For this discussion post, we are going to run a hypothesis test using the Z-distribution. Read the following:  

The average salary for a registered nurse in Las Vegas is claimed to be $82,000 with a standard deviation of $14,500. To determine if this information is accurate, we sampled 80 registered nurses working in Las Vegas and find their average salary to be $85,025. Test this at the α = 0.05 level. 

Discussion Prompts

Answer the following questions in your initial post: 

What are the hypotheses based on the words given in the problem? 
What is our Z?
What is the P-value?
Based on your p-value and alpha, what conclusion will we make? Do we have evidence that the claim is false?
Are there any other variables that could potentially impact the salary of a registered nurse? What are some methods of sampling we can use to ensure we have a representative population.

### Hypothesis Testing Using the Z-distribution

**Given Data:**
- Population mean, $ \mu_0 = 82,000 $
- Population standard deviation, $ \sigma = 14,500 $
- Sample size, $ n = 80 $
- Sample mean, $ \bar{x} = 85,025 $
- Significance level, $ \alpha = 0.05 $

### 1. **Formulating Hypotheses**

We want to test whether the average salary of registered nurses in Las Vegas is different from $82,000. This is a two-tailed test.

- **Null Hypothesis (H₀):** $ \mu = 82,000 $ (The average salary is $82,000, as claimed.)
- **Alternative Hypothesis (H₁):** $ \mu \neq 82,000 $ (The average salary is different from $82,000.)

### 2. **Finding the Z-score**

The Z-score formula for hypothesis testing is:

\[
Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Plugging in the values:

\[
Z = \frac{85,025 - 82,000}{\frac{14,500}{\sqrt{80}}}
\]

Let's calculate this:

\[
Z = \frac{3,025}{\frac{14,500}{8.944}} = \frac{3,025}{1,621.19} \approx 1.87
\]

### 3. **P-value Calculation**

Since this is a two-tailed test, we need to find the area under the Z-distribution for $ |Z| = 1.87 $. Using standard Z-tables, or a normal distribution calculator, we find the probability for Z = 1.87 is about 0.0308 for one tail.

For a two-tailed test:

\[
P\text{-value} = 2 \times 0.0308 = 0.0616
\]

### 4. **Conclusion Based on P-value and Alpha**

- $ P\text{-value} = 0.0616 $
- $ \alpha = 0.05 $

Since the p-value (0.0616) is greater than the significance level $ \alpha = 0.05 $, we **fail to reject the null hypothesis**.

### **Conclusion:**
There is **not enough evidence** to conclude that the average salary for registered nurses in Las Vegas is different from $82,000 at the 5% significance level. Hence, the claim cannot be considered false based on this sample data.

### 5. **Other Variables Affecting Salary**

Several factors could affect the salary of a registered nurse, such as:
- Experience level
- Type of healthcare facility (e.g., hospital, private clinic)
- Shift differentials (e.g., night shifts)
- Education level or specializations
- Geographic location within Las Vegas

### **Methods of Sampling to Ensure Representativeness**

1. **Simple Random Sampling:** Every nurse has an equal chance of being selected.
2. **Stratified Sampling:** Divide the nurses into subgroups (e.g., by experience or department) and sample proportionally from each.
3. **Systematic Sampling:** Choose every nth nurse from a list.
4. **Cluster Sampling:** Randomly select entire departments or facilities.
5. **Convenience Sampling (less ideal):** Choose nurses that are easiest to access, though this can introduce bias.

### **Tip:**  
When using a two-tailed test, always remember to double the one-tailed p-value to account for both directions of the test.

---

**Related Questions:**
1. How do you calculate the confidence interval for this sample mean?
2. What would happen if the sample size were smaller, say 30? Would it affect the Z-distribution?
3. How can increasing the significance level (e.g., $ \alpha = 0.10 $) change our conclusions?
4. What is the critical value of Z at the 5% significance level for a two-tailed test?
5. What are some non-parametric alternatives to the Z-test if the salary data is not normally distributed?

For this discussion post, we are going to run a hypothesis test using the Z-distribution. Read the following:

The average salary for a registered nurse in Las Vegas is claimed to be $82,000 with a standard deviation of $14,500. To determine if this information is accurate, we sampled 80 registered nurses working in Las Vegas and find their average salary to be $85,025. Test this at the α = 0.05 level.

What are the hypotheses based on the words given in the problem?
What is our Z?
What is the P-value?
Based on your p-value and alpha, what conclusion will we make? Do we have evidence that the claim is false?
Are there any other variables that could potentially impact the salary of a registered nurse? What are some methods of sampling we can use to ensure we have a representative population.

Learn how to test the claim that the average salary of registered nurses in Las Vegas is $82,000 using hypothesis testing and the Z-distribution. With a sample of 80 nurses, an average salary of $85,025, and a standard deviation of $14,500, determine the Z-score, p-value, and conclusion at a 0.05 significance level.

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