A student scores 96 on a test. The class average is 84, with a standard deviation of 4 points. What percentage of the class scored below this student?
I understand the 96-84/4 part to obtain a z-score of 3. What I do not understand is, if I do not have access to a z-score to area table, how can I obtain the area and then obtain the area specific to students under this student's score?
There is no easy way. The actual math formula is simple, but hard to do by hand. I suggest that you just have to know the values for 1,2,3 std from the mean. Just as it is best if you just know the trig values for common angles.
Without more information I'd use a normal curve distribution and notice the earned score is 3 standard deviations from the mean and the student earning a 96% is in the 99.9 percentile.
From the mean on a normal curve the 1st standard deviation is 34.1, then 13.6, then 2.1 percentages with leaves only .1% beyond the 3rd standard deviation.
That's my answer and I'm sticking to it😆
The answe is 99.85%. There is .15% on each side of the normal curve past three standard deviations.
To obtain the area under the normal curve and find the percentage of the class that scored below this student, you can use a standard normal distribution table or a statistical calculator. However, if you don't have access to these resources, there is an alternative approach called the z-score conversion method. By converting the z-score to a percentile rank, you can find the area under the normal curve and determine the percentage of the class that scored below this student.
To convert the z-score to a percentile rank, you can use a cumulative distribution function (CDF). For this case:
1. Start by calculating the z-score: (96 - 84) / 4 = 3.
2. Use a programming language or software (such as Python, R, or a scientific calculator) to find the cumulative distribution function (CDF) for a standard normal distribution with a z-score of 3.
3. The result of the CDF will give you the percentile rank. This indicates the percentage of values that fall below the z-score of 3.
4. Subtract the percentile rank from 1 to obtain the percentage of the class that scored below this student.
Alternatively, if you have a programming background, you can use programming libraries like scipy.stats in Python or the pnorm function in R to directly calculate the area. These libraries provide functions to calculate probabilities under the normal curve based on the z-score.
It is important to note that using a standard normal distribution table or a statistical calculator is more efficient and accurate. However, in the absence of these resources, the z-score conversion method or programming with statistical libraries can be used as an alternative approach.