# Conducting a Sign Test

• 1.

A certain pharmaceutical company claims that on average (referring to the median) its newly released pain reliever is effective for at least eight (8) days before symptoms may recur for pains resulting from a certain type of injury. The times (in days) until symptoms recurred in a random sample of {eq}n = 10 {/eq} patients who took this pain reliever after experiencing some pain due to such an injury are given in the accompanying table. Conduct a Sign Test to determine if there is sufficient evidence against the claim of this pharmaceutical company, then find the p-value of this test from the choices given.

• 0.62

• 0.38

• 0.83

• 0.17

• 2.

A high-school English teacher claims that the {eq}n = 10 {/eq} students of her class post, on average (median), at least a dozen (12) times on Facebook (FB) each week. The number of FB posts that the students of her class posted the past week are shown in the given table. Conduct a Sign Test to determine if there is evidence contrary to the teacher's claim, then find the p-value of this test from among the given choices.

• 0.989

• 0.0107

• 0.999

• 0.001

• 3.

A manufacturer of an ultra-hold hair gel claims that on average (median) its gel should have a holding power of at least 15 hours. A random sample of {eq}n = 10 {/eq} effective holding times are given in the accompanying table. Verify the manufacturer's claim by conducting a Sign Test to determine if there is evidence to the contrary, then compute for the p-value of this test.

• 0.95

• 0.05

• 0.83

• 0.17

• 4.

A candy manufacturer produces candy bags that each contain 30 pieces of candies. The candies in the bag are of only two types: red- or blue-colored. The manufacturer claims that on average the number of red and blue candies in each bag are equal. The accompanying table gives the number of red-colored candies in a random sample of {eq}n = 10 {/eq} such candy bags. What is the p-value of the Sign Test conducted to determine if there is evidence contrary to the manufacturer's claim (that the median number of red candies in a bag is 15)?

• 0.75

• 0.62

• 0.25

• 0.38

• 5.

A calculus teacher claims that a certain lengthy problem set will require, on average (median), no more than 15 hours to complete if the students use a computer algebra software. The required completing times of {eq}n = 10 {/eq} students based on a survey are shown in the given table. Conduct a Sign Test to determine whether there is evidence that the students' performance was inconsistent with their teacher's claim and find the p-value of this test.

• 0.95

• 0.83

• 0.17

• 0.05

• 6.

The scores of {eq}n = 11 {/eq} students in a 60-item first quiz of a high-school Algebra class are shown in the accompanying table. Conduct a Sign Test to determine whether there is evidence contrary to the claim that the median raw score of the students is equal to 30 points, then find the p-value of the test from among the choices given.

• 0.23

• 0.89

• 0.11

• 0.77

• 7.

The teacher of a high-school statistics course claims that a lengthy problem set will require not more than two weeks (14 days) to accomplish. The times (in days) it took a random sample of {eq}n = 11 {/eq} students of this course to complete the problem are given in the accompanying table. Conduct a Sign Test to determine whether there is evidence that the students' performance is contrary to the teacher's claim and find the p-value of this test.

• 0.89

• 0.97

• 0.03

• 0.11

• 8.

A courier service claims that it delivers ordered items, on average (median), within ten {eq}10 {/eq} days from the moment the order was placed. A random sample of {eq}n = 11 {/eq} delivery times of this courier service are shown in the accompanying table. Conduct a Sign Test to determine if there is evidence contrary to this courier service's claim and determine the p-value of the test. (Note that observations that are equal in value to the claimed median are excluded from the random sample.)

• 0.23

• 0.77

• 0.05

• 0.50

• 9.

A teacher administered a 60-item quiz to {eq}n = 11 {/eq} students of a high-school algebra course. The 60-item quiz is in fact two 30-item quizzes that were especially combined and the teacher wanted to determine if the two versions of the exam were on average of the same difficulty. The raw scores of the students in each of the two 30-item versions of the exam are shown in the accompanying table. Calculate the p-value for a Sign Test that may be used to determine whether there is evidence to the claim that the two exam versions are of the same difficulty. (Note that tied observation pairs should be disregarded from the comparison.)

• 0.98

• 0.82

• 0.18

• 0.02

• 10.

