Estimation

\boxed{ \text{Estimators: } \begin{cases} \bar{X} &= \hat{\mu} \\ s &= \hat{\sigma} \end{cases} } \\ \tiny \textit{sample mean estimates $\mu$; } \textit{sample sd estimates $\sigma$: }

Vocab

Estimation: Using a sample statistic to predict the value of unknown population parameter(s).

• Estimator: The sample statistic used for the estimation.
• Estimate: The value of the estimation.

Point Estimate: Estimate of a population parameter that is a single numerical value.

Interval Estimate: Interval around the point estimate likely to contain the corresponding population parameter.

Standard Error

\boxed{ \text{Standard Error: } SE_{\bar{X}} = \frac{s}{\sqrt{n}} }

On Sample Error v. Sample Deviation:

• SE measures uncertainty in a sample statistic.
• SD measures dispersion of data.

On n

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

• Raising n reduces size of error margin.
  • Increasing n can be costly or unethical.
• If getting n for a SE, round n up.

Confidence Intervals of \mu

\boxed{ \text{C.I. of $\mu$: } \bar{X} \pm t_{\alpha / 2, n - 1} \frac{s}{\sqrt{n}} } \\ \small\textit{when $\sigma$ is known}

More on t-distribution:

• Heavier tails than z.
• Approaches normal (z) as n grows.
• Uses df (degree of freedom)

On Empirical Rule v. Confidence Interval

• CI: “We’re X% confidence that \mu is between x and y.”
• Empirical Rule: “X% of data is between x and y.”
  • Wider than CI.

Hypothesis

Null Hypothesis (H_0): Presumed to be true initially.

Alternative Hypothesis (H_1): \bar{H_0}, what we hope to prove.

Note: Write conclusions in terms of H_0

• e.g., “accept H_0” or “reject H_0”

On Types of Errors:

Verdict	H_0 is true	H_1 is true
Accept H_0	Okay	Type II Error
Reject H_0	Type I Error	Okay

\text{What $\alpha$ and $\beta$ mean: } \begin{cases} \alpha &= P(\text{committing Type I error}) \\ \beta &= P(\text{committing Type II error}) \end{cases}

Hypothesis Testing for \mu (\sigma unknown)

1. Define \mu, H_0, H_1, and \alpha
   • (aka: population parameter, hypotheses, and level of significance)
2. Check if t-test can be used:
   1. Sample is from normal distribution, or
   2. CLT can be used.
3. Compute \bar{X} and s
4. Compute T

\boxed{ \text{Test Statistic: } T = \frac{ \bar{X} - c }{ s / \sqrt{n} } \sim t(n-1) } \\ \small\textit{under $H_0 : \mu = c$}

5. Compare T with critical values to make decision

\boxed{ \text{Reject $H_0$ if: } \begin{cases} H_1 : u \ne c &\quad |T| > t_{\alpha / 2 , n - 1} &\quad \text{(two-tailed)} \\ H_1 : u > c &\quad T > t_{\alpha , n - 1} &\quad \text{(right-tailed)} \\ H_1 : u < c &\quad T < - t_{\alpha , n - 1} &\quad \text{(left-tailed)} \end{cases} }

On Telling Tails:

• Check H_1 against the above cases (or, draw the curve)

On Two-Tail Tests and Confidence Intervals:

• CI and two-tailed tests use the same quantities.
• If H_0’s \mu \in CI, then accept H_0.
  • (as long as the CI and two-tailed test have same \alpha)

P-Value

p-value: Measure of evidence against H_0

• aka: tail area in the direction of H_1
• Smaller p-value means more evidence against H_0

\boxed{ \begin{aligned} \text{p-value: }& \begin{cases} H_1 : u \ne c &\quad \text{p-value} = 2 p(t > |T|) \\ H_1 : u > c &\quad \text{p-value} = p(t > T) \\ H_1 : u < c &\quad \text{p-value} = p(t < T) \end{cases} \\~\\ &\qquad \text{if p-value < $\alpha$, reject $H_0$} \end{aligned} }