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The question in the image asks you to select the appropriate **test statistic** for a hypothesis test based on different sampling scenarios, specifically focusing on testing the **population mean**. Let's break down the steps for each part of the question:

### General Problem Setup:
- The hypothesis test concerns whether the population mean daily profit of a convenience store is **less than 495 dollars**.
- A **sample** of daily profits is drawn, with a mean of **503 dollars** and a standard deviation of **76 dollars**.

### Part (a) - Sample size 14, normally distributed population with **unknown** standard deviation:
When the population standard deviation is **unknown**, and the sample size is **small** (n < 30), the **t-test** should be used. Therefore:
- **Test statistic**: t
- **Formula**: 
\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\]
Where:
  - \(\bar{x} = 503\) (sample mean)
  - \(\mu = 495\) (hypothesized population mean)
  - \(s = 76\) (sample standard deviation)
  - \(n = 14\) (sample size)

### Part (b) - Sample size 110, **non-normally distributed** population with **known** standard deviation of 75:
When the sample size is **large** (n > 30), even if the population is non-normal, by the **Central Limit Theorem**, the sampling distribution of the mean is approximately normal. Since the population standard deviation is **known**, we use the **z-test**:
- **Test statistic**: z
- **Formula**: 
\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]
Where:
  - \(\bar{x} = 503\) (sample mean)
  - \(\mu = 495\) (hypothesized population mean)
  - \(\sigma = 75\) (population standard deviation)
  - \(n = 110\) (sample size)

Would you like me to calculate the actual test statistics for both parts?

---

Here are 5 related questions to help expand on this:

1. How do we determine whether to use the **z-test** or **t-test** in hypothesis testing?
2. Why is the **Central Limit Theorem** important in large sample hypothesis testing?
3. What assumptions do we make when using the **t-test**?
4. How do we interpret the p-value in hypothesis testing?
5. What is the difference between a **one-tailed** and **two-tailed** test?

**Tip:** The **t-distribution** is more spread out than the **z-distribution** when sample sizes are small, which accounts for the increased uncertainty in estimating the population standard deviation.

We want to conduct a hypothesis test of the claim that the population mean daily profit of a convenience store is less than 495 dollars. The sample has a mean of 503 dollars and a standard deviation of 76 dollars. For each sampling scenario, choose an appropriate test statistic and calculate the value.

Learn how to perform hypothesis tests for a population mean using t-tests and z-tests. The problem includes scenarios with small and large sample sizes, as well as unknown and known population standard deviations. Apply the formulas and interpret results step-by-step.

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