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Complete Question
Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 11. Use the empirical rule to determine the following:
a. What percentage of people has an IQ score between 89 and 111?
b. What percentage of people has an IQ score less than 89 or greater than 111?
c. What percentage of people has an IQ score greater than 133?
Answer:
a) 68%
b) 32%
c) 0.15%
Step-by-step explanation:
The Empirical rule formula:
1) 68% of data falls within 1 standard deviation from the mean - that means between μ - σ and μ + σ .
2)95% of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ .
3) 99.7% of data falls within 3 standard deviations from the mean - between μ – 3σ and μ + 3σ .
a. What percentage of people has an IQ score between 89 and 111?
Using :μ - σ , μ + σ
μ = Mean = 100
σ = Standard deviation = 11
μ - σ
100 - 11 = 89
μ + σ
100 + 11 = 111
Hence, from the above calculation, this data falls within 1 standard deviation from the mean - that means between μ - σ and μ + σ . Hence, it satisfied the first rule in the Empirical formula.
Therefore, 68% of data falls within 1 standard deviation from the mean.
b. What percentage of people has an IQ score less than 89 or greater than 111?
Using :μ - σ , μ + σ
μ = Mean = 100
σ = Standard deviation = 11
μ - σ
100 - 11 = 89
μ + σ
100 + 11 = 111
Hence, from the above calculation,68% of this data falls within 1 standard deviation from the mean - that means between μ - σ and μ + σ . Hence, it satisfied the first rule in the Empirical formula.
But we are told to find the percentage of people that score less than 89 and above 111
100 - 68 = 32%
c. What percentage of people has an IQ score greater than 133?
We apply the third rule of the Empirical rule formula
99.7% of data falls within 3 standard deviations from the mean - between μ – 3σ and μ + 3σ .
μ = Mean = 100
σ = Standard deviation = 11
μ + 3σ
100 + 3(11)
100 + 33
= 133
Since we are to find the percentage greater than 133
Hence: 100 - 99.7%
= 0.3%
The percentage greater than 133(the right hand side)
= 0.3%/2
= 0.15%
