The distribution of grades of 50 students in a certain subject is normally distributed with a mean of 86 and a standard deviation of 12. 6. 7. If the passing grade is 75, how many failed in the subject?
standard deviation of 12. 6. 7 <-----????
anyway
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http://davidmlane.com/hyperstat/z_table.html
To determine how many students failed the subject, we need to find the number of students whose grades are below the passing grade.
Step 1: Convert the problem into a standardized form.
To do this, we need to convert the mean and standard deviation into a standard normal distribution, which has a mean of 0 and a standard deviation of 1.
To standardize the distribution, we use the formula:
z = (X - μ) / σ
Where:
z = standard score
X = raw score
μ = population mean
σ = population standard deviation
In this case, we have:
μ = 86
σ = 12
Step 2: Calculate the z-score for the passing grade.
Given that the passing grade is 75, we can calculate the z-score using the formula from step 1:
z = (X - μ) / σ
z = (75 - 86) / 12
z = -11 / 12
z ≈ -0.917
Step 3: Determine the proportion of students below the passing grade.
Since the distribution is assumed to be normally distributed, we can use a standard normal distribution table or a statistical calculator to find the proportion of students below the z-score of -0.917.
Using a standard normal distribution table or calculator, we find that the proportion of students below -0.917 is approximately 0.179.
Step 4: Calculate the number of students who failed.
To find the number of students who failed, we multiply the proportion from step 3 by the total number of students:
Number of students who failed = Proportion × Total number of students
Number of students who failed = 0.179 × 50
Number of students who failed ≈ 8.95
Since we can't have a fraction of a student, we round down to the nearest whole number.
Therefore, approximately 9 students failed the subject.