Theorists have suggested that there may be a link between childhood trauma and involvement in the criminal justice system. These theorists think that rehabilitation for youth involved in the criminal justice system would be more effective if it targeted individual traumas in addition to known criminal risk factors. A researcher that works at one of the institutions for youth decided to test this theory choosing n=30 youth who were incarcerated and administered a questionnaire assessing their history of traumatic events. The researcher knew that youth in the general population tend to score on average a μ = 5 (the s is not known) on the interval scale measuring trauma. The researcher is unsure whether this group of offenders will score higher or lower and after administering the scale found the mean to be M = 7 and with a variance of s2 = 4. Conduct all of the steps of hypothesis testing using an alpha = .05.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...