Determine the critical value(s) for the following scenario: $H_a:{\mu }_1>{\mu }_2$, $\alpha =0.01$, $n_1=16$, $n_2=10$. Assume the samples are independent, random, from normal populations with equal variances.

Two manufacturing plants produce the same component. A random sample from plant X yields $n_1=25$, ${\bar{x}}_1=45.2$, $s_1=5.4$. A random sample from plant Y yields $n_2=30$, ${\bar{x}}_2=42.6$, $s_2=6.1$. Assume the populations are approximately normal with equal variances. Construct a $95\%$ confidence interval for ${\mu }_1-{\mu }_2$.

Two independent samples are drawn to compare variability.

Sample A: {$10$, $12$, $9$, $11$, $13$, $10$, $14$, $8$, $12$}

Sample B: {$7$, $9$, $6$, $8$, $7$, $9$, $5$, $8$}

Use the "count five" test: compute medians, absolute deviations from medians, maxima of absolute deviations $M_1$, $M_2$, counts $c_1$, $c_2$ where $c_1=\#\{d_{1i}:d_{1i}>M_2\}$ and $c_2=\#\{d_{2j}:d_{2j}>M_1\}$, and reject the null hypothesis (equal variances) if $\max (c_1,c_2)\ge 5$. Which conclusion is correct?

Suppose we are comparing the average daily exercise minutes for individuals in three different age groups. Independent random samples of size $6$ were taken from each age group. The computed test statistic is $F=25.77$ for testing whether the mean daily exercise minutes differ among the three groups. Assume normality of the populations and equal population variances. Use a significance level of $\alpha =0.10$. What conclusion can be drawn about the mean daily exercise minutes across the groups?

A logistics company is comparing the delivery times of two different routes. They believe that route $1$ is generally slower than or equal to route $2$ in terms of average time, but a manager claims that route 1 actually has a higher average delivery time (${\mu }_1>{\mu }_2$).Test the claim at a significance level of $\alpha =0.01$. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal (${\sigma }_1^2={\sigma }_2^2$). Sample statistics are:

Route 1: ${\bar{x}}_1=52.4$, $s_1=4.15$, $n_1=15$

Route 2: ${\bar{x}}_2=47.8$, $s_2=3.90$, $n_2=15$