Conducting a Wilcoxon Signed Rank Test

  • 1.

    A certain pharmaceutical company guarantees that its newly released pain reliever lasts for at least eight (8) days (on average, referring to the median) before symptoms may recur for pains associated with certain types of injury. The table given below contains the times until recurrence (in days) based on a random sample of {eq}n = 10 {/eq} patients who took this pain relief medication. Which of the following gives the Wilcoxon Signed-Rank Test statistic and conclusion of the test of hypothesis stating that the median recurrence time is at least 8 days?

    Answers:

    • {eq}W = 42 {/eq}, reject the null hypothesis

    • {eq}W=13 {/eq}, do not reject the null hypothesis

    • {eq}W = 13 {/eq}, reject the null hypothesis

    • {eq}W = 42 {/eq}, do not reject the null hypothesis

  • 2.

    According to a high-school teacher, the {eq}n = 10 {/eq} students in her English class post, on average (referring to the median), at least a dozen (i.e., 12) Facebook (FB) posts for the five days that they are in school each week. The table given below contains the number of FB posts her students posted last week. Conduct a Wilcoxon Signed-Rank Test to determine if there is evidence inconsistent with this teacher's claim, then find the test statistic of this test of hypothesis from among the given choices.

    Answers:

    • {eq}W = 5 {/eq}, reject the null hypothesis

    • {eq}W = 50 {/eq}, do not reject the null hypothesis

    • {eq}W = 5 {/eq}, do not reject the null hypothesis

    • {eq}W = 50 {/eq}, reject the null hypothesis

  • 3.

    A company that manufactures hair gels recently came out with a new ultra-hold hair gel variety that it claims has a holding power of at least 15 hours on average (referring to the median). The given table contains the holding times based on a random sample of size {eq}n = 10 {/eq}. Conduct a Wilcoxon Signed-Rank Test to verify if the data obtained provide evidence contrary to the manufacturer's claim (that the median hold time is at least 15 hours). Which of the following gives the test statistic and conclusion of this hypothesis test?

    Answers:

    • {eq}W = 11 {/eq}, do not reject the null hypothesis.

    • {eq}W = 11 {/eq}, reject the null hypothesis.

    • {eq}W = 44 {/eq}, do not reject the null hypothesis.

    • {eq}W = 44 {/eq}, reject the null hypothesis.

  • 4.

    A candy manufacturer produces a certain 30-piece candy bag. Each candy bag is a mix of either red- or blue-colored candies. The manufacturer claims that the numbers of red and blue candies are on average equal in each bag. A random sample of {eq}n = 10 {/eq} such candy bags revealed the following number of red-colored candies in the sampled bags. Conduct a Wilcoxon Signed-Rank Test to determine if the obtained data provide evidence contrary to the manufacturer's claim (i.e., determine if the median number of red candies in a bag is equal to 15). Which of the following gives the value of the test statistic and the corresponding conclusion for this hypothesis test?

    Answers:

    • {eq}W = 23 {/eq}, do not reject the null hypothesis

    • {eq}W = 23 {/eq}, reject the null hypothesis.

    • {eq}W = 35 {/eq}, do not reject the null hypothesis

    • {eq}W = 35 {/eq}, reject the null hypothesis

  • 5.

    The table given below contains the number of hours it took {eq}n = 10 {/eq} randomly chosen students to complete a certain lengthy problem set for a technology-based calculus course. The teacher for this course claims that this problem set should not require, on average (median), more than 15 hours to complete. Conduct a Wilcoxon Signed-Rank Test to determine if there is evidence from the obtained data contrary to the teacher's claim, then find the value of the test statistic from among the given choices.

    Answers:

    • {eq}W = 40 {/eq}, do not reject the null hypothesis

    • {eq}W = 40 {/eq}, reject the null hypothesis.

    • {eq}W = 15 {/eq}, reject the null hypothesis.

    • {eq}W = 15 {/eq}, do not reject the null hypothesis

  • 6.

    The table that accompanies this problem contains the scores of {eq}n = 11 {/eq} students who took a 60-item first quiz of a high-school Algebra class. Conduct a Wilcoxon Signed-Rank Test to determine if the data from the given table is consistent with the claim that the students' median raw score for this test is equal to 30 points, then find the value of the test statistic of this test of hypothesis from among the choices given.