The pre- and post-tests raw scores in a 30-item diagnostic test for a certain Introductory Chemistry laboratory course with {eq}n = 11 {/eq} students are shown in the accompanying table. Conduct a Sign Test to determine whether the performance of the students was better in the post-test than in the pre-test (by determining if there is evidence contrary to the hypothesis that a student's post-test score is only at most the student's pre-test score), then choose the p-value of this test from among the choices given.

• 0.9990

• 0.9995

• 0.0005

• 0.0010

• 11.

A teacher of an arithmetic course wanted to determine whether her {eq}n = 12 {/eq} students immediately look into their graded and returned past exams by looking into the exam results as soon as they are returned. In connection with this, the teacher decided to re-administer a 30-item diagnostic test that had been duly graded then returned to the students the previous day. The results for the two administrations are given in the accompanying table. Compute for the p-value of the Sign Test that may be used to determine whether the performance of the students in the second administration was higher than in the first administration of the exam, i.e., if there is evidence contrary to the hypothesis that a student's second test score is only at most the corresponding first test score. (Note that tied observation pairs should be excluded from the comparison.)

• 0.03

• 0.0005

• 0.006

• 0.01

• 12.

The teacher of a statistics course administered a 35-item diagnostic test twice, as a pre-test and post-test, to her class of {eq}n = 12 {/eq} students. The raw scores of the students in the pre- and post-tests are shown in the given table. Conduct a Sign Test to determine whether there is evidence contrary to the claim that the gain in the raw score from pre- to post-test is at most 4 points, then find the p-value of this Sign Test from among the choices given. (Note that tied observation pairs should be excluded from the comparison.)

• 0.38

• 0.65

• 0.35

• 0.62

• 13.

Conduct a Sign Test to determine whether there is evidence contrary to the claim that the gain from pre- to post-test is at least 7 points for the dataset shown in the given figure. The dataset is based on a 40-item diagnostic test that was taken by {eq}n = 12 {/eq} students of a college-level statistics course. Which of the following gives the p-value for this test?

• 0.05

• 0.17

• 0.95

• 0.83

• 14.

A certain 30-item diagnostic test of a high school introductory statistics course with {eq}n = 12 {/eq} enrolled students gave pre- and post-tests raw scores that are shown in the given figure. What is the p-value for a Sign Test to determine if there is evidence contrary to the claim that the post-test scores were higher by at least 7 points than the pre-test scores?

• 0.02

• 0.98

• 0.07

• 0.93

• 15.

An instructor of an advanced physics course decided to give a make-up exam for a 50-item long quiz. The scores of the {eq}n = 12 {/eq} students in the first and second (make-up) administrations are given in the accompanying figure. Conduct a Sign Test to determine whether the performance of the students improved (i.e., if there is evidence contrary to the hypothesis that the score of a student in the make-up exam is only at most the corresponding score in the first exam) and determine the p-value of the test.

• 0.98

• 0.02

• 0.003

• 0.997

• 16.

To determine whether a given coin is biased, a student decided to toss the coin 20 times and then conduct a Sign Test to test the hypothesis stating that on average 50% of the tosses will come up heads. The results of the tosses are given below. Conduct a Sign Test to test this hypothesis, then find the p-value of this test from among the given choices.

Dataset:

{eq}\text{Result of the 20 coin tosses (1 = H, 0 = T)} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline 1& 0& 1& 0& 0 & 1& 0& 1& 1& 0 \\ \hline 0& 1& 1& 1& 1 & 0& 0& 1& 1& 1 \\ \hline \end{array} {/eq}

• 0.55

• 0.50

• 0.25

• 0.75

• 17.

To test a pseudo-random number generator (PRNG) that claims to randomly generate real numbers between 0 and 1, inclusive, a student generated 20 random numbers using the PRNG. (The 20 numbers generated are shown in the table below.) The student then conducted a Sign Test to determine if there is evidence contrary to the previously stated claim by testing whether on average 50% of the numbers generated is at most 0.50. Which of the following gives the p-value of this Sign Test?

Dataset:

{eq}\text{Twenty (20) numbers generated by the PRNG} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 0.285& 0.091& 0.001& 0.606& 0.405& 0.821& 0.942& 0.074& 0.210& 0.684 \\ \hline 0.817& 0.206& 0.880& 0.577& 0.808& 0.394& 0.412& 0.275& 0.204& 0.146 \\ \hline \end{array} {/eq}

• 0.50

• 0.13

• 0.87

• 1.00

• 18.

