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

X ~ N(1600, 140)

Population mean = 1600

Variance = 140

$\sigma=\sqrt{variance}=\sqrt{140}=11.83$

$P(X<1900)=P(Z<\frac{1900-1600}{11.83})$

$P(X<1900)=P(Z<25.36)$

Using normal distribution table, we find that probability is:

$P(X<1900)=1≈100\%$

This means 100 percent of test-takers scored above and below if score is 1900.

As per the empirical rule, the percentage of students who scored between 1880 and 1920 is:

The empirical rule formula said that;

68% of data falls within 1 standard deviation from the mean - that means between $\mu-\sigma$  and $\mu+\sigma$

95% of data falls within 2 standard deviations from the mean - between $\mu-2\sigma$  and $\mu+2\sigma$

99.7% of data falls within 3 standard deviations from the mean - between $\mu-3\sigma$ and $\mu+3\sigma$

Given that;

Population mean = 1600

Variance = 140

$\sigma=\sqrt{variance}=\sqrt{140}=11.83$

$(μ−σ,μ+σ)≈(1588.2,1611.8)$

$(μ−2σ,μ+2σ)≈(1576.3,1623.7)$

$(μ−3σ,μ+3σ)≈(1564.5,1635.5)$