In the quiz in Exercises 1 and 2, the grading scheme is as follows: each right answer is awarded 5 points, and 1 point is taken off for each wrong answer. Recall that the quiz consists of 10 questions, and that in the class, the average number of right answers is 6.2 and the SD of the number of right answers is 1.4. In what follows, it will help to express a student’s score in terms of the number of right answers the student got.
(1) Find the average score of the class.
(2) Find the SD of scores of the class.
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Here is an idea, not the answer:
S[i] = 5*Xr[i] - 1*Xw[i], where S[i] is Score, Xr[i] - right answer and Xw[i] - wrong answer for the i-th student
The key here is to remember that there are exactly 10 questions each student answers, so Xw[i] = 10 - Xr[i]
Which gives us the following affine transformation:
S[i] = 6*Xw[i] - 10
To find the average score of the class, we need to determine the score (in terms of the number of right answers) for each student and then calculate the average of all the scores.
The score for each student can be calculated by multiplying the number of right answers by 5 and subtracting the number of wrong answers. Since there are a total of 10 questions, the number of wrong answers can be calculated by subtracting the number of right answers from 10.
Now let's calculate the average score of the class using the given information:
(1) Average Score of the Class:
To calculate the average score, we need to find the average score for each student and then take the average of all those individual scores.
The average number of right answers is given as 6.2, and we know that the total number of questions is 10. So, the average number of wrong answers can be calculated as 10 - 6.2 = 3.8.
Now, let's calculate the average score for each student:
Each right answer is awarded 5 points, and 1 point is taken off for each wrong answer. So, the score for each student can be calculated as:
Score = (Number of Right Answers * 5) - (Number of Wrong Answers)
= (6.2 * 5) - (3.8)
= 31 - 3.8
= 27.2
Therefore, the average score of the class is 27.2.
(2) Standard Deviation (SD) of Scores of the Class:
To calculate the SD of scores, we need to determine the individual scores for all the students and then calculate the SD of those scores.
As discussed above, the score for each student can be calculated as:
Score = (Number of Right Answers * 5) - (Number of Wrong Answers)
Given that the SD of the number of right answers is 1.4, we need to calculate the SD of the scores based on the number of right answers.
To do this, we'll calculate the highest and lowest possible scores based on the range of the number of right answers and then calculate the SD of those scores.
The highest possible number of right answers is 6.2 + 1.4 = 7.6 (as the SD is added to the average).
The lowest possible number of right answers is 6.2 - 1.4 = 4.8 (as the SD is subtracted from the average).
Now, let's calculate the highest and lowest possible scores:
For the highest possible score:
Score = (Number of Right Answers * 5) - (Number of Wrong Answers)
= (7.6 * 5) - (10 - 7.6)
= 38 - 2.4
= 35.6
For the lowest possible score:
Score = (Number of Right Answers * 5) - (Number of Wrong Answers)
= (4.8 * 5) - (10 - 4.8)
= 24 - 5.2
= 18.8
Now, let's calculate the SD using the highest and lowest possible scores [35.6, 27.2, 18.8]:
SD = √[ ( (x1 - x̄)^2 + (x2 - x̄)^2 + ... + (xn - x̄)^2 ) / n ]
= √[ ( (35.6 - 27.2)^2 + (27.2 - 27.2)^2 + (18.8 - 27.2)^2 ) / 3 ]
= √[ ( (8.4)^2 + (0)^2 + (-8.4)^2 ) / 3 ]
= √[ ( 70.56 + 0 + 70.56 ) / 3 ]
= √[ 141.12 / 3 ]
= √47.04
≈ 6.86
Therefore, the SD of the scores of the class is approximately 6.86.