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 ($H0$) and alternative ($Ha$) hypotheses.

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

  • Choose a significance level ($eta$) 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. $Z = rac{ar{x} - ext{mean}}{SE}$).

Confidence Intervals (C.I.)

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

  • CI is affected by assumptions such as normality.

  • Check conditions before constructing a valid CI.

Hypothesis Testing via C.I.

  • Set up null ($H0$) and alternative ($Ha$) hypotheses.

  • Construct CI and check if null value is contained.

    • If $H0 ext{ is in CI}$, fail to reject $H0$.

    • If $H0 ext{ is outside CI}$, reject $H0$.

Errors in Hypothesis Testing

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

  • Type II Error (β): Failing to reject $H0$ when $Ha$ is true.

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

Choosing Significance Level ($eta$)

  • 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 $eta$: 0.01 to 0.10.

Hypothesis Testing for Population Means Recap

  • Establish hypotheses:

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

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

  • Calculate point estimate of mean.

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

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