    Answers:

    • {eq}W=54 {/eq}, do not reject the null hypothesis.

    • {eq}W = 9 {/eq}, reject the null hypothesis.

    • {eq}W=54 {/eq}, reject the null hypothesis.

    • {eq}W=9 {/eq}, do not reject the null hypothesis

  • 7.

    The students of a certain statistics course were recently assigned a lengthy problem set. The teacher for this course claims that the total time (on average, referring to the median) required to complete this particular problem set should not exceed two weeks (14 days). The accompanying table shows the actual times (in days) it took a random sample of {eq}n = 11 {/eq} students to complete this particular problem set. Is there evidence from these data of completion times that is contrary to the teacher's claim? Conduct a Wilcoxon Signed-Rank Test to determine this, then find the value of the test statistic and p-value of this test of hypothesis from the choices given below.

    Answers:

    • {eq}W = 54 {/eq}, reject the null hypothesis.

    • {eq}W = 12 {/eq}, reject the null hypothesis.

    • {eq}W = 12 {/eq}, do not reject the null hypothesis.

    • {eq}W = 54 {/eq}, do not reject the null hypothesis.

  • 8.

    The table given for this problem contains a random sample of {eq}n = 11 {/eq} delivery times of a certain home-delivery service. The company running this business claims that deliveries of ordered items, on average (median), are completed within ten {eq}10 {/eq} days from the actual time when the order was placed. Based on the data from the given table is there evidence contrary to this company's claim? Conduct a Wilcoxon Signed-Rank Test to test the hypothesis that the median delivery time is at most ten (10) days, then the value of the test statistic of this test of hypothesis from the following choices. (Note that observations that are equal in value to the claimed median are excluded from the test of hypothesis.)

    Answers:

    • {eq}W = 12 {/eq}, reject the null hypothesis.

    • {eq}W= 12 {/eq}, do not reject the null hypothesis.

    • {eq}W = 16 {/eq}, reject the null hypothesis.

    • {eq}W = 16 {/eq}, do not reject the null hypothesis.

  • 9.

    A teacher wishes to determine if the two versions of a 30-item high school algebra quiz are similar in difficulty. In this regard, the teacher combined the two versions into a 60-item quiz and administered this combined quiz to the students. The exam raw scores of {eq}n = 11 {/eq} randomly chosen students are shown in the given table. Calculate the test statistic for a Wilcoxon Signed-Rank Test that may be used to determine whether the data gathered contains evidence contrary to the claim that the two exam versions are as difficult. (Notes: 1. Tied observation pairs should be disregarded from the comparison.; 2. In the given table, Diff = (Version 2 score ) - (Version 1 score).)

    Answers:

    • {eq}W = 27 {/eq}, reject the null hypothesis.

    • {eq}W = 18 {/eq}, reject the null hypothesis.

    • {eq}W = 27 {/eq}, do not reject the null hypothesis.

    • {eq}W = 18 {/eq}, do not reject the null hypothesis.

  • 10.

    The instructor of a certain beginning high-school chemistry laboratory administered a 30-item inventory as a pre-/pot-tests to determine the extent to which the students are motivated to complete such a laboratory course. The scaled scores of {eq}n = 11 {/eq} students in the pre- and post-tests are shown in the given diagram. Which of the following gives the test statistic of a Wilcoxon Signed-Rank Test that may be conducted to determine if the obtained data has evidence contrary to the hypothesis that the pre- and post-tests scaled scores are equal? (Notes: 1. Tied observation pairs should be disregarded from the comparison.; 2. In the given table, Diff = (Post-test score ) - (Pre-test score).)

    Answers:

    • {eq}W = 0 {/eq}, reject the null hypothesis.

    • {eq}W = 55 {/eq}, do not reject the null hypothesis.

    • {eq}W = 0 {/eq}, do not reject the null hypothesis.

    • {eq}W = 55 {/eq}, reject the null hypothesis.

  • 11.

