A study of the Consideration of Future Consequences(CFC) scale found a mean score of 3.51, with a standard deviation of 0.61, for the 664 students in the sample(Petrocelli, 2003). For the sake of this exercise, let’s assume
that this particular sample comprises the entire population of interest.
a. If your CFC score is 4.2, what is your z score? Use symbolic notation and the formula. Explain why this answer makes sense.
b. If your CFC score is 3.0, what is your z score? Use symbolic notation and the formula. Explain why this answer makes sense.
c. If your z score is 0, what is your CFC score? Explain
To find the z score, we use the formula:
z = (X - μ) / σ
where:
- X is the individual's CFC score
- μ is the mean score of the population (3.51 in this case)
- σ is the standard deviation of the population (0.61 in this case)
a. To find the z score when the CFC score is 4.2:
z = (4.2 - 3.51) / 0.61
= 0.69 / 0.61
≈ 1.13
So, if your CFC score is 4.2, your z score would be approximately 1.13.
Explanation: The z score measures how many standard deviations your score is above or below the mean of the population. A positive z score indicates a score above the mean, while a negative z score indicates a score below the mean. In this case, the z score of 1.13 indicates that your CFC score of 4.2 is about 1.13 standard deviations above the mean, suggesting that your score is relatively high compared to the population.
b. To find the z score when the CFC score is 3.0:
z = (3.0 - 3.51) / 0.61
= -0.51 / 0.61
≈ -0.84
So, if your CFC score is 3.0, your z score would be approximately -0.84.
Explanation: The negative z score of -0.84 indicates that your CFC score of 3.0 is about 0.84 standard deviations below the mean. This suggests that your score is relatively lower than the average CFC score of the population.
c. To find the CFC score when the z score is 0:
Here, we rearrange the formula to solve for X:
X = μ + z * σ
Substituting z = 0 and the given values:
X = 3.51 + 0 * 0.61
X = 3.51
So, if your z score is 0, your CFC score would be 3.51.
Explanation: A z score of 0 means your score is exactly equal to the mean of the population. In this case, a z score of 0 suggests that your CFC score is the same as the population mean, which is 3.51.