The marks obtained by a number of students in a certain subject are approximately normally distributed
with mean 65 and standard deviation 5. If 3 students are selected at random from this group, what is the probability that at least 1 of them would have scored above 75
http://davidmlane.com/hyperstat/z_table.html
To find the probability that at least one of the three selected students would have scored above 75, we will first find the probability that none of the three students scored above 75, and then subtract that from 1.
Let's denote X as the score of a student, which follows a normal distribution with a mean of 65 and a standard deviation of 5.
The probability that a student scores above 75 can be calculated by finding the area under the normal curve to the right of 75.
To standardize the score and find the corresponding z-score, we can use the formula:
z = (X - μ) / σ
For X = 75, μ = 65, and σ = 5, the z-score is:
z = (75 - 65) / 5 = 2
We can now look up the corresponding area to the right of z = 2 in the standard normal table, which is approximately 0.0228.
Therefore, the probability that a single student scores above 75 is 0.0228.
To find the probability that none of the three selected students score above 75, we multiply the probability for each student:
P(no student scores above 75) = 0.0228 * 0.0228 * 0.0228
Since the students are selected randomly, the probability for each student remains the same.
Now, we can calculate the probability that at least one of the three students scores above 75:
P(at least one student scores above 75) = 1 - P(no student scores above 75)
P(at least one student scores above 75) = 1 - (0.0228 * 0.0228 * 0.0228) ≈ 1 - 0.000012448 ≈ 0.999988
Therefore, the probability that at least one of the three selected students would have scored above 75 is approximately 0.999988 or 99.9988%.
To solve this problem, we need to find the probability that at least one student out of the three scored above 75.
Step 1: Find the probability that a single student scored above 75.
- To do this, we need to standardize the score using the z-score formula: z = (x - mean) / standard deviation.
- We want to find P(X > 75), so the z-score is calculated as: z = (75 - 65) / 5 = 2.
- Using a standard normal distribution table or calculator, we can find the probability associated with z = 2, which is approximately 0.0228.
Step 2: Find the probability that none of the three students scored above 75.
- Since the three students are selected at random, we can assume their scores are independent.
- The probability that a single student scored below or equal to 75 is 1 - 0.0228 = 0.9772.
- Therefore, the probability that none of the three students scored above 75 is (0.9772)^3 = 0.932.
Step 3: Find the probability that at least one student scored above 75.
- The probability that at least one student scored above 75 is the complement of the probability that none of the students scored above 75.
- P(at least one student scored above 75) = 1 - P(none of the students scored above 75) = 1 - 0.932 = 0.068.
Therefore, the probability that at least one student out of the three scored above 75 is approximately 0.068.
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 of the Z score. This is for one student.
With 3 students, the probability would be 3 times as likely. Either-or probabilities are found by adding the individual probabilities.