To determine the sample space for the random experiment where a student is asked how many points she earned on a recent 97-point test, we need to consider all possible scores that the student could have obtained.
1. Understanding the Range of Scores:
- The test has a maximum score of 97 points.
- The minimum score that can be achieved is 0 points (if the student got everything wrong or didn't attempt the test).
2. Identifying All Possible Scores:
- The student can score any number from 0 to 97, inclusive.
- This means the possible scores are: 0, 1, 2, 3, ..., up to 97.
3. Constructing the Sample Space:
- The sample space is the set of all possible outcomes.
- Hence, the sample space should include every integer starting from 0 and ending at 97.
Now, let's evaluate the given options:
- Option A: [tex]$\{1,2,3, \ldots, 97\}$[/tex] starts from 1, missing 0.
- Option B: [tex]$\{1,2,3, \ldots, 100\}$[/tex] includes scores up to 100, which is more than the maximum possible score of 97.
- Option C: [tex]$\{97\}$[/tex] indicates only one possible score of 97, ignoring all other possible scores.
- Option D: [tex]$\{0, 1, 2, \ldots, 97\}$[/tex] correctly includes every possible score from 0 to 97.
After careful consideration, the correct answer is:
- Option D: [tex]$\{0,1,2, \ldots, 97\}$[/tex]
Thus, the sample space for the random experiment is [tex]$\{0,1,2, \ldots, 97\}$[/tex].
