Formulating Hypothesis and Identifying Test Statistic

Please solve all parts and show complete solution + explanation EACH NUMBER. I want to understand how it got solved and to check my own answer as well. Thank you!

ACTIVITY 1. Please do not just state the answer, explain and give solution.

Transform the following pairs of hypotheses in symbols. Write your answer on a separate sheet of paper. 1. Ho: The sample comes from a population whose mean is 85. H1: The sample does not come from a population whose mean is 85. 2. Ho: The mean grade of grade 12 learners is 86. H, : The mean grade of grade 12 learners is less than 86. 3. Ho: The average score of a professional basketball player in a game is 25. H,: The average score of a professional basketball player in a game is more than 25. 4. Ho: The sample comes from a population whose mean is 124. Hi: The sample does not come from a population whose mean is 124. 5. Ho: The mean price of a chicken meat per kilo is P190.00. H,: The mean price of a chicken meat per kilo is not P190.00.h. The expression 'greater harvest' is a clue that this situation involves a directional test. Therefore, this is a one-tailed test. s. In symbols we can write the null and alternative hypotheses as follows: Hn:;1='i"5 H1413 ?5 Example 2. The mean grade of the learners in XYZ High School on Statistics subject is 33. a sample of 45 learners was included in an experimental study to determine if peer mentoring can affect their performance. The result reveals a mean grade of 86 with a standard deviation of '3. a. State the null and alternative hypotheses in words. b. State whether the test is two-tailed or one-tailed c. Transform the null and alternative hypotheses in symbols. Solution 2. The parameter of interest is the population mean u = 33 a. The null and alternative hypotheses are the following: H\": The sample comes from a population whose mean is :1 is 33. H1: The sample does not come from a population whose mean is u is 33. b. The expression 'can aect their performanoe'is an indication that the learner's performanoe either increases or decreases. This means that the test is non-directional. Therefore, this is a two-tailed test. e. In symbols we can write the null and alternative hypotheses as follows: Hg: :1 = 83 H1: p at 83 It is also signicant in hypotheses testing that we can determine the test statistic to he used in a given situation. a test statistic is a value that plays a vital role in the decision making whether to reject or not to reject the null hypotheses. Let us consider the following cases: Two-Tailed Test It is a non-directional test which means that there is no specified direction as to whether the parameter is greater than or less than. It is commonly denoted by the symbol #- Dne-'I'aed Test It is a test which has a specific direction for the parameter. It can be either greater than {a} or iess than [at Example 1 In a farm, the average harvest of Mr Tolentino 1s T5 mangoes per tree with a standard deviation of 3. He decided to try an inorganic fertilizer to produce a greater harvest of his mangoes. a sample of 5D trees has been applied with the fertilizer and yielded 91] mangoes per tree with a standard deviation of ti. a. State the null and alternative hypotheses in words. b. State whether the testis two-tailed or one-tailed c. Transform the null and alternative hypotheses in symbols. otation 1. The parameter of interest is the population mean p = T5 a. The null and alternative hypotheses are the following: H\": The sample comes from a population whose mean is p is 7'5. H1: The sample does not come from a population whose mean is p is 1'5. c. Transform the null and alternative hypotheses in symbols. d. Identify the test statistic to be used in testing the hypothesis. Solution 3. The parameter of interest is the population mean # = 50 a. The null and alternative hypotheses are the following: Ho: The sample comes from a population whose mean is u is 50. H,: The sample does not come from a population whose mean is u is 50. b. The expression 'significantly addresses' signifies that learner's weight either increases or decreases due to the implementation of the program. This means that the test is non-directional. Therefore, this is a two-tailed test. c. In symbols we can write the null and alternative hypotheses as follows: Ho: u = 50 Hi: u # 50 d. The sample size of 25 learners is small which less than 30 and the population standard deviation was not indicated. So, we cannot apply the Central Limit Theorem and we need to approximate that the population is normal. We also need to estimate o using the sample standard deviation s. Therefore, we will apply the t-test to test the hypothesis.Case 1: The sample is large and the population standard deviation is known. When the sample is large such that n 2 30 and the population standard deviation o is known, Central Limit Theorem is applicable. This means that the population is normally distributed. Therefore, we can use the z-test. Test statistic: Z = X -1 The first example in the previous page can be applied with Case 1. Recall that the population mean is 75, the sample mean is 90, population standard deviation is 8 and a sample size of 50 which is greater than 30. Case 2: The sample is large, but the population standard deviation is unknown. When the sample is large such that n 2 30 but the population standard deviation o is unknown, we can still apply the Central Limit Theorem. In this case, we will use the sample standard deviation s as an estimate for the population standard deviation o. Therefore, the test statistic will then be, Test statistic: Z = X -H The second example in the previous page can be applied with Case 2. Recall that the population mean is 83, the sample mean is 86, sample standard deviation is 8 and a sample size of 45 which is greater than 30. Case 3: The sample is small such that n