  The population mean annual salary for plumbers, is $46,700. A random sample of 42 plumbers is drawn from this population. What is the probability that the mean salary of the sample is less than $44,000? Assume σ = $5600.

To find the probability that the mean salary of a sample of plumbers is less than $44,000, we need to use the concept of the sampling distribution of the sample mean. Given that the sample size $ n = 42 $ is sufficiently large, we can assume that the sample mean follows a normal distribution due to the Central Limit Theorem.

### Step-by-Step Solution:

1. **Identify the given information:**

   - Population mean salary, $ \mu = 46,700 $
   - Population standard deviation, $ \sigma = 5600 $
   - Sample size, $ n = 42 $
   - Sample mean salary, $ \bar{x} = 44,000 $

2. **Calculate the standard error of the mean (SEM):**

   The standard error of the mean is calculated using the formula:

   \[
   \text{SEM} = \frac{\sigma}{\sqrt{n}}
   \]

   Substituting the given values:

   \[
   \text{SEM} = \frac{5600}{\sqrt{42}}
   \]

3. **Compute the z-score:**

   The z-score measures how many standard errors the sample mean is away from the population mean. It is calculated using the formula:

   \[
   z = \frac{\bar{x} - \mu}{\text{SEM}}
   \]

   Substituting the given values:

   \[
   z = \frac{44,000 - 46,700}{\text{SEM}}
   \]

4. **Find the probability corresponding to the z-score:**

   Once we have the z-score, we can use the standard normal distribution to find the probability that the sample mean is less than $44,000.

Let's calculate these values step-by-step.Here are the calculated values:

1. **Standard Error of the Mean (SEM):**  
   \[
   \text{SEM} \approx 864.10
   \]

2. **Z-score:**  
   \[
   z \approx -3.12
   \]

3. **Probability that the mean salary of the sample is less than $44,000:**  
   \[
   P(\bar{x} < 44,000) \approx 0.00089
   \]

### Conclusion:

The probability that the mean salary of a random sample of 42 plumbers is less than $44,000 is approximately **0.089%**.

Would you like to know more details, or do you have any questions?

### Related Questions to Explore:

1. What is the probability that the mean salary of a sample is more than $48,000?
2. How does the sample size affect the standard error of the mean?
3. What is the 95% confidence interval for the sample mean salary?
4. If the population standard deviation were different, how would that affect the probability?
5. How would you interpret the z-score of -3.12 in terms of standard deviations from the mean?

**Tip:** Always check the assumptions (like normality and sample size) when applying statistical methods!

The population mean annual salary for plumbers, is $46,700. A random sample of 42 plumbers is drawn from this population. What is the probability that the mean salary of the sample is less than $44,000? Assume σ = $5600.

Discover how to calculate the probability that the mean salary of a sample of plumbers is less than $44,000. This problem involves key statistical concepts like the sampling distribution of the mean, Z-scores, and the Central Limit Theorem. Population mean is $46,700, standard deviation is $5600, and the sample size is 42.

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