(a) Give a brief outline describing how confidence intervals for means use the normal distribution or Student's t distribution in their basic construction.

Identify the null and alternative hypothesis in the following scenario.
To determine if battery 1 lasts longer than battery 2, the mean lasting times, of the two competing batteries are compared. Twenty batteries of each type are randomly sampled and tested. Both populations have normal distributions with unknown standard deviations.
Select the correct answer below: $H_0:{\mu }_1\ge {\mu }_2,H_a:{\mu }_1<{\mu }_2$
$H_0:{\mu }_1\le -{\mu }_2,H_a:{\mu }_1>-{\mu }_2$
$H_0:{\mu }_1\ge -{\mu }_2,H_a:{\mu }_1<-{\mu }_2$
$H_0:{\mu }_1={\mu }_2,H_a:{\mu }_1\ne {\mu }_2$
$H_0:{\mu }_1\le {\mu }_2,H_a:{\mu }_1>{\mu }_2$

Refer to the Journal of Applied Psychology (Jan. 2011) study of the determinants of task performance. In addition to $x_1=$ conscientiousness score and $x_2=\{1\text{if highly complex job, 0 if not}\}$, the researchers also used $x_3=$ emotional stability score, $x_4=$ organizational citizenship behavior score, and $x_5=$ counterproductive work behavior score to model y = task performance score. One of their concerns is the level of multicollinearity in the data. A matrix of correlations for all possible pairs of independent variables follows. Based on this information, do you detect a moderate or high level of multicollinearity? If so, what are your recommendations?

$x_1{x}_2{x}_3{x}_4$

Conscientiousness $(x_1)$

Job Complexity $(x_2).13$

Emotional Stability $(x_3).62.14$

Organizational Citizenship $(x_4).24.03.24$

Counterproductive Work $(x_5)-.23-.23-.02-.62$

Suppose you take independent random samples from populations with means ${\displaystyle \mu 1\text{and}\mu 2}$ and standard deviations ${\displaystyle \sigma 1\text{and}\sigma 2}$. Furthermore, assume either that (i) both populations have normal distributions, or (ii) the sample sizes ($n_1\text{ and }{n}_2$) are large. If $X_1\text{ and }{X}_2$ are the random sample means, then how does the quantity
${\displaystyle \frac{(\overline{x_1}-\overline{x_2})-({\mu }_1-{\mu }_2)}{\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}}}}$
Give the name of the distribution and any parameters needed to describe it.