While the asymptotic (large-sample) mean of Pearson's chi-square test for an $r \times s$ contingency table equals its degrees of freedom $(r - 1)(s - 1)$, J.B.S. Haldane (1939) derived the exact mean under conditional multinomial distribution with fixed marginal totals as:

Notice that as sample size $N \to \infty$, $\frac{N}{N-1} \to 1$, recovering the standard asymptotic mean $(r - 1)(s - 1)$. Thus, Option 3 is correct.

In hypothesis testing:
Type I error: Rejecting $H_0$ when $H_0$ is true. The probability of committing a Type I error is denoted by $\alpha$, which is the level of significance of the test.
Type II error: Failing to reject $H_0$ when $H_0$ is false (probability $\beta$).
Power of the test: $1 - \beta$, the probability of rejecting $H_0$ when $H_0$ is false.
Confidence coefficient: $1 - \alpha$, the probability of accepting $H_0$ when $H_0$ is true.

Calculations:

• Total sum of all 21 numbers $= 21 \times 9 = 189$.
• If the number 19 is removed, the sum of the remaining 20 numbers is:
  $$\text{New Sum} = 189 - 19 = 170$$
• The mean of the remaining 20 numbers is:
  $$\text{New Mean} = \frac{170}{20} = 8.5$$

Two events $A$ and $B$ are mutually exclusive (disjoint) if they cannot occur simultaneously, meaning:

By definition of conditional probability: