the marks of 100 students in an examination are normally distributed with mean of 25 marks and a standard deviation of 10 marks. Given that the pass mark is 20, estimate the number of students who passed the examination.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for the Z score. Multiply by 100.
To estimate the number of students who passed the examination, we need to find the proportion of students whose marks are greater than or equal to the pass mark.
Step 1: Standardize the pass mark
To standardize the pass mark, we need to calculate the z-score. The z-score formula is given by:
z = (X - μ) / σ
where X is the value we are standardizing (in this case, the pass mark), μ is the mean of the distribution, and σ is the standard deviation.
Substituting the values into the formula:
z = (20 - 25) / 10
z = -0.5
Step 2: Find the proportion of students passing
To find the proportion of students passing, we need to find the area to the right of the z-score (-0.5) in the standard normal distribution table. This area represents the percentage of students scoring above the pass mark.
Using a standard normal distribution table or a statistical calculator, we find that the area to the right of -0.5 is approximately 0.6915.
Step 3: Calculate the number of students passing
To calculate the number of students passing, we need to multiply the proportion found in Step 2 by the total number of students (100).
Number of students passing = 0.6915 * 100
Number of students passing ≈ 69 (rounded to the nearest whole number)
Therefore, we estimate that approximately 69 students passed the examination.