Question

Motivation of drug dealers. Refer to the Applied Psychology in Criminal Justice (September 2009) investigation of the personality characteristics of drug dealers, Exercise 2.80 (p. 111). Convicted drug dealers were scored on the Wanting Recognition (WR) Scale. This scale provides a quantitative measure of a person’s level of need for approval and sensitivity to social situations. (Higher scores indicate a greater need for approval.) Based on the study results, we can assume that the WR scores for the population of convicted drug dealers have a mean of 40 and a standard deviation of 5. Suppose that in a sample of 100 people, the mean WR scale score is x = 42. Is this sample likely selected from the population of convicted drug dealers? Explain.

Short answer

The sample is not likely to have been selected from the population of convicted drug dealers.

Step-by-step solution

Given information

From the given problem, the mean $\mu =40$ and the standard deviation,$\sigma =5$ the sample size is 100.

Calculating the probability

From the central limit theorem, as the sample size is large, the sample's mean follows a normal distribution with mean ${\mu }_{\overline{x}}\left(=\mu \right)$and standard deviation.${\sigma }_{\overline{x}}\left(=\frac{\sigma }{\sqrt{n}}\right)$

The mean of the sampling distribution$\overline{x}$is the population mean .

That is,

$\begin{array}{rcl}{\mu }_{\overline{x}}& =& \mu \\ & =& 40\\ & & \end{array}$

Thus,${\mu }_{\overline{x}}=40$

From the given problem, the sample size is.$n=100$

$\begin{array}{rcl}{\sigma }_{\overline{x}}& =& \frac{\sigma }{\sqrt{n}}\\ & =& \frac{5}{\sqrt{100}}\\ & =& 0.5\\ & & \end{array}$

Thus,${\sigma }_{\overline{x}}=0.5$

$\begin{array}{rcl}P\left(\overline{x}>42\right)& =& P\left(\frac{\overline{x}-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}>\frac{42-40}{0.5}\right)\\ & =& P\left(Z>\frac{2}{0.5}\right)\\ & =& P\left(Z>4\right)\\ & & \end{array}$

$$\begin{gathered} =0.5-P\left(0<Z<4\right) \\ =0.5-0.5 \\ =0 \end{gathered}$$

Thus,$P\left(\overline{x}>42\right)\approx 0$

Here, the probability value is almost equal to 0. Thus, the sample is not likely to have been selected from the population of convicted drug dealers.