A comprehensive study of juvenile delinquency is conducted in every state-run reform school in Australia. A scale called the Antisocial Beliefs Scale is given to all juvenile delinquents. The mean (μ) of the scale is 60 and the standard deviation (σ) is 6.
Assuming that the scores are normally distributed, what PERCENTAGE of the population falls between 68 and 56?
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To find the percentage of the population that falls between 68 and 56 on the Antisocial Beliefs Scale, we need to standardize the scores using z-scores.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
z is the z-score,
x is the raw score,
μ is the mean, and
σ is the standard deviation.
For the score 68:
z1 = (68 - 60) / 6
z1 = 8 / 6
z1 = 1.33
For the score 56:
z2 = (56 - 60) / 6
z2 = -4 / 6
z2 = -0.67
Now that we have the z-scores, we can look up the corresponding area under the normal distribution curve. The percentage between 68 and 56 is the difference between the areas associated with these z-scores.
Using a standard normal distribution table or a calculator, we can look up the area for each z-score.
The area to the left of z = 1.33 is approximately 0.9080 (or 90.80%).
The area to the left of z = -0.67 is approximately 0.2514 (or 25.14%).
To find the percentage between 68 and 56, we subtract the smaller area from the larger area:
Percentage = 0.9080 - 0.2514
Percentage ≈ 0.6566
Therefore, approximately 65.66% of the population falls between 68 and 56 on the Antisocial Beliefs Scale.