Normal distribution curve illustrating test scores with mean and standard deviations for a biology exam.

Let's solve this step by step based on the normal distribution, z-scores, and the Empirical Rule (also known as the 68-95-99.7 rule).

---

Part 1: Basic Normal Distribution Percentages

In a normal distribution:

1. Below the mean?
→ Since the normal distribution is symmetric, 50% of values are below the mean.

2. Above the mean?
→ Similarly, 50% are above the mean.

3. Within one standard deviation of the mean?
→ According to the Empirical Rule: 68%

4. Within two standard deviations of the mean?
→ 95%

5. Within three standard deviations of the mean?
→ 99.7%

---

Part 2: Biology Test Scores

Given:
- Mean (μ) = 70
- Standard deviation (σ) = 5
- Total students = 2000

We'll label the normal curve with the mean and three standard deviations from the mean:

```
Mean = 70
One SD below = 70 - 5 = 65
Two SD below = 70 - 10 = 60
Three SD below = 70 - 15 = 55
One SD above = 70 + 5 = 75
Two SD above = 70 + 10 = 80
Three SD above = 70 + 15 = 85
```

So the labeled curve looks like:

```
55 60 65 70 75 80 85
|-----|-----|-----|-----|-----|-----|
-3σ -2σ -1σ μ +1σ +2σ +3σ
```

Now let’s answer each question using the Empirical Rule and z-scores.

---

a) What percentage of scores are between 65 and 75?

- 65 = 70 - 5 = μ - σ
- 75 = 70 + 5 = μ + σ
- So this is within one standard deviation of the mean.

→ 68% of scores are between 65 and 75.

✔ Answer: 68%

---

b) What percentage of scores are between 60 and 70?

- 60 = 70 - 10 = μ - 2σ
- 70 = μ

So we're looking at the area from μ - 2σ to μ.

From the Empirical Rule:
- 95% of data lies within ±2σ → so 95% between 60 and 80.
- Since it's symmetric, half of that (47.5%) is between 60 and 70.

→ 47.5%

✔ Answer: 47.5%

---

c) What percentage of scores are between 60 and 85?

- 60 = μ - 2σ
- 85 = μ + 3σ

We need the area from μ - 2σ to μ + 3σ

Break it down:
- From μ - 2σ to μ: 47.5% (as in part b)
- From μ to μ + 3σ: 49.85% (since 99.7% is within ±3σ → half is 49.85%)

Total = 47.5% + 49.85% = 97.35%

✔ Answer: 97.35%

---

d) What percentage of scores is less than 55?

- 55 = μ - 3σ

From the Empirical Rule:
- 99.7% of data is within ±3σ → so 0.3% is outside
- Half of that is below μ - 3σ → 0.3% / 2 = 0.15%

✔ Answer: 0.15%

---

e) What percentage of scores is greater than 80?

- 80 = μ + 2σ

From Empirical Rule:
- 95% within ±2σ → so 5% outside
- Half of that is above μ + 2σ → 5% / 2 = 2.5%

✔ Answer: 2.5%

---

f) Approximately how many biology students scored between 60 and 70?

From part (b), we know 47.5% of scores are between 60 and 70.

Total students = 2000

So:
> 0.475 × 2000 = 950 students

✔ Answer: 950 students

---

g) Approximately how many biology students scored between 55 and 60?

- 55 = μ - 3σ
- 60 = μ - 2σ

We want the area between μ - 3σ and μ - 2σ.

From earlier:
- Below μ - 3σ: 0.15%
- Below μ - 2σ: 2.5% (since 95% within ±2σ → 2.5% below)

So between μ - 3σ and μ - 2σ:
> 2.5% - 0.15% = 2.35%

Now calculate number of students:
> 0.0235 × 2000 = 47 students

✔ Answer: 47 students

---

✔ Final Answers Summary:

1. Below the mean: 50%
2. Above the mean: 50%
3. Within one SD: 68%
4. Within two SDs: 95%
5. Within three SDs: 99.7%

6. Biology test questions:
a) 65–75: 68%
b) 60–70: 47.5%
c) 60–85: 97.35%
d) Less than 55: 0.15%
e) Greater than 80: 2.5%
f) Between 60 and 70: 950 students
g) Between 55 and 60: 47 students

Let me know if you'd like a visual sketch or z-score calculations!