**Is there a doctor in the house?** A market research firm reported the mean annual earnings of all family practitioners in the United States was 

$178,258

. A random sample of 

55

 family practitioners in Los Angeles had mean earnings of 

*x*

=$191,790

 with a standard deviation of 

$42,627

. Do the data provide sufficient evidence to conclude that the mean salary for family practitioners in Los Angeles is greater than the national average? Use the 

=α0.01

 level of significance and the 

*P*

-value method with the TI-84 Plus calculator. Compute the value of the test statistic. Round the answer to two decimal places.

To determine if the mean salary for family practitioners in Los Angeles is greater than the national average, we need to perform a hypothesis test. We'll use the following steps:

### **Step 1: State the Hypotheses**
- **Null Hypothesis ($H_0$)**: The mean salary for family practitioners in Los Angeles is equal to the national average.
  \[
  H_0: \mu = 178,258
  \]
- **Alternative Hypothesis ($H_1$)**: The mean salary for family practitioners in Los Angeles is greater than the national average.
  \[
  H_1: \mu > 178,258
  \]

### **Step 2: Set the Significance Level**
Given: $\alpha = 0.01$

### **Step 3: Calculate the Test Statistic**
We will use the $ t $-test for a single mean, as the population standard deviation is not known.

The formula for the test statistic $ t $ is:
\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\]
Where:
- $\bar{x} = 191,790$ (sample mean)
- $\mu = 178,258$ (population mean)
- $s = 42,627$ (sample standard deviation)
- $n = 55$ (sample size)

Substituting the values:
\[
t = \frac{191,790 - 178,258}{\frac{42,627}{\sqrt{55}}}
\]

### **Step 4: Compute the Value of the Test Statistic**
Let's calculate this step-by-step.

1. **Calculate the difference between the sample mean and the population mean:**
   \[
   191,790 - 178,258 = 13,532
   \]
2. **Calculate the standard error:**
   \[
   \text{Standard Error} = \frac{42,627}{\sqrt{55}} \approx \frac{42,627}{7.4162} \approx 5,747.41
   \]
3. **Calculate the test statistic:**
   \[
   t = \frac{13,532}{5,747.41} \approx 2.35
   \]

So, the test statistic $ t \approx 2.35 $.

### **Step 5: Find the p-value**
Using a $ t $-distribution table or calculator with $ n - 1 = 54 $ degrees of freedom:

- For $ t = 2.35 $, we can find the $ p $-value.

Since this is a one-tailed test, and we want the area to the right of $ t = 2.35 $, the corresponding $ p $-value is approximately $ 0.0116 $.

### **Step 6: Make a Decision**
- Compare the $ p $-value with $ \alpha $:
  - $ p = 0.0116 $ is greater than $ \alpha = 0.01 $.

Since the $ p $-value is slightly greater than $ \alpha $, we **fail to reject the null hypothesis**.

### **Conclusion:**
The data do not provide sufficient evidence at the $ \alpha = 0.01 $ level to conclude that the mean salary for family practitioners in Los Angeles is greater than the national average.

### **Do you want more details or have any questions?**

Here are 8 related questions you might explore:

1. What is the difference between a one-tailed and two-tailed test?
2. How do we calculate the degrees of freedom in a $ t $-test?
3. What is the significance level and why is it important?
4. How is the p-value interpreted in hypothesis testing?
5. Can you use a $ z $-test instead of a $ t $-test? If so, when?
6. What is the impact of the sample size on the test statistic?
7. What assumptions are made when conducting a $ t $-test?
8. How would the conclusion change if the significance level were $ \alpha = 0.05 $ instead?

**Tip:** When conducting a hypothesis test, it's essential to ensure that the sample data meets the assumptions required for the test you're performing.

Explore hypothesis testing to determine if the mean salary for family practitioners in Los Angeles is greater than the national average. Calculate the test statistic and interpret the results with a significance level of α = 0.01. Sample mean: $191,790, Population mean: $178,258, Sample standard deviation: $42,627.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation