Calculate CI and Test Correlation Under Normality

You draw a simple random sample of size $n=64$ from a normal population whose variance is unknown. The sample mean is $102$ and the sample standard deviation is $16$. (a) Calculate a 95% confidence interval for the population mean $\mu$; give the exact formula you use and the degrees of freedom for the t critical value. (b) You observe two variables $X$ and $Y$ on $n=100$ observations and obtain a sample Pearson correlation of $r=0.25$. Test $H_0: \rho=0$ against $H_1: \rho \neq 0$ at $\alpha=0.05$. Report the test statistic, its degrees of freedom, the two-sided p-value, and your conclusion. (c) A process follows a normal distribution with mean $50$ and standard deviation $5$. Determine the threshold $t$ such that only the top $2.5\%$ of items exceed $t$, and explain what this $t$ means in context. Overview: This item assesses proficiency in statistical inference, including confidence-interval estimation for a normal mean with unknown variance, hypothesis testing for Pearson correlation (covering degrees of freedom and p-value interpretation), and computation of an upper normal quantile, within the Statistics & Math domain for data scientist positions. It is frequently used because it tests understanding of sampling distributions, the t-distribution and degrees-of-freedom ideas, correlation significance testing and p-values, and normal quantiles, combining conceptual understanding with practical use of statistical formulas and numerical calculation.