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 (8). In hypothesis testing, the normal curve that shows the acceptance region is called the beta region. \\/ The larger the value of alpha, the smaller is the value of beta. This is the region of Type I error. a = P [Type I error] = P [H, is true, Reject Ho] Region where Ho is true OL -15 This is the region of Type II error. B = P [type II error] = P [H. is false, Fail to reject Ho] Region where Ho is B false -2.5 -1.5 -0.5 0.5 1.5 2.5 To summarize the difference between the Type I and Type II errors, take a look at the table below. Null Hypothesis H. Fail to Reject H. Reject Ho Correct Decision Type I Error True - Failed to reject Ho when - Rejected H. when it is true Jit is true Type II Error Correct Decision False -Failed to reject H. when - Rejected H. when it it is false is false