Question 1174870
Let's break down how to show that p̂ (p-hat) = X/n is an unbiased estimator of p.

**1. Definitions**

* **X:** The total number of successes in n trials.
* **X ~ b(n, p):** X follows a binomial distribution with n trials and probability of success p.
* **p̂ = X/n:** The estimator for p.
* **Unbiased Estimator:** An estimator is unbiased if its expected value is equal to the true parameter it's estimating. In this case, we need to show E[p̂] = p.

**2. Expected Value of X**

* Since X ~ b(n, p), the expected value of X is:
    * E[X] = np

**3. Expected Value of p̂**

* We want to find E[p̂] = E[X/n].
* Using the linearity of expectation, we can write:
    * E[p̂] = E[X/n] = (1/n) * E[X]

**4. Substitute E[X]**

* Substitute E[X] = np into the equation:
    * E[p̂] = (1/n) * (np)

**5. Simplify**

* E[p̂] = (np)/n
* E[p̂] = p

**Conclusion**

Since E[p̂] = p, we have shown that p̂ = X/n is an unbiased estimator of p. This means that, on average, the sample proportion (p̂) will equal the true population proportion (p).