Final answer:
To find the number of students who may fail an exam, we calculate the standard deviation from the given variance and then use the z-score to estimate the probability of scoring below the passing mark. Multiplying this probability by the total number of students gives us the expected number of students failing, assuming a normal distribution of scores.
Step-by-step explanation:
In a class of 20 students, the average marks (mean) in a mathematics examination is 62, and their variance is 30. To find the number of students that may fail the exam, failing being defined as receiving less than 40 marks, we utilize the properties of the normal distribution, assuming that the exam scores are normally distributed.
The first step is to determine the standard deviation, which is the square root of the variance. Therefore, the standard deviation is √30, which is approximately 5.48. Using the z-score formula, z = (X - μ) / σ, where X is the score of interest (40 marks), μ is the mean (62), and σ is the standard deviation (5.48), we can find the z-score for a student who scores 40. The calculated z-score helps us determine the probability of a student scoring less than 40.
Since exact numbers or a z-table are not provided, we cannot calculate the exact number of students who fail. But theoretically, using a z-table, if we find the probability corresponding to the calculated z-score, we can estimate the percentage of students who might score below 40. Multiplying this percentage by the total number of students (20) gives us the expected number of students who might fail the exam.