The Empirical Rule SAT Math scores have a bell-shaped distribu...

Question

The Empirical Rule SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of 114.

Source: College Board

c. What percentage of SAT scores is greater than 743?

Accepted Answer

Welcome back, everyone. Test scores are approximately bell-shaped with a mean of mu equals 600 and the standard deviation of sigma equals 80. Using the 6895 99.7 rule, find the approximate percentage of scores that are greater than 760. A 5%, B 3.5%, C 2.5%. In the 16%. So for this problem, let's draw the sketch for this problem. What we can do is simply introduce a bell-shaped curve. We're going to label the mean value. It is equal to 600, and specifically we are looking at squares that are greater than 760, right? So anything above is the region of interest, we want to identify that percentage. Now, let's go ahead and apply the empirical rule. Now what we want to do is simply identify the number of standard deviations. And specifically that would be the Z score for the value of 760. Let's remember the formula, we take the value X, we subtract the mean, and we divide it by the standard deviation. So in this case, we take 760, we subtract 600, and divide by the standard deviation of 80, and we get 2, right? So that data point is two standard divisions above the mean value. Now, the empirical rule says that 95% of data is contained. Between Two standard divisions from the mean, right? So, essentially what we're going to do is label two standard divisions below the mean, and we know that 760 is two standard divisions above the mean. So what we can do is simply shade that region. And we can say that according to the empirical rule. All of these data points make a total of 95%. So if we're looking for the region in red. And knowing that the bell-shaped curve is symmetric relative to the mean value, we can first of all subtract 100% minus 95% to get 5%. Now, 5% corresponds to the region in red plus the region in pink, right? Because essentially, we can go to the left from 600 and to the right. To get More than 2 standard divisions below the mean, or more than 2 standard divisions above the mean, right? So, The pink region plus the red region make a total of 5%, and because they are symmetric, what we want to do is simply divide 5% by 2 to get 2.5% remaining for. The region given in red. So the answer to this problem corresponds to the answer choice C. Thank you for watching.

The Empirical Rule SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of 114.

Source: College Board

c. What percentage of SAT scores is greater than 743?

Key Concepts

Empirical Rule

Standard Deviation

Z-Score and Percentage Calculation

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