Question

Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.56 and a standard deviation of 0.38. Using the empirical rule, what percentage of the students have grade point averages that are at least 1-422 Please do not round your answer

Answer 1

The empirical rule states that 68% observations lie within 1 standard deviation from mean, that is between μ – σ and μ + σ,

95% observations lie within 2 standard deviations from mean, that is between μ – 2σ and μ + 2σ,

and 99.7% observations lie within 3 standard deviations from mean, that is between μ – 3σ and μ + 3σ.

(a) We are given $\mu=2.56 \ and \ \sigma=0.38$ .

$\mu-3\sigma=2.56-3(0.38)=1.42$

99.7% 95% 68% 34% 34% 2.35% 2.359 0.15% 13.5% 13.5% 0.1 5%

This shows, 100%-0.15%=99.85% of the students have grade point averages that are at least 1.42.

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(b) We are given $\mu=2.61 \ and \ \sigma=0.4$ .

$\\\mu-\sigma=2.61-0.4=2.21 \\\mu+\sigma=2.61+0.4=3.01$

99.7% 95% 68% 34% 34% 2.35% 2.359 0.15% 13.5% 13.5% 0.1 5% -2.21 -3.01

This shows, 68% of the students have grade point averages that are between 2.21 and 3.01.