Final Exam

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Point Estimation

Terms & Definitions

• Parameter (\theta) = unknown population value (like \mu, \sigma^2)

• Estimator (\hat{\theta}) = formula to estimate a parameter using sample data

• Point estimate = specific calculated value of \hat{\theta}

Key formulas

• Example estimators:

  \hat{\mu} = \bar{X}, \quad \hat{\sigma}^2 = s^2

Cheat Note

“A point estimate is just your best guess for an unknown number in the population.”

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Bias & Efficiency

Terms & Definitions

• Unbiased estimator → its average value equals the true parameter:

  E[\hat{\theta}] = \theta

• Efficiency → among unbiased estimators, the one with smallest variance is most efficient.

Cheat Note

“Unbiased = hits the target on average. Efficient = less bouncing around.”

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Standard Error (s.e.)

Definition

• The standard deviation of an estimator:

  s.e.(\hat{\theta}) = \sqrt{Var(\hat{\theta})}

Key formula (mean)

s.e.(\bar{X}) = \frac{\sigma}{\sqrt{n}} \quad \text{(or } \frac{s}{\sqrt{n}} \text{ if } \sigma \text{ unknown)}

Cheat Note

“How much my estimate would wiggle if I repeated the study many times.”

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Confidence Interval (C.I.)

Definition

• An interval that will capture the true parameter about 1 - \alpha of the time.

Key formula

\hat{\theta} \pm C \cdot s.e.(\hat{\theta})

For mean (known \sigma):

\bar{X} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}

For mean (unknown \sigma):

\bar{X} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}

Cheat Note

“A plausible range where the true value probably lies. Wider → less precise.”

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Sample Size Planning

Key formula

n = \left( \frac{z_{\alpha/2} \cdot \sigma}{e} \right)^2

Where e = desired margin of error.

Cheat Note

“Want more precise estimates? Increase n.”

🔎 Chapter 10: Hypothesis Testing Basics

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Hypotheses

Terms

• H_0 = null hypothesis (status quo, no effect)

• H_a = alternative hypothesis (what you’re testing for)

Cheat Note

“H_0: nothing’s changed. H_a: something’s going on.”

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Types of Tests

• Two-sided → testing for any change (\neq)

• One-sided lower → testing if smaller (<)

• One-sided upper → testing if bigger (>)

Cheat Note

“Pick the tail based on the question direction.”

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Errors