Indicate whether the given statement could apply to a data set consisting of 1,000 values that are all different.

a. The 29th percentile is greater than the 30th percentile.
b. The median is greater than the first quartile.
c. The third quartile is greater than the first quartile.
d. The mean is equal to the median.
e. The range is zero

b and c are correct.

To answer these questions, you need to understand the concepts of percentiles, quartiles, median, mean, and range. I will explain each of them and then provide the answer to each statement.

1. Percentiles: Percentiles divide a data set into 100 equal parts. For example, the 29th percentile indicates that 29% of the values in the data set are below that value.

2. Quartiles: Quartiles divide a data set into four equal parts. The first quartile (Q1) is the value below which 25% of the data falls, the second quartile (Q2) is the median which divides the data into two equal parts, and the third quartile (Q3) is the value below which 75% of the data falls.

3. Median: The median is the middle value of a data set when it is arranged in ascending order. If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of the two middle values.

4. Mean: The mean is the average of a data set and is calculated by summing up all the values and dividing by the total number of values.

5. Range: The range is the difference between the maximum and minimum values in a data set.

Now, let's answer the given statements:

a. The 29th percentile is greater than the 30th percentile.
To answer this statement, you need to understand that percentiles are calculated based on the order of the data values. In a data set with 1,000 different values, if all the values are unique, it is impossible for the 29th percentile to be greater than the 30th percentile. Every percentile represents a unique value in the data set, so the 29th percentile will always be smaller than or equal to the 30th percentile. Therefore, the answer is no.

b. The median is greater than the first quartile.
To answer this statement, you need to understand that the median divides the data set into two equal parts, while the first quartile divides the data set into four equal parts. In a data set with 1,000 different values, the median will be the value in the middle, and the first quartile will be the value below which 25% of the data falls. Since the median is the middle value, it will always be greater than the first quartile. Therefore, the answer is yes.

c. The third quartile is greater than the first quartile.
To answer this statement, you need to understand that the third quartile is the value below which 75% of the data falls, while the first quartile is the value below which 25% of the data falls. In a data set with 1,000 different values, the third quartile will be the value in the middle of the second half of the data set, and the first quartile will be the value in the middle of the first half of the data set. Since the second half of the data set contains higher values than the first half, the third quartile will always be greater than the first quartile. Therefore, the answer is yes.

d. The mean is equal to the median.
To answer this statement, you need to understand that the mean is the average of the data set, while the median is the middle value of the data set. In a data set with 1,000 different values, the mean and median could be equal if the data set is symmetrically distributed. However, since the data set consists of 1,000 different values, it is unlikely to have a symmetric distribution, and therefore it is unlikely for the mean to be equal to the median. Therefore, the answer is no.

e. The range is zero.
To answer this statement, you need to understand that the range is the difference between the maximum and minimum values in a data set. In a data set with 1,000 different values, if all the values are different, it is impossible for the range to be zero. There will always be a difference between the highest and lowest value. Therefore, the answer is no.