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calculate and get final answers for each part

Let's solve each part step-by-step:

### Step 1: Calculate the Expected Grade (Part a)

To calculate the expected grade, we multiply each score by its weight (proportion of grade) and then sum these products:

\[
\text{Expected Grade} = (56 \times 0.15) + (73.5 \times 0.27) + (73 \times 0.24) + (39.5 \times 0.34)
\]

Let's calculate each term:

1. **Assignments:** \( 56 \times 0.15 = 8.4 \)
2. **Quizzes:** \( 73.5 \times 0.27 = 19.845 \)
3. **Midterms:** \( 73 \times 0.24 = 17.52 \)
4. **Final:** \( 39.5 \times 0.34 = 13.43 \)

Now add these up:

\[
\text{Expected Grade} = 8.4 + 19.845 + 17.52 + 13.43 = 59.195
\]

So, the **expected grade** is **59.195%**.

---

### Step 2: Calculate the Standard Deviation of Expected Grade (Part b)

To find the standard deviation, we need to calculate the variance first. This requires computing the squared deviation of each score from the expected grade, weighting each by its proportion, and summing these values.

The formula for the variance is:

\[
\text{Variance} = \sum \left( (\text{Score} - \text{Expected Grade})^2 \times \text{Proportion} \right)
\]

Then, the standard deviation is the square root of the variance.

Calculating each term:

1. **Assignments:** \( (56 - 59.195)^2 \times 0.15 = (-3.195)^2 \times 0.15 = 10.207025 \times 0.15 = 1.53105375 \)
2. **Quizzes:** \( (73.5 - 59.195)^2 \times 0.27 = (14.305)^2 \times 0.27 = 204.651025 \times 0.27 = 55.25577675 \)
3. **Midterms:** \( (73 - 59.195)^2 \times 0.24 = (13.805)^2 \times 0.24 = 190.633025 \times 0.24 = 45.752046 \)
4. **Final:** \( (39.5 - 59.195)^2 \times 0.34 = (-19.695)^2 \times 0.34 = 387.007025 \times 0.34 = 131.5823885 \)

Summing these values gives us the variance:

\[
\text{Variance} = 1.53105375 + 55.25577675 + 45.752046 + 131.5823885 = 234.121265
\]

The standard deviation is:

\[
\text{Standard Deviation} = \sqrt{234.121265} \approx 15.2964
\]

So, the **standard deviation** is approximately **15.2964%**.

---

### Step 3: Calculate the Expected Profit (Part c)

The expected profit is based on the additional income per grade point, the scholarship, and tuition costs.

1. **Expected additional income:** \( 59.195 \times 19.5 = 1154.3025 \)
2. **Total cost:** \( 1150 - 450 = 700 \)
3. **Profit:** \( 1154.3025 - 700 = 454.3025 \)

So, the **expected profit** is approximately **454.30** dollars.

---

### Summary of Answers

- **(a)** Expected Grade: **59.195%**
- **(b)** Standard Deviation of Expected Grade: **15.2964%**
- **(c)** Expected Profit: **454.30 dollars**

Do you need a detailed breakdown of any step? 

---

### Related Questions

1. How would the expected grade change if one of the scores increased?
2. How would increasing the weight of the final exam impact the standard deviation?
3. If tuition costs increased, how would that affect the profit calculation?
4. How can expected grade be recalculated if additional assessments are introduced?
5. What is the importance of calculating standard deviation for understanding grade distribution?

---

**Tip:** When calculating expected values, always ensure weights add up to 1 for accuracy.

A student is trying to calculate their final grade. The student estimates their performance based on a table of marks in various categories (Assignments, Quizzes, Midterms, and Final), with specific proportions of the grade. They need to calculate (a) the expected grade, (b) the standard deviation of the expected grade, and (c) the expected profit based on earnings per grade point and additional costs.

Learn how to calculate a student’s expected final grade, the standard deviation of their performance, and the expected profit based on earnings per grade point. This problem covers weighted averages, variance, and expected value, ideal for students in Grades 10-12.

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