Question 22

The mean is equal to zero with a standard deviation of 1.

A basic assumption of a z-score is that the distribution has a mean of zero and a standard deviation of one, allowing for the standardization of scores across different distributions.

A) The mean is equal to zero with a standard deviation of 2.

This option is incorrect because a z-score specifically standardizes data such that the mean is zero and the standard deviation is one, not two. A standard deviation of two would alter the interpretation of the z-scores and their relationship to the standard normal distribution.

B) The mean is equal to zero with a standard deviation of 1.

This option is correct, as it accurately describes the standard normal distribution, which is the basis for calculating z-scores. Z-scores represent how many standard deviations a data point is from the mean, which is defined as zero in this context.

C) Outlier data points are critical to a z-score calculation.

This option is incorrect because while outliers can affect z-scores, they are not critical to the calculation itself. Z-scores can still be computed regardless of the presence of outliers, although these points may skew the interpretation of the z-scores.

D) Outlier data points must be eliminated from a z-score calculation.

This option is also incorrect. While it may be beneficial to address outliers in some analyses, they do not have to be eliminated for a z-score calculation. Z-scores can be calculated with all data points, including outliers, although the presence of outliers can influence the mean and standard deviation.

Conclusion

The correct answer, that the mean is equal to zero with a standard deviation of one, is fundamental to understanding z-scores and their application in statistics. All other options either misstate the parameters of the z-score or misinterpret the role of outliers in the calculation. Thus, option B stands out as the only accurate representation of the assumptions underlying z-scores.