To determine how many questions would be considered unusual to get correct by guessing on a 13-question multiple choice test with 5 possible answers for each question, we can use the binomial distribution.
- Number of trials (n): 13 questions
- Probability of success (p): \( \frac{1}{5} = 0.2 \) (because there is one correct answer out of 5 choices)
- Probability of failure (q): \( 1 - p = 0.8 \)
The expected number of correct answers when guessing is given by:
\[ \text{Expected value (E)} = n \cdot p = 13 \cdot 0.2 = 2.6 \]
To find the range that is considered "usual," we can calculate the standard deviation of this binomial distribution:
\[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{13 \cdot 0.2 \cdot 0.8} = \sqrt{2.08} \approx 1.44 \]
In this context, an unusual outcome is typically defined as being more than 2 standard deviations away from the expected value. Therefore:
\[ \text{Usual range} = E \pm 2\sigma = 2.6 \pm 2(1.44) \]
Calculating this gives:
\[ E - 2\sigma = 2.6 - 2.88 \approx -0.28 \quad \text{(not meaningful since correct answers can't be negative)} \] \[ E + 2\sigma = 2.6 + 2.88 \approx 5.48 \]
This means that getting more than approximately 5.5 questions correct would be unusual. Rounding to the nearest whole number, we see that it would be unusual to get 6 or more questions correct by guessing.
Thus, the final answer is:
6