Statistics The table shows Tom’s Fall 2022 MA200 grades from each category and the corresponding weight. Homework Quiz Test Final Exam Xi (points) 100 90 70 80 Wi (weight) 10% 20% 40% 30% Note: Write down the formula for each of the questions before showing your work step by step and then highlight your final answer with a box. 1. Calculate the direct mean. Suppose the final exam has not started yet, how many points Tom will need to make such that his final direct average is 90? 2. Calculate the weighted mean. Suppose the final exam has not started yet, how many points Tom will need to make such that his final weighted mean is 90? 3. What mean gives Tom higher final average, direct mean or weighted mean? Why?

Let's solve this problem step by step. We will address each question, first writing the formula needed, and then going through the calculations before arriving at the final answer.

---

### 1. **Calculate the Direct Mean**  
The **direct mean** is the average of all grades without considering the weights.

#### Formula:
\[
\text{Direct Mean} = \frac{\sum X_i}{n}
\]
Where:
- \( X_i \) are the individual points (grades)
- \( n \) is the number of categories.

For this problem:
\[
\text{Direct Mean} = \frac{100 + 90 + 70 + 80}{4} = \frac{340}{4} = 85
\]

#### Final Answer:
\[
\boxed{85}
\]

Now, suppose the final exam has not started yet, and Tom wants his final direct average to be 90. We need to calculate how many points he needs on the final exam to achieve this.

Let the final exam score be \( x \).

#### Formula for direct mean with \( x \):
\[
\frac{100 + 90 + 70 + x}{4} = 90
\]

Solving for \( x \):
\[
\frac{260 + x}{4} = 90
\]
\[
260 + x = 360
\]
\[
x = 360 - 260 = 100
\]

#### Final Answer:
\[
\boxed{100}
\]

---

### 2. **Calculate the Weighted Mean**  
The **weighted mean** takes into account the weights assigned to each category.

#### Formula:
\[
\text{Weighted Mean} = \sum (X_i \times W_i)
\]
Where:
- \( X_i \) are the individual points (grades)
- \( W_i \) are the corresponding weights.

For this problem:
\[
\text{Weighted Mean} = (100 \times 0.10) + (90 \times 0.20) + (70 \times 0.40) + (80 \times 0.30)
\]

Step-by-step calculation:
\[
\text{Weighted Mean} = 10 + 18 + 28 + 24 = 80
\]

#### Final Answer:
\[
\boxed{80}
\]

Now, suppose the final exam has not started yet, and Tom wants his final weighted mean to be 90. We need to calculate how many points he needs on the final exam to achieve this.

Let the final exam score be \( x \).

#### Formula for weighted mean with \( x \):
\[
0.10(100) + 0.20(90) + 0.40(70) + 0.30(x) = 90
\]

Solving for \( x \):
\[
10 + 18 + 28 + 0.30x = 90
\]
\[
56 + 0.30x = 90
\]
\[
0.30x = 90 - 56 = 34
\]
\[
x = \frac{34}{0.30} \approx 113.33
\]

Since a score over 100 is impossible, Tom cannot achieve a weighted average of 90.

#### Final Answer:
\[
\boxed{x = 113.33 \text{ (Impossible)}}
\]

---

### 3. **Which Mean Gives a Higher Final Average? Why?**

#### Comparison:
- The **direct mean** calculated was 85.
- The **weighted mean** calculated was 80.

The direct mean gives Tom a **higher final average** because the weighted mean incorporates the weights of each category, giving more importance to lower scores (like the test with 70 points weighted at 40%), which brings the overall average down.

#### Final Answer:
\[
\boxed{\text{Direct Mean gives a higher final average.}}
\]
This is because the weighted mean gives more importance to categories where Tom scored lower, such as the test.

---

### Do you have any questions, or would you like more details on any part of this?

Here are 5 related questions:
1. How is the weighted mean affected by changes in the weight distribution?
2. What happens to the weighted mean if Tom scores higher on categories with greater weights?
3. How can Tom improve his weighted mean if his test scores are low?
4. Why is it important to consider weights in academic grading systems?
5. How would the direct and weighted means compare if Tom's scores were more consistent across all categories?

**Tip:** Always pay attention to the weights of each category when interpreting averages. A high score in a heavily weighted category can have a significant impact on the final result.

Learn how to calculate both the direct and weighted mean of Tom's Fall 2022 MA200 grades. This step-by-step guide explains key statistical concepts like direct mean and weighted mean using a real-world example of homework, quiz, test, and final exam scores. Suitable for students in Grades 9-12.

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