    A teacher of an arithmetic course wanting to know if the students in her class immediately review graded and returned short quizzes, decided to re-administer a similar short quiz a day after the previous graded short quiz was returned. The scores of her {eq}n = 12 {/eq} students in the first and re-administered short quiz that had a maximum total score of 30 points are shown in the accompanying table below. Conduct a Wilcoxon Signed Rank Test to determine if there is any indication from the data gathered that the scores in the re-administered short quiz are only as high as those in the first quiz. Find the test statistic of this test of hypothesis from among the choices that follow. (Notes: 1. Tied observation pairs should be disregarded from the comparison.; 2. In the given table, Diff = (2nd Quiz score ) - (1st Quiz score).)

    Answers:

    • {eq}W = 5 {/eq}, do not reject the null hypothesis.

    • {eq}W = 5 {/eq}, reject the null hypothesis.

    • {eq}W = 61 {/eq}, reject the null hypothesis.

    • {eq}W = 61 {/eq}, do not reject the null hypothesis.

  • 12.

    Using the data from the table given, conduct a Wilcoxon Signed-Rank Test to determine if there is any indication from the data contrary to the claim that the gain in the raw score from pre- to post-test is at most 4 points. (The scores in the table are the pre- and post-test scores of {eq}n = 12 {/eq} students in a high-school statistics diagnostic test. The maximum total score in this exam is 35 points.) After conducting the test of hypothesis, find the value of the test statistic from among the choices given. (Notes: 1. Tied observation pairs should be disregarded from the comparison.; 2. In the given table, Diff - 4 = (Post-test score ) - (Pre-test score) - 4.)

    Answers:

    • {eq}W = 29 {/eq}, do not reject the null hypothesis.

    • {eq}W = 26 {/eq}, reject the null hypothesis.

    • {eq}W = 26 {/eq}, do not reject the null hypothesis.

    • {eq}W = 29 {/eq}, reject the null hypothesis.

  • 13.

    The dataset shown in the table below are pre- and post-test scores of {eq}n = 12 {/eq} students in a certain college-level statistics quiz that had a maximum total score of 40 points. Conduct a Wilcoxon Signed-Rank Test to test the claim that the gain in score from pre- to post-test is at least 7 points. Which of the following gives the test statistic of this test of hypothesis? (Note: In the given table, Diff - 7 = (Post-test score ) - (Pre-test score) - 7.)

    Answers:

    • {eq}W = 20 {/eq}, do not reject the null hypothesis.

    • {eq}W = 58 {/eq}, do not reject the null hypothesis.

    • {eq}W = 58 {/eq}, reject the null hypothesis.

    • {eq}W = 20 {/eq}, reject the null hypothesis.

  • 14.

    A random sample of {eq}n = 12 {/eq} enrolled students were chosen from among those that took a diagnostic test to determine the extent of previous knowledge for a high school introductory statistics course. The obtained data are shown in the accompanying table below. What is the value of the test statistic of a Wilcoxon Signed-Rank Test conducted to test the claim that the differences in the score: post-test less pre-test, was at least 7 points on average (median)? (Note: In the given table, Diff - 7 = (Post-test score ) - (Pre-test score) - 7.)

    Answers:

    • {eq}W = 12 {/eq}, reject the null hypothesis.

    • {eq}W = 66 {/eq}, reject the null hypothesis.

    • {eq}W = 12 {/eq}, do not reject the null hypothesis.

    • {eq}W = 66 {/eq}, do not reject the null hypothesis.

  • 15.

    Considering that the {eq}n = 12 {/eq} students of his advanced physics course did not fare well during a midterms exam, an instructor decided to administer a make-up exam. The scores of the {eq}n = 12 {/eq} students for the two exam administrations are given in the table below. The first and make-up exams each contains 50 items. Using a Wilcoxon Signed-Rank Test conduct a test to determine if the obtained data support the hypothesis that the performance of the students in the make-up exam is at least their performance in the original midterm exam, then find the value of the test statistic from among the choices given. (Note: In the given table, Diff = (Make-up exam score ) - (1st Exam score).)

    Answers:

    • {eq}W = 75 {/eq}, do not reject the null hypothesis.

    • {eq}W = 3 {/eq}, reject the null hypothesis.

    • {eq}W = 75 {/eq}, reject the null hypothesis.

    • {eq}W = 3 {/eq}, do not reject the null hypothesis.

  • 16.

    The systolic blood pressure (in mm Hg) of a patient during the past {eq}n = 10 {/eq} times that it was taken is shown in the given table. Use a Wilcoxon Signed-Rank Test to determine whether these data provide evidence contrary to the claim that the patient's systolic blood pressure is on average (referring to the median) at most 120 mm Hg. Which of the following gives the test statistic and an estimate of the corresponding p-value of the test of hypothesis that you conducted?

