What is the probability that an SRS of 100 first-time car buyers will purchase a car with a mean price greater than $13,000?
A. 0.0832
B. 0.1576
C. 0.2660
D. 0.3140
5. Suppose the starting annual salary of people in the workforce with a high school diploma or less is N($26,000, $5200).
If an SRS of 50 people is collected, what is the probability their mean starting annual salary is between $27,000 and $28,000?
A. 0.0837
B. 0.1672
С. 0.2504
D. 0.3298

Let's approach each question step-by-step.

### 1. **Probability of an SRS of 100 first-time car buyers purchasing a car with a mean price greater than $13,000**

Let:
- $ \mu = 12,500 $ (population mean price of first-time car buyers)
- $ \sigma = 3000 $ (population standard deviation)
- $ n = 100 $ (sample size)

Since the sample size is large, we can use the Central Limit Theorem to approximate the sampling distribution of the sample mean $ \bar{X} $ as normally distributed with mean $ \mu = 12,500 $ and standard deviation $ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} $.

#### Step 1: Calculate the standard deviation of the sample mean:
\[
\sigma_{\bar{X}} = \frac{3000}{\sqrt{100}} = 300
\]

#### Step 2: Calculate the $ Z $-score for $ \bar{X} = 13,000 $:
\[
Z = \frac{\bar{X} - \mu}{\sigma_{\bar{X}}} = \frac{13,000 - 12,500}{300} = \frac{500}{300} \approx 1.67
\]

#### Step 3: Find the probability $ P(\bar{X} > 13,000) $:
Using standard normal tables, $ P(Z > 1.67) \approx 0.0475 $.

**Answer:** The probability is approximately 0.0475, which does not match the given options exactly. However, let's select the closest option:
- **Correct Answer: None of the given options is a close match for 0.0475.**

---

### 2. **Probability of an SRS of 50 people having a mean starting annual salary between $27,000 and $28,000**

Let:
- $ \mu = 26,000 $ (population mean salary)
- $ \sigma = 5200 $ (population standard deviation)
- $ n = 50 $ (sample size)

The sampling distribution of the sample mean $ \bar{X} $ is approximately normal with mean $ \mu = 26,000 $ and standard deviation $ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} $.

#### Step 1: Calculate the standard deviation of the sample mean:
\[
\sigma_{\bar{X}} = \frac{5200}{\sqrt{50}} \approx 735.71
\]

#### Step 2: Calculate the $ Z $-scores for $ \bar{X} = 27,000 $ and $ \bar{X} = 28,000 $:
For $ \bar{X} = 27,000 $:
\[
Z = \frac{27,000 - 26,000}{735.71} \approx 1.36
\]

For $ \bar{X} = 28,000 $:
\[
Z = \frac{28,000 - 26,000}{735.71} \approx 2.72
\]

#### Step 3: Find the probability $ P(27,000 < \bar{X} < 28,000) $:
Using standard normal tables:
- $ P(Z < 2.72) \approx 0.9967 $
- $ P(Z < 1.36) \approx 0.9131 $

Thus, $ P(27,000 < \bar{X} < 28,000) = 0.9967 - 0.9131 = 0.0836 $.

**Answer:** The probability is approximately 0.0836, which is closest to option A.
- **Correct Answer: A. 0.0837**

---

If you have any questions or want further details, feel free to ask.

---

### Additional Practice Questions
1. What is the probability of selecting a sample with a mean above $12,000?
2. If the sample size is increased to 200, how does it affect the probability calculation in question 1?
3. What if the standard deviation was 4000 instead of 3000 in question 1?
4. How would you calculate a confidence interval for the mean car price?
5. Why does increasing the sample size reduce the standard error?

---

**Tip:** For probability questions, calculating the $ Z $-score is crucial. It standardizes values, allowing us to use the standard normal distribution for probabilities.

What is the probability that an SRS of 100 first-time car buyers will purchase a car with a mean price greater than $13,000?
A. 0.0832
B. 0.1576
C. 0.2660
D. 0.3140

Suppose the starting annual salary of people in the workforce with a high school diploma or less is N($26,000, $5200).
If an SRS of 50 people is collected, what is the probability their mean starting annual salary is between $27,000 and $28,000?
A. 0.0837
B. 0.1672
C. 0.2504
D. 0.3298

Explore the probability that a random sample of 100 car buyers will have a mean purchase price above $13,000 and calculate the probability that a sample of 50 workers with high school diplomas has a mean starting salary between $27,000 and $28,000. This problem covers key concepts in normal distribution and the Central Limit Theorem.

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