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1681One-Sample z-Test with Known VolatilityA signal's average daily alpha over 25 days is 0.8, and the daily standard deviation is known to be 2. Test H 0: =0 against a two-sided alternative at the 5% level using a z-test.统计简单derivation未尝试免费1682One-Sided t-Test on a Small SampleFor n=10 independent observations, the sample mean is 1.5 and the sample standard deviation is 2. Test H 0: =0 against H 1: >0 at the 5% level using a t-test.统计简单derivation未尝试免费1683Two-Sample z-Test for Mean DifferenceTwo independent books produce average daily PnL of 102 and 100 over 100 days each. The daily standard deviation is known to be 5 for each book. Test whether the means are equal using a two-sided z-test.统计中等derivation未尝试面试订阅1684Chi-Square Goodness-of-Fit for a DieA six-sided die is rolled 60 times, and the counts are (14,8,11,9,10,8). Test fairness using a chi-square goodness-of-fit test.统计简单derivation未尝试免费1685Power of a One-Sided z-TestConsider testing H 0: =0 versus H 1: >0 with known =10 and sample size n=25 at significance level =0.05. What is the power when the true mean is =4?统计中等derivation未尝试面试订阅1686Sample Size for a z-TestYou want a two-sided z-test of H 0: =0 with known =4 to detect a true mean of 1 at 5% significance and 80% power. Approximately what sample size is needed?统计中等derivation未尝试面试订阅1687Paired t-Test on Trade-Cost DifferencesFor 16 paired executions, the mean cost difference (strategy A minus strategy B) is -1.2 and the sample standard deviation of the differences is 1.5. Test whether strategy A has lower mean cost using a 5% one-sided paired t-test.统计简单derivation未尝试免费1688Chi-Square Test for a Normal VarianceAssume normal data with n=20 and sample variance s 2=9. Test H 0: 2=4 using the classical chi-square variance test.统计中等derivation未尝试面试订阅1689Approximate Fair-Coin Test with a Normal ApproximationA coin is flipped 20 times and lands heads 14 times. Use the normal approximation to test fairness at the 5% two-sided level.统计简单derivation未尝试免费1690Likelihood-Ratio Test for an Exponential RateSuppose X 1,\dots,X 10 \sim Exp ( ) and X i=5. Use a likelihood-ratio test of H 0: =1 against the unrestricted alternative.统计困难derivation未尝试面试订阅1691Confidence Interval and Test DecisionA two-sided test for a coefficient gives z = 1.8. Without doing a separate interval calculation, what can you say about whether the 95% confidence interval contains zero?统计简单essay未尝试免费1692Type I vs Type II Tradeoff from Changing a ThresholdA one-sided z-test for H 0: =0 with n=25 and known variance 1 rejects when X>c. Compare the thresholds c=0.33 and c=0.66 in terms of Type I error and power when the true mean is 0.5.统计困难derivation未尝试面试订阅1693Chi-Square Test of Independence in a 2x2 TableA 2x2 contingency table of outcomes is \begin pmatrix 30 & 20\\ 10 & 40\end pmatrix . Test independence using the classical chi-square test.统计中等derivation未尝试面试订阅1694One-Sided vs Two-Sided p-ValuesA test statistic is z=2.1 under a standard normal null. What are the one-sided and two-sided p-values, approximately?统计简单derivation未尝试免费1695Which Classical Test Should You Use?You have a sample of size 9 from a population you are willing to model as normal, but the population variance is unknown. You want to test whether the mean equals zero. Should you use a z-test or a t-test, and why?统计简单essay未尝试免费1696Expected False Finalists After a Lookback FunnelA desk studies 30 genuinely null signals. For each signal it tries 4 lookback windows, advances the signal if any in-sample p-value is below 10%, and then requires a fresh holdout p-value below 5%. Assuming independence under the null, what is the expected number of false finalists?统计简单derivation未尝试免费1697Any False Sector Winner After Within-Sector MiningA desk has 12 sectors, each containing 5 genuinely null variants. In each sector it keeps only the smallest p-value, and it flags the sector if that winning p-value is below 1%. Assuming independence, what is the probability at least one sector is falsely flagged?统计简单derivation未尝试免费1698Bonferroni Cutoff After Collapsing to Effective FamiliesA research grid contains 60 model variants, but the desk argues they amount to only 15 effectively distinct families. If it wants family-wise error at most 10% using a Bonferroni family-level rule, what p-value cutoff should it apply to each effective family?统计中等derivation未尝试面试订阅1699Sidak Cutoff for 18 Effective Research ChoicesA desk believes its many correlated parameter tweaks boil down to 18 effectively independent research choices. What Sidak per-choice cutoff controls family-wise error at 5%?统计简单derivation未尝试免费1700BH q Range That Keeps Exactly Three FamiliesFive family-level winners have ordered p-values 0.004, 0.011, 0.018, 0.031, and 0.070. For what range of BH target levels q would Benjamini-Hochberg keep exactly the first three discoveries?统计中等derivation未尝试面试订阅