    Answers:

    • {eq}T_+ = 45; \text{p-value} = 0.96 {/eq}

    • {eq}T_- = 10; \text{p-value} = 0.04 {/eq}

    • {eq}T_- = 45; \text{p-value} = 0.04 {/eq}

    • {eq}T_+ = 10; \text{p-value} = 0.96 {/eq}

  • 17.

    The table given below includes the {eq}n = 10 {/eq} most recent diastolic blood pressure (in mm Hg) readings of a certain patient. Conduct a Wilcoxon Signed-Rank Test to determine if these blood pressure readings are contrary to the claim that on average the systolic blood pressure reading of this patient is at most 80 mm Hg. Find the test statistic and the corresponding p-value of the test that you conducted from the choices given.

    Answers:

    • {eq}T_+ = 26; \text{p-value} = 0.54 {/eq}

    • {eq}T_- = 29; \text{p-value} = 0.46 {/eq}

    • {eq}T_+ = 29; \text{p-value} = 0.54 {/eq}

    • {eq}T_- = 26; \text{p-value} = 0.46 {/eq}

  • 18.

    A certain brand of quick-drying cement is said to take less than three (3) hours to dry, on average (referring to the median time). The data that are shown in the accompanying table show {eq}n = 10 {/eq} drying times (in hours). Is there any statistical evidence in these data that is contrary to the counter-claim, i.e., that the drying time is at least 3 hours? To answer this question, conduct a Wilcoxon Signed-Rank Test to verify the claim (i.e., that the median drying time is at most 3 hours), then find the value of the test statistic and corresponding p-value from the choices given.

    Answers:

    • {eq}T_+ = 25; \text{p-value} = 0.62 {/eq}

    • {eq}T_+ = 30; \text{p-value} = 0.62 {/eq}

    • {eq}T_- = 30; \text{p-value} = 0.38 {/eq}

    • {eq}T_- = 25; \text{p-value} = 0.38 {/eq}

  • 19.

    A certain school tries to maintain a balance in the number of male and female students in each section of admitted students each year. The percentage of female students in {eq}n = 10 {/eq} randomly selected sections of this year's incoming freshmen are shown in the given table. Conduct a Wilcoxon Signed-Rank Test to determine if the data provide any evidence contrary to the claim that on average (median) 50% of the students in each section are females, then find the value of the obtained test statistic and the corresponding p-value for this hypothesis test from the given choices.

    Answers:

    • {eq}T_+ = 24; \text{p-value} = 0.77 {/eq}

    • {eq}T_- = 24; \text{p-value} = 0.23 {/eq}

    • {eq}T_+ = 31; \text{p-value} = 0.77 {/eq}

    • {eq}T_- = 31; \text{p-value} = 0.23 {/eq}

  • 20.

    The table that is provided for this question gives the number of hours it took {eq}n = 10 {/eq} randomly selected students to complete a lengthy problem set for a statistics course. Is there evidence from these data contrary to the claim that on average (referring to the median) it took the student only at most a week (7 days) to complete this particular problem set? Conduct a Wilcoxon Signed-Rank Test to answer this, then find the value of the test statistic and corresponding p-value from the following choices.

    Answers:

    • {eq}T_- = 41; \text{p-value} = 0.10 {/eq}

    • {eq}T_- = 14; \text{p-value} = 0.10 {/eq}

    • {eq}T_+ = 14; \text{p-value} = 0.90 {/eq}

    • {eq}T_+ = 41; \text{p-value} = 0.90 {/eq}

  • 21.

    The table below shows the raw scores of {eq}n = 11 {/eq} randomly selected students in a 50-item departmental exam. Conduct a Wilcoxon Signed-Rank Test to determine if these data provide statistical evidence contrary to the claim that the median raw score is at least 25 points. Which of the following gives the value of the test statistic and corresponding p-value of the hypothesis test that was conducted?

    Answers:

    • {eq}T_- = 53; \text{p-value} = 0.03 {/eq}

    • {eq}T_+ = 53; \text{p-value} = 0.97 {/eq}

    • {eq}T_+ = 13; \text{p-value} = 0.97 {/eq}

    • {eq}T_- = 13; \text{p-value} = 0.03 {/eq}

  • 22.

