To solve the problem in the image, let's address each part step by step:

Problem Setup:

• Population defect rate ($p$) = 1% or 0.01.
• Sample size ($n$) = 4400.
• Sample proportion of defects is denoted by $\hat{P}$.

(a) Find the mean of $\hat{P}$:

The mean of the sample proportion ($E[\hat{P}]$) is equal to the population proportion ($p$):

$E[\hat{P}] = p = 0.01$

(b) Find the standard deviation of $\hat{P}$:

The standard deviation of the sample proportion ($\sigma_{\hat{P}}$) is given by:

$\sigma_{\hat{P}} = \sqrt{\frac{p(1-p)}{n}}$

Substitute the values:

$\sigma_{\hat{P}} = \sqrt{\frac{0.01(1 - 0.01)}{4400}} = \sqrt{\frac{0.01 \cdot 0.99}{4400}} = \sqrt{\frac{0.0099}{4400}} \approx \sqrt{0.00000225} \approx 0.0015$

(c) Compute $P(\hat{P} \geq 0.0125)$:

To calculate this, we need the z-score:

$z = \frac{\hat{P} - p}{\sigma_{\hat{P}}}$

Substitute the values:

$z = \frac{0.0125 - 0.01}{0.0015} = \frac{0.0025}{0.0015} \approx 1.67$

Now, find the probability using the z-table or standard normal distribution. The probability that corresponds to $z = 1.67$ is approximately 0.9525.

Thus, the probability of exceeding this is:

$P(\hat{P} \geq 0.0125) = 1 - 0.9525 = 0.0475$

Final Answers:

1. (a) Mean = $0.01$.
2. (b) Standard deviation = $0.0015$.
3. (c) Probability = $0.0475$ (rounded to four decimal places).

Would you like me to explain any part in more detail or provide a breakdown of the z-score calculation? Here are 5 related questions to expand your understanding:

1. What happens to the standard deviation if the sample size increases?
2. How do we interpret the z-score in this problem?
3. What would the probability be for $P(\hat{P} \leq 0.0125)$?
4. Why is the standard error important in sampling distributions?
5. How would the calculation change if $p$ were 2% instead of 1%?

Tip: When dealing with proportions, always ensure the sample size $n$ is large enough for the normal approximation to hold ($np \geq 10$ and $n(1-p) \geq 10$).