SOLUTION: Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.62 and a standard deviation of 0.4 Using the empirica

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Question 1197769: Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.62 and a standard deviation of 0.4 Using the empirical rule, what percentage of the students have grade point averages that are at least 1.82? Please do not round your answer.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

mu = 2.62 = mean
sigma = 0.4 = standard deviation

Let's find the z score when x = 1.82
z = (x - mu)/sigma
z = (1.82 - 2.62)/0.4
z = -2
This score is exactly 2 standard deviations below the mean.

The task of computing P(X > 1.82) is equivalent to finding P(Z > -2).

The empirical rule says roughly 95% of the normal distribution is within 2 standard deviations of the mean.

That means there is roughly 2.5% of the distribution in the left tail since (100%-95%)/2 = 2.5%
The remaining portion is then 100% - 2.5% = 97.5%

Or basically there's 95% in the middle and 2.5% in the right tail, so 95% + 2.5% = 97.5%
and P(Z > -2) = 0.975

Check out this diagram below


Answer: Approximately 97.5%