Final answer:
Z-scores, or z*, separate percentages of data in a normal distribution. For 95%, 85.3%, 97%, and 93%, the respective z* values are approximately ±1.96, ±1.44, ±2.17, and ±1.81.
Step-by-step explanation:
The student's question is about finding the z* values that separate the given percentages of z values in a normal distribution. This is a concept that is fundamental in statistics when dealing with the standard normal distribution. Below are the answers for each condition described:
- (a) For the middle 95% of z values, the z* can be found approximately at ±1.96, because 95% of the data in a normal distribution lies within 1.96 standard deviations from the mean.
- (b) To find the middle 85.3% of z values, z* is approximately ±1.44. This is because, looking at a z-table or using statistical software, 85.3% of the data lies within 1.44 standard deviations from the mean.
- (c) For the middle 97% of z values, z* is approximately at ±2.17, as from a z-table or statistical software, we know that 97% of the data lies within 2.17 standard deviations from the mean.
- (d) To separate the middle 93% of all z values from the most extreme 7%, z* is approximately ±1.81. This is according to standardized z-tables that show 93% of values lying within 1.81 standard deviations from the mean.
It's important to note that these z* values are rounded to two decimal places, as requested by the student.