To address this problem, let’s break it down:

The question asks for determining whether the statements are true or false regarding the estimation of $p$, the population proportion of successes. The population proportion $p$ is estimated using the point estimate, which is the sample proportion.

The formula for the sample proportion is:

$\hat{p} = \frac{x}{n}$

Where:

• $x$ = the number of successes in the sample
• $n$ = the sample size

Now, let’s evaluate each option:

1. Option A: "False, to estimate the value of $p$, use the point estimate $p = x$."

   • This is incorrect because $\hat{p}$ is not equal to $x$, but rather $\hat{p} = \frac{x}{n}$.

2. Option B: "False, to estimate the value of $p$, use the point estimate $\hat{p} = \frac{x}{n}$."

   • This is the correct formula for the sample proportion, so this statement should be True, not false.

3. Option C: "True, to estimate the value of $p$, use the point estimate $\hat{p} = x$."

   • This is incorrect because the correct formula is $\hat{p} = \frac{x}{n}$, not $\hat{p} = x$.

4. Option D: "False, to estimate the value of $p$, use the point estimate $\hat{p} = \frac{n}{q}$."

   • This is incorrect because $\frac{n}{q}$ is not related to the formula for the sample proportion.

Correct Answer:

The correct response based on the options provided is Option B should be True, but the statement of 'False' makes it misleading. Adjust the problem statement as needed.

Would you like to further discuss proportions or need help calculating examples?

Expanding Questions:

1. How is the formula $\hat{p} = \frac{x}{n}$ derived?
2. What are common errors made when calculating sample proportions?
3. How do you interpret the difference between sample proportion and population proportion?
4. Can you explain why options C and D are mathematically incorrect?
5. How does the sample size $n$ influence the accuracy of $\hat{p}$ as an estimate of $p$?

Tip:

Always verify formulas and their interpretations carefully in statistics problems, especially when dealing with sample versus population parameters!