A student takes a 50 question true/false test.

What is the expected number of correct answers if the student guesses on every question?

Let X = number of questions students gets correct on the test.

n =  . Enter the Exact Value.

p =  . Write as a decimal, one decimal place. Example 0.3

=  . Enter the Exact Value.

=  . Write as a decimal, rounded to 3 decimal places. For example: 0.3456 enter as 0.346

Expected Number of Successes =  Enter the Exact Value.

Using mean and standard deviation, what is the lowest number of correct answers that would be considered usual? Round to the nearest whole number.

Would a student correctly scoring 35 correct out of 50 on the test be unusually high?

Enter Y for yes or N for no.

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).

---

### 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**

---

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.

A student takes a 50 question true/false test. What is the expected number of correct answers if the student guesses on every question? Let X = number of questions students gets correct on the test. n =  . Enter the Exact Value. p =  . Write as a decimal, one decimal place. Example 0.3 =  . Enter the Exact Value. =  . Write as a decimal, rounded to 3 decimal places. For example: 0.3456 enter as 0.346 Expected Number of Successes =  Enter the Exact Value. Using mean and standard deviation, what is the lowest number of correct answers that would be considered usual? Round to the nearest whole number. Would a student correctly scoring 35 correct out of 50 on the test be unusually high? Enter Y for yes or N for no.

Discover how to calculate the expected number of correct answers on a 50-question true/false test if a student guesses. Learn the mean, standard deviation, and whether scoring 35 correct answers would be unusually high. This problem explores key probability concepts like the binomial distribution and standard deviation, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation