Question

Assuming the mean and SD are the same for this year’s national exam, what percentage of test-takers scored above and below you if your score is 1900 and given that X ~ N(1600, 140). Using the empirical rule, find the percentage of students who scored between 1880 and 1920?

Answer 1

Given that $X\sim N(1600, 140).$

Then

$$\mu=1600, \sigma^2=140, \sigma=\sqrt{140}$$ $$P(X<1900)=P(Z<\dfrac{1900-\mu}{\sigma})$$ $$=P(Z<\dfrac{1900-1600}{\sqrt{140}})\approx P(Z<25.3546)$$ $$\approx1(100\%)$$

The empirical rule formula:

68% of data falls within 1 standard deviation from the mean - that means between $μ - σ$  and $μ + σ.$

95% of data falls within 2 standard deviations from the mean - between $μ – 2σ$  and $μ + 2σ.$

99.7% of data falls within 3 standard deviations from the mean - between $μ - 3σ$ and $μ + 3σ.$

Given

$$\mu=1600, \sigma=\sqrt{140}\approx11.83$$

Then

$$(\mu-\sigma, \mu+\sigma)\approx(1588.2, 1611.8)$$ $$(\mu-2\sigma, \mu+2\sigma)\approx(1576.3, 1623.7)$$ $$(\mu-3\sigma, \mu+3\sigma)\approx(1564.5, 1635.5)$$

We cannot use the empirical rule to find the percentage of students who scored between 1880 and 1920.