In a sample of 1000 cases, the mean of a certain test is 14 and standard deviation is 2.5. Assuming the distribution to be normal, find (i) How many students score between 12 and 15? (ii)How many score above 18?(iii) How many score below 18
To solve the questions, we first need to convert the given values to z-scores using the formula:
z = (X - μ) / σ
Where:
X = the value we are interested in
μ = the mean
σ = the standard deviation
(i) For the students scoring between 12 and 15:
z1 = (12 - 14) / 2.5 = -0.8
z2 = (15 - 14) / 2.5 = 0.4
Now, we need to find the area between these two z-scores using a standard normal distribution table or calculator. The area between -0.8 and 0.4 is the same as the area between 0.4 and -0.8, which is approximately 0.4332.
Therefore, approximately 43.32% of students score between 12 and 15.
(ii) For the students scoring above 18:
z = (18 - 14) / 2.5 = 1.6
Now, we need to find the area to the right of the z-score of 1.6 which is the same as finding the area to the left of -1.6 which is approximately 0.0548.
Therefore, approximately 5.48% of students score above 18.
(iii) For the students scoring below 18:
Since we have already calculated the area to the right of the z-score of 1.6 as 0.0548 in part (ii), the area to the left (below) 18 is 1 - 0.0548 = 0.9452.
Therefore, approximately 94.52% of students score below 18.