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The problem shown in the image involves calculating certain statistics for the sampling distribution of the proportion of defective light bulbs in a sample size of 4400 bulbs, where the true defect rate is 1%. Let's address each part:

### Part (a) - Find the mean of \(\hat{p}\)

The mean of the sampling distribution of the proportion \(\hat{p}\) is given by the true population proportion \(p\). Since the defect rate is 1%, the mean of \(\hat{p}\) is:
\[ \mu_{\hat{p}} = p = 0.01 \]

### Part (b) - Find the standard deviation of \(\hat{p}\)

The standard deviation of the sampling distribution of the proportion \(\hat{p}\), denoted as \(\sigma_{\hat{p}}\), is calculated using the formula:
\[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \]
where \(p\) is the true proportion of defects (0.01) and \(n\) is the sample size (4400). Plugging in the values:
\[ \sigma_{\hat{p}} = \sqrt{\frac{0.01 \times 0.99}{4400}} \]

### Part (c) - Compute an approximation for \(P(\hat{p} \geq 0.0125)\)

This is a probability calculation that \(\hat{p}\), the sample proportion, is at least 0.0125. Assuming the sampling distribution is approximately normal, we use the standard normal distribution after standardizing \(\hat{p}\).

1. Calculate the z-score:
\[ Z = \frac{0.0125 - \mu_{\hat{p}}}{\sigma_{\hat{p}}} \]

2. Use the z-score to find the probability from standard normal tables or a calculator.

Let's compute these values now.Here are the calculated results:

### Part (b) - Standard deviation of \(\hat{p}\)
The standard deviation of \(\hat{p}\) is approximately \(0.0015\).

### Part (c) - Approximation for \(P(\hat{p} \geq 0.0125)\)
The probability that the sample proportion of defective bulbs \(\hat{p}\) is at least 0.0125 is about \(0.0478\) (rounded to four decimal places).

Would you like more details on any of these calculations, or do you have any other questions?

Here are some follow-up topics we could discuss:
1. What changes if the defect rate differs from the 1% assumption?
2. How does the sample size affect the standard deviation of \(\hat{p}\)?
3. What would the probability be if a different threshold, say 1.5%, was used instead of 1.25%?
4. Can you compare this situation to a different sampling scenario with a larger or smaller sample size?
5. What other statistical measures could be relevant when analyzing defects in manufacturing?

**Tip:** Always check the normality assumption in real-world scenarios, especially when dealing with proportions near 0 or 1, or when the sample size is small.

The records of a light bulb manufacturer show that, when the manufacturing machinery is working correctly, the defect rate is 1%. The manufacturer's control department periodically tests samples of the bulbs, and when 1.25% or more are defective, they call repair technicians for service. The control department is going to take a random sample of 4400 light bulbs. Let p̂ be the proportion of defective light bulbs in the sample. Answer the following: (a) Find the mean of p̂. (b) Find the standard deviation of p̂. (c) Compute an approximation for P(p̂ ≥ 0.0125), which is the probability that repair technicians will be called.

Calculate the mean, standard deviation, and probability for the proportion of defective light bulbs in a random sample of 4400 bulbs. Learn how to find P(p̂ ≥ 0.0125) using normal approximation, suitable for high school and college students.

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