Hypothesis Testing Cont'd

Hypothesis Testing Overview

• Define the research question and parameter of interest.

• Decide on one-sided or two-sided test.

• Establish null ($H<em>0$) and alternative ($H</em>a$) hypotheses.

• Identify the appropriate test statistic or point estimate; check assumptions.

• Choose a significance level ($\beta$) and test hypothesis; methods include:

  • P-value vs. significance level.

  • Z-score vs. critical value.

  • Confidence interval.

• Make decisions based on results and interpret in context.

Key Concepts in Hypothesis Testing

• Decision Rules: Compare P-value to significance level.

  • One-sided test: $P$ value = $Pr(Z > |z|)$

  • Two-sided test: $P$ value = $2 * Pr(Z > |z|)$

• Use of Z-score vs. critical value (e.g. $Z = rac{\bar{x} - ext{mean}}{SE}$).

Confidence Intervals (C.I.)

• For a 95% CI: $(\bar{x} - 1.96 * SE, \bar{x} + 1.96 * SE)$

• CI is affected by assumptions such as normality.

• Check conditions before constructing a valid CI.

Hypothesis Testing via C.I.

• Set up null ($H<em>0$) and alternative ($H</em>a$) hypotheses.

• Construct CI and check if null value is contained.

  • If $H<em>0 ext{ is in CI}$, fail to reject $H</em>0$.

  • If $H<em>0 ext{ is outside CI}$, reject $H</em>0$.

Errors in Hypothesis Testing

• Type I Error (α): Rejecting $H_0$ when it is true.

• Type II Error (β): Failing to reject $H<em>0$ when $H</em>a$ is true.

• Balancing error rates is crucial; often Type I is considered more serious.

Choosing Significance Level ($\beta$)

• Commonly set at 0.05; adjust based on consequences:

  • If Type I error is costly, lower significance level (e.g., 0.01).

  • If Type II error is more critical, higher significance level (e.g., 0.10).

• Typical range for $\beta$: 0.01 to 0.10.

Hypothesis Testing for Population Means Recap

• Establish hypotheses:

  • Null: $H_0: \bar{x} = ext{null value}$

  • Alternative: $H_a: \bar{x} >$, $< $, or $<br>eq$ null value.

• Calculate point estimate of mean.

• Check assumptions (independence, sample size $<br>\geq 30$ if data is skewed).

• Compute z-score, p-value, or CI as needed to perform the test, and conclude accordingly.