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1664Weight on the Noisier Signal to Hit a Variance TargetTwo independent unbiased estimators T1 and T2 have variances 1 and 4. In the blend W = wT2 + (1-w)T1, what is the smallest positive w that makes Var(W) exactly 0.9?统计简单derivation未尝试免费1665Interval Where a Fixed Benchmark Beats a Noisy Unbiased SignalA noisy unbiased signal X satisfies X ~ N(theta, 0.25). A fallback benchmark always reports 1.2. For what values of theta does the fixed benchmark have lower MSE than X?统计简单derivation未尝试免费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1711Backtest Slide Misreads p = 0.03A PM deck says: ‘The event-study p-value is 0.03, so there is a 97% probability the signal is real.’ What is the statistical mistake?统计简单essay未尝试免费1712Statistical Significance Is Not Economic SignificanceWith millions of observations, a strategy's average edge is estimated at 0.2 basis points and is highly statistically significant. Why is 'highly significant' not enough to conclude the strategy matters economically?统计简单essay未尝试免费1713No Significance Is Not No EffectA small-cap execution pilot reports p = 0.18 and the note says ‘there is no effect, so we should stop the project.’ Why is that too strong?统计简单essay未尝试免费