    A teacher claims that the senior high-school students of their school are responsible enough to study at least five (5) hours on average during weekends to supplement the time that they spend studying outside of normal class hours. A survey of {eq}n = 11 {/eq} randomly selected senior-year students gave the responses shown in the table below. Using a Wilcoxon Signed-Rank Test conduct a test of hypothesis to determine if the gathered data support the teacher's claim, then identify the choice below that gives the value of the test statistic and corresponding p-value of this test of hypothesis.

    Answers:

    • {eq}T_+ = 42; \text{p-value} = 0.79 {/eq}

    • {eq}T_- = 24; \text{p-value} = 0.21 {/eq}

    • {eq}T_+ = 24; \text{p-value} = 0.79 {/eq}

    • {eq}T_- = 42; \text{p-value} = 0.21 {/eq}

  • 23.

    A certain fast-food store that offers home delivery service claims that it takes on average not more than 45 minutes for a delivery process to complete (measured from the time the order was called). The table given for this problem contains the {eq}n = 11 {/eq} randomly chosen delivery times from a reasonable amount of recent delivery times. Determine if these data provide evidence contrary to the company's claim by conducting a Wilcoxon Signed-Rank Test. Which of the following gives the value of the test statistic and the corresponding p-value of this hypothesis test? (Note: Exclude observations that are equal in value to the claimed median from the test of hypothesis.)

    Answers:

    • {eq}T_+ = 28; \text{p-value} = 0.07 {/eq}

    • {eq}T_- = 28; \text{p-value} = 0.93 {/eq}

    • {eq}T_+ = 0; \text{p-value} = 0.07 {/eq}

    • {eq}T_- = 0; \text{p-value} = 0.93 {/eq}

  • 24.

    The data in the accompanying table gives the scores of {eq}n = 11 {/eq} in two versions of a 25-item exam. Conduct a Wilcoxon Signed-Rank Test to determine if the data provide statistical evidence contrary to the claim that these two versions are of equal difficulty, then find the value of the test statistic and the corresponding p-value from the given choices. (Notes: 1. Tied observation pairs should be disregarded from the comparison.; 2. In the given table, Diff = (Version 2 score ) - (Version 1 score).)

    Answers:

    • {eq}T_+ = 18; \text{p-value} = 0.35 {/eq}

    • {eq}T_- = 18; \text{p-value} = 0.65 {/eq}

    • {eq}T_- = 27; \text{p-value} = 0.65 {/eq}

    • {eq}T_+ = 27; \text{p-value} = 0.35 {/eq}

  • 25.

    The table below contains the pre- and post-test scaled scores of {eq}n = 11 {/eq} randomly selected students in a mathematics anxiety inventory that was administered to determine the effect of exposure to one year of high school algebra on students' mathematics anxiety (scaled scores are proportional to levels of anxiety). Conduct a Wilcoxon Signed-Rank Test to determine if the data provide any statistical evidence contrary to the hypothesis that the median of the pre- and post-test scores are equal. Which of the following gives the test statistic and p-value of this hypothesis test? (Note: In the given table, Diff = (Post-test score ) - (Pre-test score).)

    Answers:

    • {eq}T_+ = 43; \text{p-value} = 0.13 {/eq}

    • {eq}T_- = 12; \text{p-value} = 0.87 {/eq}

    • {eq}T_- = 43; \text{p-value} = 0.87 {/eq}

    • {eq}T_+ = 12; \text{p-value} = 0.13 {/eq}

  • 26.

    A teacher of a first-year high school algebra course wanted to determine how responsible are his students when it comes to reviewing the mistakes that they committed in the returned and graded short quizzes. In order to do so, he decided to give a short quiz a day after it was returned graded to the students. The results of the first and second test scores of {eq}n = 12 {/eq} randomly selected students are shown in the table provided (the short quiz had a maximum total of 30 points). Is there statistical evidence from the gathered data contrary to the hypothesis that the performance of the students in the re-exam is only as good as that in the first exam? Conduct a Wilcoxon Signed-Rank Test to test this hypothesis, then identify the choice below that contains the test statistic and the corresponding p-value of the test that you conducted? (Notes: 1. Tied observation pairs should be disregarded from the comparison.; 2. In the given table, Diff = (2nd Quiz score ) - (1st Quiz score).)

