The records of a light bulb manufacturer show that, when the manufacturing machinery is working correctly, the defect rate (due to imperfections in the material used) is 1%. The manufacturer's control department periodically tests samples of the bulbs, and when 1.5% 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 p be the proportion of defective light bulbs in the sample assuming the machinery is working correctly. What is the mean and standard deviation of p? What is the probability that, assuming the machinery is working correctly, the repair technicians will be called?

We can solve this problem using concepts from statistics, specifically dealing with proportions in sampling.

### Step 1: Mean and Standard Deviation of the Proportion \( p \)

Let \( p \) represent the proportion of defective light bulbs in the sample, and assume the machinery is working correctly, so the defect rate is 1% or \( p_0 = 0.01 \).

- The mean \( \mu_p \) of the sample proportion \( p \) is simply the population proportion of defective bulbs:

\[
\mu_p = p_0 = 0.01
\]

- The standard deviation \( \sigma_p \) of the sample proportion \( p \) is given by the formula for the standard deviation of a proportion in a sample:

\[
\sigma_p = \sqrt{\frac{p_0 (1 - p_0)}{n}}
\]

where \( n \) is the sample size. In this case, \( n = 4400 \) and \( p_0 = 0.01 \). Plugging in the values:

\[
\sigma_p = \sqrt{\frac{0.01(1 - 0.01)}{4400}} = \sqrt{\frac{0.01 \times 0.99}{4400}} = \sqrt{\frac{0.0099}{4400}} \approx \sqrt{2.25 \times 10^{-6}} \approx 0.0015
\]

Thus, the mean and standard deviation of \( p \) are:
- \( \mu_p = 0.01 \)
- \( \sigma_p \approx 0.0015 \)

### Step 2: Probability that Technicians Will Be Called

The control department will call the repair technicians if 1.5% or more of the bulbs in the sample are defective, which corresponds to \( p \geq 0.015 \).

We now need to calculate the probability that \( p \geq 0.015 \). Since \( p \) follows a normal distribution (thanks to the Central Limit Theorem), we can standardize the variable \( p \) to compute the probability using the standard normal distribution (z-distribution).

The z-score is given by:

\[
z = \frac{p - \mu_p}{\sigma_p}
\]

For \( p = 0.015 \):

\[
z = \frac{0.015 - 0.01}{0.0015} = \frac{0.005}{0.0015} \approx 3.33
\]

Now, we look up the z-score of 3.33 in standard normal distribution tables or use a calculator to find the corresponding probability. The probability that \( p \) is less than 0.015 corresponds to the cumulative probability for \( z = 3.33 \), which is approximately 0.9996.

Thus, the probability that \( p \geq 0.015 \) (i.e., the technicians are called) is:

\[
P(p \geq 0.015) = 1 - 0.9996 = 0.0004
\]

So, the probability that the technicians will be called is approximately 0.04%.

### Summary
- **Mean of \( p \):** 0.01
- **Standard deviation of \( p \):** 0.0015
- **Probability that the technicians will be called:** 0.0004 or 0.04%

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Would you like further details on any of the steps? 

Here are five related questions to think about:
1. What assumptions allow us to use the normal approximation for this problem?
2. How would the probability change if the sample size were larger?
3. How would the defect rate affect the standard deviation if it were higher?
4. What is the z-score if the repair department sets the threshold at 2% defect rate instead?
5. How could the manufacturer improve the defect rate?

**Tip:** When working with sampling problems, always check if the sample size is large enough to apply the normal approximation for proportions.

The records of a light bulb manufacturer show that, when the manufacturing machinery is working correctly, the defect rate (due to imperfections in the material used) is 1%. The manufacturer's control department periodically tests samples of the bulbs, and when 1.5% 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 p be the proportion of defective light bulbs in the sample assuming the machinery is working correctly. What is the mean and standard deviation of p? What is the probability that, assuming the machinery is working correctly, the repair technicians will be called?

Solve a statistics problem involving defective light bulbs where the defect rate is 1%. Learn how to calculate the mean, standard deviation, and the probability that technicians will be called for service based on a sample size of 4400. This problem uses proportions, the normal distribution, and the Central Limit Theorem. Suitable for high school statistics students.

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