Significance tests #1 - Questions and Answers

Question 1 For a test of Ho: p = 0.5, the z test statistic equals 1.74. Find the p-value for Ha: p>0.5. α) 0.0446 b) 0.0409 c) 0.892 d) 0.9591 e) 0.0818 f) 0.9554 Answer Step 1 Solution: Using standard normal table z = 1.74 This Right tailed test p(z > 1.74) 1-0.9591 = 0.0409 Correct optioan is 0.0409 Question 2 Suppose you conduct a test and your p-value turns out to be 0.08. What can you conclude? Reject Ho at a = 0.05 but not at a = 0.10 a) Reject Ho at a = 0.05 but not at a = 0.10 b) Reject Ho at a = 0.01 but not at a = 0.05 c) Reject Ho at a = 0.10 but not at a = 0.05 d) Reject Ho at alpha equal to 0.10, 0.05, and 0.01 e) Do not reject Ho at a equal to 0.10, 0.05, or 0.01 Answer If a p-value is less than a equal to significance level (α), then we reject null hypothesis. Case 1 - if α = 0.05 We fail to reject H₀ (Null Hypothesis) Case 2 - if α = 0.01 We fail to reject H₀ Case 3 - if α = 0.10 So, the correct option is, c) Reject H₀ at α = 0.10 but not at α = 0.05 Question 3 (a) In formulating hypotheses for a statistical test of significance, the null hypothesis is often A. the probability of observing the data you actually obtained B. a statement that the data are all 0. C. a statement of "no effect" or "no difference". D. 0.05 (b) In testing hypotheses, which of the following would be strong evidence against the null hypothesis? A. Using a small level of significance. B. Obtaining data with a large P-value. C. Obtaining data with a small P -value. D. Using a large level of significance. (c) The P-value of a test of a null hypothesis is A. the probability, assuming the null hypothesis is false, that the test statistic will take a value at least as extreme as that actually observed. B. the probability the null hypothesis is false. C. the probability the null hypothesis is true. D. the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as extreme as that actually observed. Answer (α) In hypothesis testing the for statistical significance test, the null hypothesis is the statement which states that there is no effect of the treatment or there is no significance difference. Correct answer: Option C. a statement of "no effect" or "no difference" (b) When the level of significance is increased there would be a strong evidence against the null hypothesis because the null hypothesis is rejected in p-value is less than significance level. Correct answer: Option D. Using a large level of significance. (c) The p-value of the hypothesis is likelihood that the test statistic that would take a value at least as extreme as that actually observed assuming that the null hypothesis is true. Correct answer: Option D. the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as extreme as that actually observed. Question 4 Your math test scores are 68, 78, 90, and 91. What is the lowest score you can earn on the next test and still achieve an average of at least 85? Getting an Answer Solve your inequality to find the lowest score you can earn on the next test and still achieve an average of at least 85. What score do you need to earn? Answer 1 Step 1 (68+78+90+91+x) / 5 ≥ 85 68+78+90+91+x ≥ 5 × 85 327 + x ≥ 425 x ≥ 98 So, the minimum score required to have an average of at lest 85 is 98. Answer 2 Step 1 Math test scores are 68, 78, 90 and 91 Average of five scores have to be at least 85. Step 2 Let the unknown score be x Average = Total marks / number of counts For at least, one have to use ≥ Therefore (68+78+90+91+x) / 5 ≥ 85 327 + x ≥ 425 x ≥ 98 So the lowest score is 98. Answer 3 Step 1: Let's denote the score you need to earn on the next test as x. The average score of all the tests can be calculated by adding up all the scores and dividing by the number of tests. Since there are 4 tests with scores 68, 78, 90, and 91, the average is given by: (68+78+90+91+x) / 5 To achieve an average of at least 85, this expression must be greater than or equal to 85. Therefore, we can write the inequality: (68+78+90+91+x) / 5 ≥ 85 Step 2: To solve this inequality, we can first multiply both sides by 5 to eliminate the fraction: 68+78+90 +91 + x ≥ 85×5 Simplifying the right side: 68+78+90 +91 + x ≥ 425 Combining like terms: x+327≥ 425 Step 3: Next, we isolate the variable x by subtracting 327 from both sides of the inequality: x ≥ 425-327 x ≥ 98 Therefore, the lowest score you need to earn on the next test to achieve an average of at least 85 is 98. Answer 4 Let's denote the score on the next test as x. The average of all the test scores should be at least 85, so we can write the inequality as: (68+78+90+91 + x) / 5 ≥ 85 To solve this inequality, we can start by simplifying the left side of the equation: (327 + x) / 5 ≥ 85 Next, we can multiply both sides of the inequality by 5 to eliminate the fraction: 327 + x ≥ 425 To isolate x, we subtract 327 from both sides of the inequality: x ≥ 425 - 327 Simplifying further, we have: x ≥ 98 Therefore, the lowest score you need to earn on the next test to achieve an average of at least 85 is 98. Question 5 For a test of Ho: p = 0.5, the z test statistic equals -1.74. Find the p-value for Ha: p < 0.5. α. 0.0892 b. 0.0446 c. 0.0409 d. 0.9554 e. 0.0818 Answer 1 Solution:- p value = 0.1286 Since we have a two-tailed test, the P-value is the probability that the z statistic is less than -1.52 or greater than 1.52. We use the z Distribution Calculator to find P(t < -1.52) = 0.0643, and P(t > 1.52) = 0.0643 Thus, the P-value = 0.0643 + 0.0643 = 0.1286 Interpret results. Since the P-value (0.1286) is greater than the significance level (0.05), we cannot reject the null hypothesis. From the test we conclude that we have sufficient evidence to support the claim that p = 0.5. Answer 2 From the given information, Null hypothesis: H₀:p=0.5 Alternative hypothesis: Hₐ:p<0.5 That is, the alternative hypothesis is one-sided hypothesis. Hence, the p-value is obtained for left tailed test. From the given information, the value of test statistic is -1.74. p=P(z