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Answer:
a) 16% of students have an SAT math score greater than 615.
b) 2.5% of students have an SAT math score greater than 715.
c) 34% of students have an SAT math score between 415 and 515.
d)
e)
Step-by-step explanation:
To solve this question, we have to understand the normal probability distribution and the empirical rule.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Empirical rule
The Empirical Rule states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
(a) What percentage of students have an SAT math score greater than 615?
615 is one standard deviation above the mean.
68% of the measures are within 1 standard deviation of the mean. The other 32% are more than 1 standard deviation from the mean. The normal probability distribution is symmetric. So of those 32%, 16% are more than 1 standard deviation above the mean and 16% more then 1 standard deviation below the mean.
So, 16% of students have an SAT math score greater than 615.
(b) What percentage of students have an SAT math score greater than 715?
715 is two standard deviations above the mean.
95% of the measures are within 2 standard deviations of the mean. The other 5% are more than 2 standard deviations from the mean. The normal probability distribution is symmetric. So of those 5%, 2.5% are more than 2 standard deviations above the mean and 2.5% more then 2 standard deviations below the mean.
So, 2.5% of students have an SAT math score greater than 715.
(c) What percentage of students have an SAT math score between 415 and 515?
415 is one standard deviation below the mean.
515 is the mean
68% of the measures are within 1 standard deviation of the mean. The normal probability distribution is symmetric, which means that of these 68%, 34% are within 1 standard deviation below the mean and the mean, and 34% are within the mean and 1 standard deviation above the mean.
So, 34% of students have an SAT math score between 415 and 515.
(d) What is the z-score for student with an SAT math score of 620?
We have that:
This is Z when X = 620. So
(e) What is the z-score for a student with an SAT math score of 405?
We have that:
This is Z when X = 405. So