State the null hypothesis represented by the following graph. Then, state the alternative hypothesis and sketch its graph.

For a left-tailed $t$-test with $\alpha =0.025$ and a sample size of $n=20$, find the critical value and rejection region.

A quality control manager claims that the variance in the lifespans of a certain type of lightbulb is exactly $36$ $\text{hours}$. To test this claim, a random sample of $18$ bulbs is selected, and the sample variance is found to be $42$ ${\text{hours}}^2$. At the $\alpha =0.05$ significance level, test the manager's claim. Assume the lifespans are normally distributed.

A Kruskal–Wallis test is used to compare the wait time distributions at three different service centers. The calculated test statistic is: $H\approx 8.37$

At the $\alpha =0.05$ significance level, the critical value from the chi-square distribution with $df=2$ is: ${\chi }_{0.05,2}^2=5.991$

Based on this information, should the null hypothesis be rejected?

A researcher investigates how the temperature of an oven (in ${}^{\circ }\text{C}$) affects the time (in minutes) needed to bake a loaf of bread. After collecting data from several test bakes, a regression equation $\hat{y}=80.5-0.12x$ was developed, where $x$ is the oven temperature. What does the symbol $\hat{y}$ represent in this equation?

A researcher collects data on the number of hours spent exercising per week and cholesterol levels for a random sample of $35$ adults. The Spearman rank correlation coefficient calculated from the data is $r_s=-0.38$. At $\alpha =0.10$, is there a significant correlation between hours of exercise and cholesterol levels? Use a two-tailed test.

Given a confidence level of $0.90$ and a sample size of $22$, what are the critical values for the $t$-distribution?

Researchers evaluated two study techniques for passing a certification exam. Out of $210$ students using active recall, $62$ passed the exam. Out of $205$ students using re-reading, $38$ passed. Does the difference in pass rates between the two techniques appear to have practical significance?

Determine whether the following hypothesis test is left-tailed, right-tailed, or two-tailed.

$H_0:\mu \ge 12.7$

A quality engineer is comparing the strength variability of steel rods produced by two different rolling mills. From each mill she takes an independent sample of rods, computes the sample variances $s_1^2$ and $s_2^2$, and forms the $F$ test statistic $F=\frac{s_1^2}{s_2^2}$.

True or False: If $F$ is very close to $1$, then the two samples have essentially the same variability.