Suppose a college's SAT scores are normally distributed with mean 1000 and standard deviation 1 for students above 1200. What percentage of students is eligible for scholarships
It would help if you proofread your questions before you posted them.
SD typo? Does 1200 make them eligible?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. Multiply by 100.
To find the percentage of students eligible for scholarships, we need to determine the z-score for the SAT score of 1200 and calculate the area under the normal curve to the right of that z-score.
First, we need to calculate the z-score using the formula:
z = (x - μ) / σ
where:
x = SAT score
μ = mean
σ = standard deviation
In this case, the SAT score is 1200, the mean is 1000, and the standard deviation is 1000. Plugging these values into the formula:
z = (1200 - 1000) / 1000
z = 0.2
Next, we need to find the area to the right of the z-score of 0.2. We can look up this value in the standard normal distribution table or use a calculator. Using a calculator, we find that the area to the right of 0.2 is approximately 0.4207.
Since the percentage is the area under the curve multiplied by 100, the percentage of students above 1200 SAT score who are eligible for scholarships is:
0.4207 * 100 = 42.07%
Therefore, approximately 42.07% of students are eligible for scholarships.
To find the percentage of students who are eligible for scholarships, we need to determine the percentage of students who score above a certain threshold. In this case, the threshold is 1200 SAT score.
Since the SAT scores are normally distributed with mean 1000 and standard deviation 1 for students above 1200, we can use a standard normal distribution table, or a calculator with a normal distribution function, to determine the proportion of students beyond the threshold.
Here are the steps to calculate the percentage of students eligible for scholarships:
1. Calculate the z-score: To convert the SAT score of 1200 into a z-score, subtract the mean (1000) from the threshold score (1200) and divide it by the standard deviation (1):
z = (1200 - 1000) / 1 = 200 / 1 = 200.
2. Look up the z-score on a standard normal distribution table: Find the corresponding probability or percentage associated with the z-score you calculated. The z-score of 200 will have an extremely small probability attached to it.
3. Interpret the result: The percentage you obtained from the table or calculator represents the proportion of students who score above the threshold of 1200. Subtract this proportion from 1 (100%) to find the percentage of students eligible for scholarships.
It is important to note that the given standard deviation of 1 for students above 1200 is likely a typo or an incorrect statement, as a standard deviation of 1 implies very little variability. In reality, the standard deviation of SAT scores is much larger.