Use the given models
The normal-tail question assumes a standard-normal null statistic. The precision question assumes independent observations with fixed sigma. The correction question assumes valid p-values for a prespecified family. These conditions belong to the exercises and are not claims about observed investment data.
Explain the choice before opening the answer
Identify the planned direction in the first task, the square-root relationship in the second and the family threshold in the third. The explanations distinguish the selected answer from each distractor.
Try the questions
An upper-sided test was planned. Its normal statistic is −2. Which p-value is appropriate?
- A.
About 0.02275
- B.
About 0.97725
- C.
About 0.04550
Answer and explanation
Answer: B
Keep the alternative fixed and calculate its tail using the signed statistic.
- A
This is the lower tail at −2, which addresses the opposite alternative.
- B
The planned upper tail is P(Z ≥ −2), approximately 0.97725.
- C
This is the symmetric two-sided value, not the planned upper test.
Under independent observations with fixed sigma, what sample-size multiplier halves the mean SE?
- A.
Two
- B.
Four
- C.
One half
Answer and explanation
Answer: B
Solve sigma/√n_new = (sigma/√n_old)/2, giving n_new = 4 n_old.
- A
Doubling n multiplies SE by 1/√2, not by 1/2.
- B
The square root of four is two, so sigma/√n is halved.
- C
Reducing n increases SE under the fixed independent model.
A Bonferroni family uses alpha 0.05 and 20 tests. Does raw p = 0.004 meet its rule?
- A.
Yes, because 0.004 is below 0.05
- B.
Yes, because p/20 is 0.0002
- C.
No, because 0.004 exceeds 0.0025
Answer and explanation
Answer: C
Compare raw p with alpha/m or capped m p with alpha, using valid prespecified tests.
- A
This comparison ignores the family adjustment.
- B
Dividing the p-value moves the correction in the wrong direction.
- C
The divided threshold is 0.05/20 = 0.0025; adjusted p is 0.08.
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