Large group of students took a test in English and the final grades have a mean of 75 and a standard deviation of 8. If we can approximate the distribution of these grades by a normal distribution. What percentage of the students
(ii) Should pass the test (grades >= 60)
(iii) Should fail the test(grades <60?
To find the percentage of students who should pass the test (grades >= 60) and fail the test (grades < 60), we need to use the standard normal distribution.
First, let's find the z-scores for the passing grade (60) and the mean (75) using the formula:
z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation.
For passing grade (60):
z1 = (60 - 75) / 8 = -1.875
To find the area under the normal curve to the left of z1, we can look up the corresponding value in the standard normal distribution table. In this case, the table will provide the percentage of students who scored below the passing grade.
The value of -1.875 is between -1.9 and -1.8 in the table. By interpolating between these two values, we can estimate the area to be approximately 0.0301.
To convert this to a percentage, we multiply by 100:
0.0301 * 100 = 3.01%
Therefore, approximately 3.01% of the students should fail the test (grades < 60).
To find the percentage of students who should pass the test (grades >= 60), we subtract the percentage of students who should fail from 100%:
100% - 3.01% = 96.99%
Therefore, approximately 96.99% of the students should pass the test (grades >= 60).
To find the percentage of students who should pass the test, we need to calculate the area under the normal distribution curve that corresponds to grades greater than or equal to 60.
Step 1: Standardize the cutoff grade (60) using the standard deviation and mean of the distribution.
Z-score = (X - mean) / standard deviation
Z-score = (60 - 75) / 8
Z-score = -1.875
Step 2: Use the Z-table or a statistical software to find the area to the left of the Z-score (-1.875).
From the Z-table, the area to the left of -1.875 is approximately 0.0304.
Step 3: Calculate the percentage by subtracting the area to the left from 1.
Percentage passed = 1 - 0.0304 = 0.9696 or 96.96%
Therefore, approximately 96.96% of the students should pass the test (grades >= 60).
For the percentage of students who should fail the test (grades <60), we can find it by subtracting the percentage of students who passed from 100%.
Percentage failed = 100% - Percentage passed
Percentage failed = 100% - 96.96%
Percentage failed = 3.04%
Therefore, approximately 3.04% of the students should fail the test (grades <60).