6.22 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
a. To find the z score for a CFC score of 4.2, we can use the formula:
z = (x - μ) / σ
Where:
x = CFC score
μ = mean score
σ = standard deviation
Plugging in the values:
z = (4.2 - 3.51) / 0.61
z = 0.69 / 0.61
z ≈ 1.13
So, if your CFC score is 4.2, your z score is approximately 1.13. This z score tells us that your score is about 1.13 standard deviations above the mean.
Now why does this answer make sense? Well, the z score essentially tells us how far away a particular score is from the mean in terms of standard deviations. In this case, a z score of 1.13 means your CFC score is above average, indicating a higher level of consideration of future consequences compared to the sample population.
b. Using the same formula, for a CFC score of 3.0:
z = (3.0 - 3.51) / 0.61
z = -0.51 / 0.61
z ≈ -0.84
Therefore, if your CFC score is 3.0, your z score is approximately -0.84. This means your CFC score is about 0.84 standard deviations below the mean. So your score is below average in terms of consideration of future consequences compared to the sample population.
c. If the z score is 0, it means the CFC score is exactly at the mean.
Using the formula to find the CFC score, where z is 0:
0 = (x - 3.51) / 0.61
Simplifying the equation:
0 = x - 3.51
x = 3.51
Therefore, if your z score is 0, your CFC score is exactly 3.51, which is the mean score for the sample population. This means your score is right at the average level of consideration of future consequences.
To calculate the z-score, we use the formula:
z = (X - μ) / σ
Where:
- X is the individual score
- μ is the mean of the population
- σ is the standard deviation of the population
a. Let's calculate the z-score when the CFC score is 4.2:
X = 4.2
μ = 3.51
σ = 0.61
z = (4.2 - 3.51) / 0.61
z = 0.69 / 0.61
z ≈ 1.13
Therefore, the z-score when the CFC score is 4.2 is approximately 1.13.
Explanation: The z-score indicates how many standard deviations a particular score is above or below the mean. A positive z-score means the score is above the mean, and a negative z-score means the score is below the mean. In this case, a z-score of 1.13 suggests that a CFC score of 4.2 is 1.13 standard deviations above the mean. This implies that the individual has a higher CFC score compared to the average student in the sample.
b. Let's calculate the z-score when the CFC score is 3.0:
X = 3.0
μ = 3.51
σ = 0.61
z = (3.0 - 3.51) / 0.61
z = -0.51 / 0.61
z ≈ -0.84
Therefore, the z-score when the CFC score is 3.0 is approximately -0.84.
Explanation: A z-score of -0.84 suggests that a CFC score of 3.0 is 0.84 standard deviations below the mean. This implies that the individual has a lower CFC score compared to the average student in the sample.
c. If your z-score is 0, it means your CFC score is equal to the mean of the population, which is 3.51.
Explanation: A z-score of 0 indicates that the score is exactly at the mean of the population. In this case, a z-score of 0 suggests that the individual has the same CFC score as the average student in the sample, which is 3.51.