Final answer:
To determine the percentage of scores above or below a raw score using a Z-score and normal curve table, calculate the Z-score, look up the corresponding area in the Z-table, and then translate it back to a raw score using the mean and standard deviation of your data set. The correct multiple-choice option is (A).
Step-by-step explanation:
Understanding Z-Scores and the Normal Distribution Curve
The steps for figuring out the percentage of scores above or below a particular raw score or Z-score using the normal curve table involve several key procedures. To determine this percentage, you first calculate the Z-score, which reflects how many standard deviations a data point is from the mean. Then, using a normal curve table, also known as a Z-table, you can find the area under the curve that corresponds to this Z-score. The Z-score is calculated using the formula Z = (X - μ) / σ, where X is the value, μ (mu) is the mean, and σ (sigma) is the standard deviation.
For a positive Z-score, representing a value above the mean, you would use the Z-table to find the area to the left of the Z-score under the curve, and subtract that value from 1 to get the percentage of scores above it. Conversely, for a negative Z-score, indicating a value below the mean, the Z-table gives you directly the percentage of scores that are lower than the given score. This process utilizes the properties of the standard normal distribution, which has a mean of 0 and standard deviation of 1. Z-scores allow comparison of scores from different data sets with various means and standard deviations, making it a versatile tool in statistics.
To illustrate, for a Z-score of 1.28, which corresponds to an area under the curve of about 0.9 to the left, this indicates that approximately 90% of the data falls below this Z-score. Knowing the mean and standard deviation, you can then translate this standardized score back to a raw score in the context of your specific data set.
Understanding the empirical rule, also known as the 68-95-99.7 rule, is also helpful when interpreting Z-scores. This rule states that approximately 68% of data falls within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations. The correct multiple-choice option is (A).