A statistics teacher usually administers a special 10-item diagnostic test at the beginning of each semester. The raw scores on this diagnostic test of a randomly selected set of 20 students that have taken this diagnostic test are shown in the table below. Conduct a Sign Test on this dataset to test whether the median score of the students in this diagnostic test is six (6) points, then identify which of the choices below gives the p-value of this Sign Test. (Note: Observations that are equal to the claimed median score should be discarded.)

Dataset:

{eq}\text{Raw scores of 20 students} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 2& 6& 3& 3& 8& 4& 3& 10& 0& 2 \\ \hline 5& 3& 2& 2& 8& 6& 4& 8& 0& 4 \\ \hline \end{array} {/eq}

• 1.00

• 0.03

• 0.00

• 0.97

• 19.

The table below shows the number of yellow candies per bag in a random sample of {eq}n = 10 {/eq} candy bags from a manufacturing company that claims that on average half of the candies in each 30-piece candy bag are yellow while the other half is blue. Find the p-value of the SIgn Test that is conducted to test the manufacturer's claim, i.e., the claim that on average the median number of yellow candies in a 30-piece bag is equal to 15.

Dataset:

{eq}\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Bag ID} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text{No. of yellow candies} & 10& 14& 11& 19& 13& 16& 16& 17& 11& 16 \\ \hline \end{array} {/eq}

• 1.00

• 0.62

• 0.50

• 0.38

• 20.

The table below gives the number of hours that it took {eq}n = 10 {/eq} randomly selected students of a certain statistics course to finish one of their assigned problem sets. The teacher claims that on average it will not take a student more than 12 hours (the median time) to complete this problem set. Using the data provided, conduct a Sign Test to test the teacher's claim, then find the p-value of this Sign Test from among the choices given.

(Note: Observations that are equal to the claimed median should be discarded.)

Dataset:

{eq}\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Student ID} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text{Completion time (in hours)} & 2& 10& 12& 12& 14& 14& 18& 18& 18& 33 \\ \hline \end{array} {/eq}

• 0.14

• 0.04

• 0.96

• 0.86

• 21.

One can decide on the fairness of an ordinary coin by tossing it a number of times and then determining if on average 50% of the tosses come up heads. A student tossed a certain coin 20 times; the results of these tosses are given below. Conduct a Sign Test to test the hypothesis that on average the number of heads is half of the tosses. Which of the following gives the p-value of this test of hypothesis?

Dataset:

{eq}\text{Result of the 20 coin tosses (1 = H, 0 = T)} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 1& 0& 0& 0& 0& 1& 1& 1& 1& 0 \\ \hline 0& 1& 0& 1& 0& 0& 0& 0& 0& 0 \\ \hline \end{array} {/eq}

• 0.94

• 0.74

• 0.26

• 0.06

• 22.

A certain pseudo-random number generator (PRNG) claims to randomly generate real numbers between 0 and 1, inclusive. To put this claim to a test, a student used this PRNG to generate 20 random numbers (the 20 numbers generated are shown in the table below), then conducted a Sign Test to test the hypothesis that on average 50% of the numbers generated by this PRNG are at most 0.50. Find the p-value of this test of hypothesis from among the given choices.

dataset:

{eq}\text{Twenty (20) numbers from the PRNG} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 0.590& 0.811& 0.218& 0.742& 0.536& 0.984& 0.122& 0.741& 0.805& 0.889 \\ \hline 0.705& 0.412& 0.498& 0.593& 0.695& 0.675& 0.892& 0.637& 0.132& 0.359 \\ \hline \end{array} {/eq}

• 0.94

• 0.12

• 0.06

• 0.88

• 23.

A statistics teacher claims that the median score in one of her favorite 10-item short quiz is at least six (6) points. The raw scores of a randomly selected set of 20 students who previously took this short quiz are shown in the given table. Conduct a Sign Test to verify the teacher's claim, then find the p-value of this test of hypothesis from among the choices given. (Note: Observations that are equal to the claimed median score should be discarded.)

Dataset:

{eq}\text{Raw scores of 20 students} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 7& 9& 8& 7& 4& 4& 10& 1& 5& 7 \\ \hline 7& 2& 10& 5& 2& 6& 4& 2& 5& 9 \\ \hline \end{array} {/eq}

• 0.68

• 1.00

• 0.50

• 0.32

• 24.

