For a two-tailed hypothesis test with a test statistic value of z = 2.09, the p-value is _____.

Question

For a two-tailed hypothesis test with a test statistic value of z = 2.09, the p-value is   _____.

BlackTom AI · Accepted Answer

Question restatement: For a two-tailed hypothesis test with a test statistic value of z = 2.09, the p-value is _____.

Option A: .0404
- Interpretation: This value is close to 0.04, which would correspond to a p-value around 4%.
- Why this is not correct: A z-score of 2.09 corresponds to a tail area of about 0.009 on one side (since 2.09 is beyond the 0.975 quantile but not far beyond). Doubling that tail area for a two-tailed test gives roughly 0.018, which is notably smaller than 0.0404. The mismatch suggests this option overestimates the two-tailed p-value for z = 2.09.

Option B: .0202
- Interpretation: This value is around 2.0%, which would be half of the approximate two-tailed p-value for a z around 2.09.
- Why this is not correct: For a two-tailed p-value, you should take the full combined tail area on both sides. Since the one-tail area is near 0.009, doubling it should yield around 0.018, not 0.0202. The discrepancy arises from the precise tail area calculation, but this option is slightly lower than the expected two-tailed value for z = 2.09.

Option C: .0366
- Interpretation: This value is about 3.66%, which aligns with a two-tailed p-value calculation for a z-score slightly above 2.0.
- Why this is plausible: Using a standard normal table or calculator, Phi(2.09) ≈ 0.9817. The one-tail area is 1 − 0.9817 ≈ 0.0183. Doubling gives 0.0366. This matches the standard two-tailed p-value computation for z = 2.09.

Option D: .4899
- Interpretation: This value is around 0.49, i.e., nearly 50%, which would indicate an almost non-informative test result.
- Why this is not correct: A z-value of 2.09 lies far into the tail; such a large p-value would imply the observed statistic is highly compatible with the null, which is inconsistent with the observed z exceeding 2 standard deviations. This option grossly overstates the p-value and is not consistent with the standard normal distribution tail areas.

Synthesis: To determine the p-value for a two-tailed z-test, compute the one-tail probability P(Z > 2.09) and double it. Using the standard normal distribution, P(Z > 2.09) ≈ 0.0183, and doubling yields approximately 0.0366. Therefore, the option that corresponds to this calculation is the one that matches 0.0366, while the others misestimate the tail areas for this z-score.

For a two-tailed hypothesis test with a test statistic value of z = 2.09, the p-value is _____.单项选择题

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