A comprehensive study of juvenile delinquency is conducted in every state-run reform school in Canada. A scale called the Antisocial Beliefs Scale is given to all juvenile delinquents. The mean (μ) of the scale is 71 and the standard deviation (σ) is 5.
Assuming that the scores are normally distributed, what PERCENTAGE of the population falls between 81 and 66?
Same process as following post from Alecia.
To find the percentage of the population that falls between two values (81 and 66) in a normally distributed data set, you can use the properties of the standard normal distribution or Z-scores.
1. Calculate the Z-score for each value.
The Z-score represents the number of standard deviations a data point is away from the mean. The formula to calculate the Z-score is:
Z = (X - μ) / σ
Z1 = (81 - 71) / 5
Z2 = (66 - 71) / 5
Z1 = 10 / 5 = 2
Z2 = -5 / 5 = -1
2. Look up the corresponding values for the Z-scores in the standard normal distribution table or use a calculator.
For Z = 2, the table gives a value of approximately 0.9772
For Z = -1, the table gives a value of approximately 0.1587
3. Calculate the percentage between the two Z-scores.
The percentage between two Z-scores can be found by subtracting the smaller value from the larger value and multiplying by 100.
Percentage = (larger_Z - smaller_Z) * 100
Percentage = (0.9772 - 0.1587) * 100
Percentage = 0.8185 * 100
Percentage = 81.85%
Therefore, approximately 81.85% of the population falls between the scores of 81 and 66 on the Antisocial Beliefs Scale in the state-run reform schools in Canada.