To determine how many times you would expect to roll a 6 out of 60 rolls of a number cube, you start by calculating the expected number of 6's. Since the probability of rolling a 6 is [tex]\(\frac{1}{6}\)[/tex], and you have 60 rolls, you calculate:
[tex]\[
\text{Expected number of 6's} = 60 \times \frac{1}{6} = 10
\][/tex]
Next, you consider what range of results would be reasonable to expect, given some tolerance for variability in a small number of trials. Suppose a reasonable tolerance for questioning the model is set to [tex]\(\pm 2\)[/tex].
You calculate the upper and lower limits of this tolerance range:
- Upper tolerance limit: [tex]\(10 + 2 = 12\)[/tex]
- Lower tolerance limit: [tex]\(10 - 2 = 8\)[/tex]
This means that you would expect the number of 6's in a fair model to fall in the range from 8 to 12. Now, let’s evaluate each of the given options to see which ones fall within or outside this tolerance range.
- Option A: 6 (falls outside the range 8-12)
- Option B: 14 (falls outside the range 8-12)
- Option C: 20 (falls outside the range 8-12)
- Option D: 10 (falls within the range 8-12)
From this evaluation:
- Option A (6): This would make you question the model.
- Option B (14): This would make you question the model.
- Option C (20): This would make you question the model.
- Option D (10): This would not make you question the model.
So, the only number of 6's that falls within the acceptable range and would not make you question your model is option D, 10. Hence, options 6, 14, and 20 would all make you question your model.
