Answer :
By using the 68-95-99.7 Rule, the percent of scores in the data set that are expected to be below a score of 120 is 84.1%.
What is the 68-95-99.7 Rule?
In Mathematics, the 68-95-99.7 Rule is sometimes referred to as the Empirical Rule and it can be defined as a statistical rule which states that:
The middle 68% of a normal distribution would be within one (1) standard deviation of its mean i.e between μ - σ and μ + σ.
The middle 95% of a normal distribution would be within two (2) standard deviations of its mean i.e between μ - 2σ and μ + 2σ.
The middle 99.7% of a normal distribution would be within three (3) standard deviations of its mean i.e between μ - 3σ and μ + 3σ.
At the middle 68% of the normal distribution, we would apply the 68-95-99.7 Rule to the given data as follows;
μ - σ = 108 - 12 = 96.
μ - σ = 108 + 12 = 120.
Where:
- μ is the sample mean.
- x is the score.
- σ is the standard deviation.
Now, we can determine the percent of scores (P) in the data set that are expected to be below a score (x) of 120 by using the probabilities and standard distribution table;
P(x < 120) = P((x - μ)/σ < (120 - 108)/12)
P(x < 120) = P((x - μ)/σ < 12/12)
P(x < 120) = P(z < 1)
P(x < 120) = 0.841
P(x < 120) = 84.1%
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