Marks in an examination are distributed normally with a mean of 55 and standard deviation of 15. 200 students sit the exam. (a) If the passing mark is 40, what is the number of students expected to fail? (b) If you score 75 marks on the exam what do you expect your ranking to be among all 200 students? (c) If you want to score among the top 10 performers in this exam, what is the minimum mark you should aim for
A. Use same process, but multiply proportion by 200.
B. Same process, but convert to rank.
C. top ten = 10/200 = 1/20 = .05 = proportion
55
To answer these questions, we need to use the concept of the normal distribution. The normal distribution is a mathematical model that describes the distribution of data points around a mean with a certain standard deviation. In this case, the marks in the examination are distributed normally with a mean of 55 and a standard deviation of 15.
(a) To find the number of students expected to fail, we need to calculate the probability of scoring less than the passing mark of 40. We can do this by finding the area under the normal distribution curve to the left of 40. Here's how you can calculate it:
1. Find the z-score: The z-score represents the number of standard deviations a data point is from the mean. In this case, we can calculate the z-score for the passing mark of 40 using the formula:
z = (x - μ) / σ
where x is the passing mark (40), μ is the mean (55), and σ is the standard deviation (15).
Substituting the values, we get:
z = (40 - 55) / 15
z = -1
2. Find the probability: Once we have the z-score, we can use a standard normal distribution table or a calculator to find the probability associated with that z-score. The probability (P) of scoring less than 40 can be found using the table or calculator.
By looking up the z-score of -1 in the standard normal distribution table or using a calculator, you will find that P(z < -1) is approximately 0.1587.
3. Calculate the number of students expected to fail: To find the number of students expected to fail, we multiply the probability by the total number of students (200):
Number of students expected to fail = P(z < -1) * Total number of students
Number of students expected to fail = 0.1587 * 200
Number of students expected to fail ≈ 31.74
Therefore, approximately 31.74 students are expected to fail the exam.
(b) To estimate your ranking among all 200 students based on a score of 75, we need to find the probability of scoring less than 75. Follow these steps:
1. Calculate the z-score: Using the formula as mentioned above, we find the z-score for a score of 75:
z = (75 - 55) / 15
z = 20 / 15
z = 1.33
2. Find the probability: Look up the z-score of 1.33 in the standard normal distribution table or use a calculator to find P(z < 1.33).
By referring to the table or using a calculator, you will find that P(z < 1.33) is approximately 0.908.
3. Calculate your ranking: To estimate your ranking among the 200 students, we multiply the probability by the total number of students and then subtract it from the total number of students:
Ranking ≈ (1 - P(z < 1.33)) * Total number of students
Ranking ≈ (1 - 0.908) * 200
Ranking ≈ 0.092 * 200
Ranking ≈ 18.4
Therefore, you can expect your ranking to be approximately 18.4 among the 200 students.
(c) To score among the top 10 performers in the exam, we need to find the minimum mark required. Here's how you can do it:
1. Calculate the z-score corresponding to the top 10 percentile: The top 10 percentile corresponds to a z-score that leaves 90% of the distribution to the left. We can find this z-score using the standard normal distribution table or a calculator.
By referring to the table or using a calculator, you will find that the z-score for the top 10 percentile is approximately 1.28.
2. Calculate the minimum mark: Using the formula mentioned earlier, we can calculate the minimum mark required for a top 10 performance:
Minimum mark = (z * σ) + μ
Minimum mark = (1.28 * 15) + 55
Minimum mark ≈ 73.2
Therefore, you should aim for a minimum mark of approximately 73.2 to score among the top 10 performers in the exam.