Testing of Hypotheses: Null and Alternative Hypotheses, Type I and II Errors, Significance Level, Large Sample Tests (Z-test)

A fundamental theorem of Simple Random Sampling Without Replacement (SRSWOR) states that the probability of any specified unit being selected at the $k$-th draw is identically equal to $\frac{1}{N}$ for all $k = 1, 2, \dots, N$.

Proof: Probability of a specific unit $u_i$ being drawn at the $k$-th draw = $P(\text{not drawn in first } k-1 \text{ draws}) \times P(\text{drawn at } k\text{-th draw | not drawn earlier})$

Analysis of errors in hypothesis testing:

• Incorrect: Rejecting $H_0$ when it is false is a correct decision (Power of the test, $1 - \beta$).
• Incorrect: Rejecting $H_0$ when it is true is Type-I error, not Type-II error.
• Incorrect: Type-II error occurs when one fails to reject (accepts) $H_0$ when it is false.
• Correct: By definition, a Type-I error consists of rejecting the null hypothesis $H_0$ when $H_0$ is actually true. Its probability is denoted by $\alpha$ (level of significance).

Analysis of critical regions in statistical tests:

• Correct: When the alternative hypothesis specifies a value greater than the hypothesized parameter ($H_1: \phi \gt 5$), the rejection region lies entirely in the upper (right) tail of the test statistic's sampling distribution, making it a right-tailed test.
• Incorrect: A left-tailed test corresponds to $H_1: \phi \lt 5$.
• Incorrect: 'Center tailed' is non-standard; tests with $H_1: \phi \ne 5$ are two-tailed.
• Incorrect: 'Cross tailed' is not a statistical concept.

Analysis of p-value decision criteria:
The significance level is $\alpha = 5\% = 0.05$.
The universal statistical decision rule states:
Reject $H_0$ if $p\text{-value} \le \alpha$.
Fail to reject $H_0$ if $p\text{-value} \gt \alpha$.

• Incorrect: $0.15 \gt 0.05$, fails to reject $H_0$.
• Incorrect: $0.10 \gt 0.05$, fails to reject $H_0$.
• Incorrect: $0.06 \gt 0.05$, fails to reject $H_0$.
• Correct: $0.046 \lt 0.05$, which falls strictly within the critical rejection region, leading to the rejection of the null hypothesis $H_0$.