Blue & Red candy bags come in packages of 30 candies. The manufacturer claims that on average half of the candies are blue and half are red in such 30-piece candy bags. The number of blue candies in a random sample of {eq}n = 10 {/eq} such candy bags are shown in the table below. Conduct a Sign Test to verify this manufacturer's claim (i.e., that the median number of blue candies in each bag is 15), then find the p-value of this hypothesis from among the choices given. (Note: Observations that are equal to the claimed median should be discarded.)

Dataset:

{eq}\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Bag ID} & 1 & 2 & 3 &4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text{Number of blue candies} &19& 11& 15& 17& 16& 16& 14& 11& 15& 18 \\ \hline \end{array} {/eq}

• 0.36

• 0.73

• 0.27

• 0.64

• 25.

The table below gives the pre- and post-test scores in a certain 15-item diagnostic test of {eq}n = 11 {/eq} randomly selected students that are enrolled in an AP statistics course. Conduct a Sign Test to verify if the post-test score is at least 5 points higher than the post-test score. Which of the following gives the p-value for this test of hypothesis? (Notes: 1. Tied observations should be discarded.; 2. Diff - 5 = (Post-test score) - (Pre-test score) - 5.)

• 0.0327

• 0.9453

• 0.9673

• 0.0547

• 26.

A teacher of an arithmetic course administered two versions of a 25-item diagnostic test, one after the other, to verify if the two versions have the same difficulty. The table below gives the scores of a randomly selected set of {eq}n = 12 {/eq} students who took both versions of the test. Conduct a Sign Test to determine if the scores of a student in both versions are equal on average, then find the p-value for this test of hypothesis from among the given choices. (Note: Diff = (Second version score) - (First version score).)

• 0.39

• 0.61

• 0.77

• 0.23

• 27.

Third-grade (8-9 years old) students are expected to be able to read about 107 to 162 words per minute (wpm). The table below contains the reading rates (in wpm) of {eq}n = 12 {/eq} randomly selected 3rd Grade students before (pre) and after (post) the first quarter (Q1) in a certain school. Using this dataset, conduct a Sign Test to determine if the reading rate of these 3rd Grade students has improved by at least 7 wpm during the first quarter in this school. Which of the following gives the p-value of this test of hypothesis? (Notes: 1. Tied observations should be discarded.; 2. Diff - 7 = (Post-Q1 reading rate) - (Pre-Q1 reading rate) - 7.)

• 0.39

• 0.38

• 0.61

• 0.62

• 28.

The table below gives the reading rates (in wpm or words per minute) of {eq}n = 12 {/eq} randomly selected 1st Grade (6-7 years old) students from a certain primary school before (pre) and after (post) four quarters (Q1 to Q4) in Grade 1. Conduct a Sign Test to determine whether there is evidence contrary to the claim that the reading rates of these 1st Grade students have increased by at least 50 wpm during a span of four quarters. The p-value for this test of hypothesis is equal to _____.

(Notes: 1. Tied observations should be discarded.; 2. Diff - 50 = (Post-Q4 reading rate) - (Pre-Q1 reading rate) - 50.)

• 0.89

• 0.97

• 0.03

• 0.11

• 29.

The pre- and post-test scores of {eq}n = 12 {/eq} randomly selected enrolled students in a certain 30-item diagnostic test are shown in the table below. Determine the p-value for a Sign Test conducted to verify the claim that the gain in score from pre- to post-test is at least 10 points. (Notes: 1. Tied observations should be discarded.; 2. Diff - 10 = (Post-test score) - (Pre-test score) - 10.)

• 0.50

• 0.27

• 0.00

• 0.73

• 30.

After the {eq}n = 12 {/eq} students enrolled in his advanced statistics course scored poorly in a 45-item long exam, an instructor decided to administer a make-up exam. The table below shows the scores of the students in the 1st and make-up exams. The instructor claims that the gain in points between the make-up exam and 1st exam is at least 10 points. Conduct a Sign Test to verify this claim, then identify the p-value of this test of hypothesis from among the choices given. (Note: DIff - 10 = (Make-up score) - (1st Exam score) - 10.)