When calculating sample means, the type of command or formula you use depends on whether the population standard deviation ($\sigma$) is known or unknown. Here’s an overview of the approaches for both cases:

---

1. Population Standard Deviation Known

• Use the Z-Test for sample mean: $Z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}$

  • $\bar{X}$: Sample mean
  • $\mu$: Population mean (hypothesized)
  • $\sigma$: Population standard deviation
  • $n$: Sample size

  Purpose: This is used when $\sigma$ is known and the sampling distribution is assumed to be normal (or the sample size is large enough for the Central Limit Theorem to apply).

---

2. Population Standard Deviation Unknown

• Use the T-Test for sample mean: $T = \frac{\bar{X} - \mu}{\frac{s}{\sqrt{n}}}$

  • $\bar{X}$: Sample mean
  • $\mu$: Population mean (hypothesized)
  • $s$: Sample standard deviation
  • $n$: Sample size

  Purpose: This is used when $\sigma$ is unknown, so you estimate it using the sample standard deviation ($s$). The $t$-distribution accounts for extra uncertainty, especially in smaller samples ($n < 30$).

---

Summary of Key Commands or Decisions:

1. Known $\sigma$: Z-test
2. Unknown $\sigma$: T-test
3. Use of Central Limit Theorem:
   • If $n \geq 30$, assume normal distribution regardless of population shape.
4. Paired or Independent Samples:
   • Consider paired $t$-test or two-sample tests depending on data structure.
5. Confidence Intervals:
   • Use Z or T formulas for constructing confidence intervals, depending on whether $\sigma$ is known.

---

Would you like an example or more detailed steps?

Relative Questions:

1. How do Z and T tests differ in practical applications?
2. What conditions ensure the validity of a Z-test?
3. Why does the T-test use degrees of freedom?
4. How does sample size influence the choice between Z and T tests?
5. What happens when neither $\sigma$ nor $s$ is available?

Tip:

Always verify the assumptions of normality and independence in your data before proceeding with Z or T tests.