https://file.aiquickdraw.com/mathbot/file/17304428001574njgln0z.jpg

calculate and get final answers for each part

Let's solve each part of this problem step-by-step.

### Given Information
The problem states that the final exam marks are normally distributed with:
- Mean \( \mu = 63 \)%
- Standard deviation \( \sigma = 10 \)%

### Solution

#### (a) Probability that a random chosen final exam mark will be at most 71.3%
To find this probability, we calculate the z-score for 71.3% and use the standard normal distribution table.

1. **Calculate the z-score**:
   \[
   z = \frac{X - \mu}{\sigma} = \frac{71.3 - 63}{10} = \frac{8.3}{10} = 0.83
   \]

2. **Find the probability**:
   Using the standard normal distribution table or calculator, we look up the probability for \( z = 0.83 \).
   \[
   P(Z \leq 0.83) \approx 0.79673
   \]

   **Answer**: The probability that a random final exam mark will be at most 71.3% is approximately **0.79673**.

---

#### (b) Proportion of students who will not pass the statistics final exam (if passing mark is 50%)
Here, we need to find the probability that a student scores less than 50%. 

1. **Calculate the z-score**:
   \[
   z = \frac{X - \mu}{\sigma} = \frac{50 - 63}{10} = \frac{-13}{10} = -1.3
   \]

2. **Find the probability**:
   Using the standard normal distribution table, we look up the probability for \( z = -1.3 \).
   \[
   P(Z \leq -1.3) \approx 0.09680
   \]

   **Answer**: The proportion of students who will not pass the statistics final exam is approximately **0.09680** (or 9.68%).

---

#### (c) Maximum final exam mark needed to receive a letter grade of at most an F (bottom 3.5% of students)
In this part, we need to find the score corresponding to the bottom 3.5% of the distribution, which means finding the z-score that corresponds to the 3.5th percentile.

1. **Find the z-score for the 3.5th percentile**:
   From the standard normal distribution table, the z-score corresponding to the 3.5th percentile is approximately \( z = -1.81 \).

2. **Calculate the exam mark**:
   Using the z-score formula, solve for \( X \):
   \[
   X = \mu + z \cdot \sigma = 63 + (-1.81) \cdot 10 = 63 - 18.1 = 44.9
   \]

   **Answer**: The maximum final exam mark needed to earn a letter grade of at most an F is approximately **44.9%**.

---

### Summary of Answers
- **(a)** Probability of scoring at most 71.3%: **0.79673**
- **(b)** Proportion of students not passing (below 50%): **0.09680**
- **(c)** Maximum mark for bottom 3.5% (F grade): **44.9%**

Would you like a deeper breakdown of any part of these calculations?

---

### Additional Related Questions
1. How would the probability in (a) change if the mean was higher?
2. What score corresponds to the top 10% of students?
3. How would you calculate the probability for any given percentage using the cumulative distribution function?
4. What is the z-score formula and when is it typically used in statistics?
5. How can standard deviation affect the distribution of grades?

### Tip
When solving normal distribution problems, always start by calculating the z-score to standardize your value. This makes it easier to find probabilities using the standard normal table or calculator.

The professor of a Statistics class has stated that, historically, the distribution of final exam grades in the course resembles a Normal distribution with a mean final exam mark of μ = 63% and a standard deviation of σ = 10%. (a) What is the probability that a random chosen final exam mark in this course will be at most 71.3%? (b) In order to pass this course, a student must have a final exam mark of at least 50%. What proportion of students will not pass the statistics final exam? (c) The bottom 3.5% of students writing the final exam will receive a letter grade of at most an F in the course. Find the maximum final exam mark needed on the statistics final to earn a letter grade of at most an F in the course.

Solve a statistics problem involving a normal distribution of final exam grades with a mean of 63% and standard deviation of 10%. Learn how to calculate the probability of scoring at most 71.3%, the proportion of students scoring below 50%, and the mark corresponding to the bottom 3.5% of grades. Suitable for college-level statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation