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kotegsom [21]
3 years ago
9

In a class of 25 students, 24 of them took an exam in class and 1 student took a make-up exam the following day. The professor g

raded the first batch of 24 exams and found an average score of 76 points with a standard deviation of 8.5 points. The student who took the make-up the following day scored 66 points on the exam.
a) Does the new student's score increase or decrease the average?
b) The new average is:
(round to two decimal places)
c) Does the new student's score increase or decrease the standard deviation of the scores?
Mathematics
1 answer:
Vlad [161]3 years ago
5 0

Answer:

If the teacher includes the score of the student who took the make-up test

a) The new student´s score decrease the average.

b) The new average is 75.6

c) The new student´s score increase the standard deviation.

Step-by-step explanation:

a) The concept of average implies the ratio between the sum of all scores (observations) and the number of scores (population, N).

At first we have an average of 76, if we add another score two things will happen: if the new score is greater than the average, the average will increase but, if the score is lower than the average, the average will decrease. In this case, as the new score 66<76, the average will decrease.

b)

1) At first we have:

Average = (score 1 + score 2 +....+ score 24)/ 24 (N)

⇒ 76 = (score 1 + score 2 +....+ score 24)/24

⇒ 76 x 24 = (score 1 + score 2 +....+ score 24)

⇒ 1824 = (score 1 + score 2 +....+ score 24)

2) When the student´s score (66) is added we have tha N=25 and

(score 1 + score 2 +....+ score 25) = 1824 + 66

⇒ (score 1 + score 2 +....+ score 25) = 1890.

Now we can calculate the new average

Average = (score 1 + score 2 +....+ score n)/25

⇒ Average = 1890/25 ⇒ Average = 75.6.

The new average is 75.6

c) The concept of standar deviation give us an idea of the range of the scores and is represented as (mean ± standard deviation) = [mean - standard deviation ; mean + standard deviation]. In this case we have (76 ± 8.5) = [ 67.5 ; 84.5]. This implies that all scores are included in this range. So if we add a score that it is not included in the range the standard deviation will increase.

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For a, this is just the median and the range of the data. Median is the middle number after all of them are in order, and the range is the highest number minus the smallest number.

If we arrange them all in order:

2, 4, 5, 5, 5, 6, 7, 8, 9, 9, 10, 10, 11, 12, 14, 15

The middle ones are 8 and 9, because there is an even number of results. That means the median is 8.5.

The range is 15 - 2, which is 13.

For b, 21 is an outlier because it is outside the set of movies watched. If 21 was a value in the table, that means that the order is:

2, 4, 5, 5, 5, 6, 7, 8, 9, 9, 10, 10, 11, 12, 14, 15, 21

Meaning that the middle value is now 9 (because of the odd number of values) and the range is 21 - 2, which is 19. The range would increase because the 21 is greater than all the other values (outlier), so the highest number is now 21. The median would also increase because you added a higher number to the list of numbers, so that the middle would shift one place higher.

Hope I helped!

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