Question

Suppose a MTH instructor is teaching two sections of the course, and administers an exam. The instructor grades the exams, and calculates the mean exam score to be 65 for section 1 and 83 for section 2.

a- Do you (not the instructor) have enough information to calculate the overall mean for all students enrolled on either section? Explain

b- Suppose section 1 has 35 students and section 2 has 25 students. Calculate the overall mean. Is the overall mean closer to 65 or to 83?

c- Give an example of two section sizes n1 and n2 for which the overall mean is more than 81 (show your calculation)

Answer 1

Here for section 1, mean exam score= 65

for section 2 , mean exam score 83

(a) No, we don't have enough information to calculate the overall mean for all students enrolled on either section as we don't know the number of students in each section.

(b) Section 1, number of students = 35

Section 2, Number of students = 25

total students = 35 + 25 = 60

Overall mean = (Number of students in section 1 * Mean exam score of section 1 + Number of students in section 2 * Mean exam score of section 2)/Total students

= (35 * 65 + 25 * 83)/60

= 72.5

Here the overall mean is near to 65 than 83.

(c) Here the overall mean is more than 81.

so here if two section sizes are n1 and n2

so here overall mean > 81

(n₁ * 65 + n₂* 83)/(n₁ + n₂) > 81

65 n₁ + 83n₂ > 81 n₁ + 81n₂

2n₂> 16 n₁

n₂> 8n₁

so for values where n₂ is more than 8 times the value of n₁ overall mean is greater than 81.

so here Let say n₂ = 49 and n₁ = 6

so,

overall mean = (49 * 83 + 6 * 65)/(49 + 6) = 81.04 > 81