Calculate CI and Test Correlation Under Normality
Overview
This question evaluates competency in statistical inference, covering confidence-interval estimation for a normal mean with unknown variance, hypothesis testing for Pearson correlation (including degrees of freedom and p-value interpretation), and calculation of an upper normal quantile, within the domain of Statistics & Math for data scientist roles. It is commonly asked because it probes understanding of sampling distributions, the t-distribution and degrees-of-freedom concepts, correlation significance testing and p-values, and normal quantiles, reflecting a blend of conceptual understanding and practical application of statistical formulas and numeric computation.
Inference on a Mean, Significance of a Correlation, and a Normal Quantile
Assume standard parametric conditions (normality as stated). Show formulas, identify degrees of freedom where relevant, and give clear numeric answers.
(a) 95% CI for a Normal Mean (σ unknown)
You have a simple random sample of size n = 64 from a normal population with unknown variance. The sample mean is x̄ = 102 and the sample standard deviation is s = 16. Compute a 95% confidence interval for the population mean μ. State the exact formula you use and the t critical value's degrees of freedom.
(b) Test Significance of a Correlation
You measure two variables X and Y on n = 100 observations and obtain a sample Pearson correlation r = 0.25. Test H0: ρ = 0 versus H1: ρ ≠ 0 at α = 0.05. Show the test statistic, its degrees of freedom, the two-sided p-value, and your conclusion.
(c) Upper 2.5% Threshold of a Normal Process
A process is normally distributed with mean μ = 50 and standard deviation σ = 5. Find the threshold t such that only the top 2.5% of items exceed t, and interpret this t in context.
Solution
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