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.
To find the average score of the class, we need to calculate the average number of right answers for each student and then convert it to a score.
1. To calculate the average score, we start by finding the average number of right answers. The given information tells us that the average number of right answers is 6.2. Since each right answer is worth 5 points, we can find the average score by multiplying the average number of right answers by 5.
Average score = Average number of right answers * 5 = 6.2 * 5 = 31
Therefore, the average score of the class is 31.
To find the standard deviation (SD) of scores of the class, we need to convert the standard deviation of the number of right answers to a standard deviation of scores.
2. Since each right answer is worth 5 points and each wrong answer deducts 1 point, the standard deviation of scores will not be the same as the standard deviation of the number of right answers. To find the SD of scores, we need to consider the effect of both the number of right answers and wrong answers.
The SD of the number of right answers is given as 1.4. Since each right answer is worth 5 points, the variation due to the number of right answers will be multiplied by 5. However, for the variation due to wrong answers, we need to consider that 1 wrong answer deducts 1 point.
To calculate the SD of the scores, we will consider both the contribution from the number of right answers and the contribution from wrong answers.
Let's denote the standard deviation of the number of right answers as σ_right and the number of wrong answers as σ_wrong.
The variation due to the number of right answers will be (5 * σ_right)^2.
The variation due to wrong answers will be (1 * σ_wrong)^2.
Now, we need to square root the sum of these variations to get the final SD of scores:
SD of scores = √[(5 * σ_right)^2 + (1 * σ_wrong)^2]
Given that σ_right = 1.4, we need to find the value of σ_wrong to calculate the SD of scores.
Please provide the value of σ_wrong, or if it is not given, any additional information related to wrong answers.