Critical values will depend on the significance level that is selected.

c08.p077n.q014

Question

Critical values will depend on the significance level that is selected.

c08.p077n.q014

BlackTom AI · Accepted Answer

Critical values are the cutoff points on the sampling distribution that separate the acceptance region from the rejection region; they are determined by the chosen significance level (alpha) and the test’s directionality and distribution.

Relevant concepts:
- Significance level (α) is the probability of a Type I error (rejecting a true null hypothesis).
- The critical region must have probability α under the null; for a two-tailed test this probability splits into two tails (α/2 each), for a one-tailed test it sits entirely in one tail.
- The sampling distribution (e.g., standard normal Z, Student’s t with df) determines the specific numerical critical value corresponding to the tail area.

Step-by-step reasoning and example calculations:
1. For a two-tailed Z-test with α = 0.05: split α into two tails, each with area 0.025. The critical values are the z-scores with cumulative probabilities 0.025 and 0.975. Using the standard normal inverse:
   - z0.975 ≈ 1.95996 → commonly rounded to ±1.96.
   Thus critical values = −1.96 and +1.96.
2. For a two-tailed Z-test with α = 0.01: each tail has 0.005. z0.995 ≈ 2.5758 → critical values ≈ ±2.576.
3. For a one-tailed Z-test with α = 0.05: the critical z satisfies cumulative probability 0.95, z0.95 ≈ 1.645 (only one cutoff).

These calculations show that changing α from 0.05 to 0.01 moves the critical values from about ±1.96 to about ±2.576 (two-tailed), and using one-tailed vs two-tailed changes the numerical cutoff as well. Additionally, using a t-distribution (small sample sizes) yields different critical values that depend on degrees of freedom.

Why the correct answer is True: since the critical value is defined to produce a tail area equal to α (or α/2), any change in the significance level directly changes the critical cutoff. The statement is therefore True.

Other option (False) is incorrect because it denies the explicit dependence of critical values on α (and on test type/distribution). Confirmed final answer: True.

Critical values will depend on the significance level that is selected. c08.p077n.q014判断题

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