A sample of five "$8$-hour" workdays in a bank had actual lengths (in minutes) of $468, 449, 494, 496, 479$. Compute the sample variance and construct a $95\%$ confidence interval for the population standard deviation $\sigma$, assuming that the workday lengths are normaly distributed.
A study of the diameter at breast height of Acer Saccharinum gave sample variance $s^2 = 50.4\, cm^2$ based on a sample of $22$ observations. Assuming that the sample comes from a normal population, test $H_0: \sigma = 6\, cm$ versus $H_1: \sigma > 6\, cm$ at the $5\%$ level of significance.
Two independent sampling stations are chosen for a study, one located downstream from an acid mine discharge point and the other located upstream. For $12$ samples collected at the downstream station the species diversity index has mean value $\bar x_1 = 3.11$ and st. dev. $s_1=0.771$, while $10$ samples collected at the downstream station had $\bar x_2=2.04$ and $s_2=0.448$. Test $H_0: \mu_1 = \mu_2$ versus $H_1: \mu_1>\mu_2$ at the $5\%$ level of significance assuming that the populations are normally distributed with equal variances. Make sure to draw a conclusion in the context of the problem.
The deterioration of many municipal pipeline networks across the country is a growing concern. The following data on the tensile strength (psi) of pipelines both when a certain fusion process was used and when this process was not used.$$ \begin{array}{c|cccccccccc} \text{No fusion} & 2748 & 2700 & 2655 & 2822 & 2511 & 3149 & 3257 & 3213 & 3220 & 2753\\ \text{Fused} & 3027 & 3356 & 3359 & 3297 & 3125 & 2910 & 2889 & 2902 \end{array} $$Assuming that the two poluations are normally distributed test the hypothesis that there fusion does not affect the tensile strength at the $\alpha = 0.05$ level of significance.