    Answers:

    • {eq}T_- = 5; \text{p-value} = 0.005 {/eq}

    • {eq}T_- = 61; \text{p-value} = 0.005 {/eq}

    • {eq}T_+ = 5; \text{p-value} = 0.995 {/eq}

    • {eq}T_+ = 61; \text{p-value} = 0.995 {/eq}

  • 27.

    Conduct a Wilcoxon Signed-Rank Test for the data given in the table below to determine if there is any statistical evidence contrary to the claim that the gain in score from pre-test to post-test is only at most four (4) points. (The dataset contains the pre-/post-test score of {eq}n = 12 {/eq} randomly selected students who took this diagnostic test that has a maximum total rating of 40 points.) Which of the following choices lists the test statistic corresponding p-value of the hypothesis test that you conducted? (Notes: 1. Tied observation pairs should be disregarded from the comparison.; 2. In the given table, Diff - 4 = (Post-test score ) - (Pre-test score) - 4.)

    Answers:

    • {eq}T_+ = 56; \text{p-value} = 0.98 {/eq}

    • {eq}T_- = 56; \text{p-value} = 0.02 {/eq}

    • {eq}T_- = 10; \text{p-value} = 0.02 {/eq}

    • {eq}T_+ = 10; \text{p-value} = 0.98 {/eq}

  • 28.

    A school offering a course for improving typewriting skills claims to be able to increase typing speed by at least 10 words per minute on average (referring to the median) among its 12- to 16-year old students. A random sample {eq}n = 12 {/eq} students gave the following dataset on rates (words per minute) before and after taking this course. Find from among the choices given the test statistic and p-value of the Wilcoxon Signed-Rank Test that may be conducted to determine if the data provides evidence to the school's claim? (Note: In the given table, Diff - 10 = (Post-training rate) - (Pre-training rate) - 10.)

    Answers:

    • {eq}T_- = 12; \text{p-value} = 0.01 {/eq}

    • {eq}T_+ = 66; \text{p-value} = 0.99 {/eq}

    • {eq}T_+ = 12; \text{p-value} = 0.99 {/eq}

    • {eq}T_- = 66; \text{p-value} = 0.01 {/eq}

  • 29.

    To determine whether her students have ample pre-calculus preparation for an AP Calculus course, a teacher gave a 40-item diagnostic test at the start of the semester. She then administered this same diagnostic as a post-test at the end of the semester. The start-of-semester (pre-test) and end-of-semester (post-test) scores of a random sample of {eq}n = 12 {/eq} enrolled students are shown in the given table. By conducting a Wilcoxon Signed-Rank Test, determine if the obtained data show any evidence contrary to the claim that the gain in points from pre-test to post-test is at least three (3) points. Which of the following choices given the value of the test statistic and corresponding p-value of this test? (Notes: 1. Tied observation pairs should be disregarded from the comparison.; 2. In the given table, Diff - 3 = (Post-test score ) - (Pre-test score) - 3.)

    Answers:

    • {eq}T_+ = 14; \text{p-value} = 0.92 {/eq}

    • {eq}T_- = 14; \text{p-value} = 0.08 {/eq}

    • {eq}T_- = 41; \text{p-value} = 0.08 {/eq}

    • {eq}T_+ = 41; \text{p-value} = 0.92 {/eq}

  • 30.

    Make-up exams are usually given to help improve the scores from a previously given exam in which the students didn't perform well. The table given below contains the initial raw scores of {eq}n = 12 {/eq} randomly chosen students in a 50-item exam that they took and the scores in a make-up exam. Use a Wilcoxon Signed-Rank Test to determine if the performance of the students in the make-up exam is at least as high as those in the first exam. Which of the choices below gives the test statistic and p-value of this test? (Note: In the given table, Diff = (Make-up exam score ) - (1st Exam score).)

    Answers:

    • {eq}T_- = 3; \text{p-value} = 0.001 {/eq}

    • {eq}T_+ = 75; \text{p-value} = 0.999 {/eq}

    • {eq}T_- = 75; \text{p-value} = 0.001 {/eq}

    • {eq}T_+ = 3; \text{p-value} = 0.999 {/eq}

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