https://file.aiquickdraw.com/mathbot/file/1729722545195ck7xiixv.jpg

Let's analyze the image to help you with the problem shown.

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The visible steps in your problem relate to **hypothesis testing** and involve the following:

### **Step 1: State the Hypothesis**
- This part requires identifying the **null hypothesis (H₀)** and **alternative hypothesis (H₁)**.
    - **H₀:** μ = 175 (Null hypothesis states the mean is 175)
    - **H₁:** μ ≠ 175 (Alternative hypothesis states the mean is not 175)

### **Step 2: Identify the Sampling Distribution**
- The population standard deviation (σ) is **$37**.
- The **sampling distribution** is modeled as **normally distributed** with:
  - **Mean (μ)** = 175
  - **Standard Error (SE)** = 5.78

#### **Standard Error Formula:**
\[
SE = \frac{\sigma}{\sqrt{n}}
\]
where:
- σ = 37
- n = sample size

To ensure accuracy, the sample size (n) isn’t visible in this portion. If you know the value of **n**, I can confirm the SE calculation.

### **Step 3: Find the Probability**
This step involves calculating the **z-score** to find the probability:
\[
Z = \frac{\bar{X} - \mu}{SE}
\]
- From the image:
  - **Sample mean (X̄)** = 159.4878
  - Population mean (μ) = 175
  - **Standard Error (SE)** = 5.78

#### **Calculating Z:**
\[
Z = \frac{159.4878 - 175}{5.78}
\]
\[
Z = \frac{-15.5122}{5.78} \approx -2.68
\]

### **Finding the Probability**
- You need to find **P(Z ≤ -2.68)** (using the standard normal distribution).
- Using Z-tables or a calculator:
  - **P(Z ≤ -2.68)** ≈ 0.0037

Thus, the probability is approximately **0.0037**, which suggests that observing a sample mean of **159.49** or lower is very unlikely if the true population mean is **175**.

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Based on this probability, the statistician may have enough evidence to **reject the null hypothesis** (H₀) if using a significance level of **α = 0.05** since 0.0037 < 0.05.

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Do you need further clarification on any step?  
Here are **5 related questions** you might explore:
1. How do you interpret a Z-score in hypothesis testing?
2. What is the meaning of the standard error of the mean (SE)?
3. How does the sample size affect the standard error?
4. How do you decide between a one-tailed and two-tailed test?
5. What is the impact of different significance levels (e.g., 0.01 vs. 0.05)?

**Tip:** When in doubt, always visualize your Z-score on the standard normal curve to better understand where your result lies.

Do you think the statistician should reject the mayor's claim? Why or why not?

This problem involves hypothesis testing to determine whether a statistician should reject the mayor's claim based on a sample mean, standard error, and Z-score. The solution explores key concepts such as normal distribution, Z-scores, and probability in the context of hypothesis testing.

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