Let's work through the problem step-by-step.

Step 1: Identify the distribution

Since the student is guessing on each true/false question, the situation can be modeled using the binomial distribution, where:

• $n = 50$ (the total number of questions),
• $p = 0.5$ (the probability of answering any single question correctly, since each question has two options and guessing means there's a 50% chance).

Step 2: Mean and Standard Deviation

For a binomial distribution, the mean $\mu$ and standard deviation $\sigma$ are given by:

• Mean $\mu = n \times p$
• Standard deviation $\sigma = \sqrt{n \times p \times (1 - p)}$

Let's compute these:

1. $n = 50$
2. $p = 0.5$

Now, calculate the exact values.

Calculations:

1. $\mu = 50 \times 0.5 = 25$ (Expected number of correct answers)
2. $\sigma = \sqrt{50 \times 0.5 \times 0.5} = \sqrt{12.5} \approx 3.536$ (Rounded to three decimal places)

Step 3: Usual Range

To determine what would be considered "usual," we define the usual range as being within two standard deviations of the mean. So, we calculate:

• Lowest usual number: $\mu - 2\sigma = 25 - 2 \times 3.536 = 25 - 7.072 = 17.928 \approx 18$ (Rounded to the nearest whole number)
• Highest usual number: $\mu + 2\sigma = 25 + 2 \times 3.536 = 25 + 7.072 = 32.072 \approx 32$ (Rounded to the nearest whole number)

So, a "usual" number of correct answers would be between 18 and 32.

Step 4: Is 35 correct answers unusually high?

Since 35 correct answers is greater than 32 (the upper bound of the usual range), a student getting 35 correct answers would be considered unusually high.

Thus, the answer to the last question is Y (Yes).

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Final Answers:

1. $n = 50$
2. $p = 0.5$
3. $\mu = 25$
4. $\sigma = 3.536$
5. Lowest usual number of correct answers: 18
6. Is 35 unusually high? Y

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Do you want more details or have any questions?

Related Questions:

1. What is the probability that a student gets exactly 30 questions correct by guessing?
2. What is the probability that a student gets fewer than 20 questions correct?
3. If the student guesses on 100 questions instead of 50, what would the new mean and standard deviation be?
4. How would the probabilities change if the test was a multiple-choice test with 4 options per question?
5. How can we calculate the probability that a student gets more than 40 correct answers by guessing?

Tip:

In a binomial distribution, when the probability of success is 0.5 (like in true/false tests), the distribution is symmetric around the mean, making the calculations simpler.