Question

Using the z-score of ±1.645 for the 5 percent cutoff and the z-score of ±1.96 for the 2.5 percent in the tail, identify the subject identification (ID) number for subjects who fall at or above the cutoff for the upper 2.5 percent and 5 percent of the scores and those who are at or below the lower 2.5 percent and 5 percent of the scores. Do this by comparing each participant’s z-score with the appropriate critical value (1.645, 1.96, -1.645, -1.96). To fall into the upper tail of 5% (the 95th percentile), a participant’s z-score would need to be equal to or greater than 1.645. To fall into the upper tail of 2.5% (the 97.5th percentile), the z-score would need to be equal to or greater than 1.96. For the tails at the lower end, you would look for z-scores of -1.645 or lower (5%) or -1.96 or lower (2.5%)

Answer 1

Subjects with a z-score >= 1.645 and >= 1.96 fall within the upper 5% and 2.5% of scores respectively, while those with a z-score <= -1.645 and <= -1.96 fall within the lower 5% and 2.5%. The critical value 1.645 takes into account the desired confidence level and reflects the z-score needed to capture the central 90% of data for a 5% cutoff in a normal distribution.

Step-by-step explanation:

To identify subjects who fall within the upper 2.5 percent and 5 percent of scores, or at or below the lower 2.5 percent and 5 percent of scores, you must compare each participant's z-score with the critical values: 1.645, 1.96, -1.645, and -1.96. If a participant's z-score is equal to or greater than 1.645, they fall into the upper 5 percent, and if their z-score is equal to or greater than 1.96, they fall into the upper 2.5 percent. Conversely, a z-score at or below -1.645 indicates a participant falls in the lower 5 percent and at or below -1.96 indicates they are in the lower 2.5 percent.

The critical value is determined by the desired confidence level and corresponds to the z-score that captures the central percentage of data in a normal distribution. As per the Empirical Rule, approximately 68 percent, 95 percent, and 99.7 percent of values lie within one, two, and three standard deviations from the mean, respectively. This can be calculated for any set of data given its mean and standard deviation, and the corresponding z-scores for any data point can be found using the formula (data point - mean) / standard deviation.

Using these principles, a subject's ID can be matched to their z-score to determine if they are in the upper or lower tails of the distribution based on the provided cutoffs.