To solve for the mean ($\mu$) of the test, let's analyze the problem systematically:

Step 1: Understanding the normal distribution

• A score of 77 is two standard deviations above the mean, meaning: $77 = \mu + 2\sigma$
• A score of 61 is two standard deviations below the mean, meaning: $61 = \mu - 2\sigma$

Here, $\sigma$ represents the standard deviation.

---

Step 2: Set up the equations

From the two equations above:

1. $77 = \mu + 2\sigma$
2. $61 = \mu - 2\sigma$

---

Step 3: Eliminate $\sigma$

Add the two equations together to eliminate $\sigma$: $(77) + (61) = (\mu + 2\sigma) + (\mu - 2\sigma)$ $138 = 2\mu$

---

Step 4: Solve for $\mu$

Divide both sides by 2: $\mu = \frac{138}{2} = 69$

---

Final Answer:

The mean of the test is 69.

---

Would you like further clarification or a visual explanation? 😊

Related Questions:

1. How do you find the standard deviation ($\sigma$) using the same data?
2. What are the properties of a normal distribution?
3. How do z-scores relate to standard deviations in a normal distribution?
4. Can you verify the solution by substituting $\mu$ back into the equations?
5. How do you calculate probabilities using a normal distribution?

Tip:

In a normal distribution, approximately 68% of the data falls within one standard deviation, 95% within two, and 99.7% within three!