Question

Interviewing candidates for a job. The costs associated with conducting interviews for a job opening have skyrocketed over the years. According to a Harris Interactive survey $\mathbf{211}$ of $\mathbf{502}$senior human resources, executives at U.S. companies believe that their hiring managers are interviewing too many people to find qualified candidates for the job (Business Wire, June $\mathbf{8}\mathbf{}\mathbf{2006}$).

a.Describe the population of interest in this study.

b.Identify the population parameter of interest $\mathbf{p}$.

c.Is the sample size large enough to provide a reliable estimate of $\mathbf{p}$?

d.Find and interpret an interval estimate for the true proportion of senior human resources executives who believe that their hiring managers interview too many candidates during a job search. Use a confidence levelof $\mathbf{98}\mathit{\%}$.

e.If you had constructed a$\mathbf{90}\mathit{\%}$ confidence interval, would it be wider or narrower?

Short answer

1. The population of interest is hiring managers.
2. The population parameter of interest $p$is hiring managers.
3. Here sample size is large enough to provide a reliable estimate of $p$.
4. The confidence level of $98\%$is given by(0.369, 0.472)
5. For $98\%$ confidence level, it is given as $0.14$and a narrower value.

Step-by-step solution

Given information

Here costs associated with conducting interviews for a job opening have skyrocketed over the years. In this problem given that $211$of $502$, senior human resources executives at U.S companies believe that their hiring managers are interviewing too many people to find qualified candidates for the job.

(a) Describe the population of interest

The population of interest in statistical research is the group from which the researcher wishes to draw findings. Hiring managers is the population of interest in this study.

(b) Identify the population parameter of interest

The population parameter of interest $p$is hiring managers. The parameter of interest is a statistical variable that provides additional data on the study sample or group under investigation. These criteria define and characterize a specific study group.

(c) Define sample size is large enough to provide a reliable estimate of   p

Let’s sample size=$\mathrm{n}$. Then, the sample size$n$=$502$

Then the reliable estimate is given by$\begin{array}{c}\mathrm{p}=\frac{\overline{\mathrm{X}}}{\mathrm{n}}\\ =\frac{211}{502}\\ =0.42\end{array}$

Here sample size is large than a reliable estimate. So, the sample size is large enough to provide a reliable estimate of $p$.

(d) Find an interval estimate for the true proportion of the senior human resources executives who hire managers and also find the confidence level of  98%

It is given that, n = 502 and x = 211

The formula CI for p is:

$\mathrm{CI}=\stackrel{⌢}{\mathrm{p}}\pm {\mathrm{z}}_{\frac{\mathrm{a}}{2}}\frac{\sqrt{\stackrel{⌢}{\mathrm{p}}(1-\stackrel{⌢}{\mathrm{p}})}}{\mathrm{n}}$

The z- critical value at a 0.01 level of significance is 2.33

The sample proportion is,

$\begin{array}{c}\stackrel{⌢}{\mathrm{p}}=\mathrm{x}/\mathrm{h}\\ =211/502\\ =\mathrm{0.4203.}\end{array}$

The 98% CI for p is:

$\begin{array}{c}\mathrm{CI}=\stackrel{⌢}{\mathrm{p}}\pm {\mathrm{z}}_{\frac{\mathrm{a}}{2}}\frac{\sqrt{\stackrel{⌢}{\mathrm{p}}(1-\stackrel{⌢}{\mathrm{p}})}}{\mathrm{n}}\\ =0.4203\pm 2.33\frac{\sqrt{0.4203(1-0.4203)}}{502}\\ =0.4203\pm 2.33\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}\\ =0.4203\pm (2.33\times 0.022)\\ =0.4203\pm 0.0513\end{array}$

$\begin{array}{l}=(0.4203-0.0513,0.4203+0.0513)\\ =(0.369,0.472)\end{array}$

Therefore, the 98% CI for p is (0.369, 0.472)

(e) Find the confidence level of 90%and define whether it is narrower or wider

For $90\mathrm{\%}$confidence interval, the interval estimate is given by:

$\begin{array}{c}\mathrm{z}=\overline{\mathrm{X}}\pm 1.64\times \frac{\mathrm{\sigma }}{\sqrt{\mathrm{n}}}\\ =211\pm 1.64\times \frac{1}{\sqrt{502}}\\ =211.07\end{array}$

And $210.93$

$\begin{array}{l}\text{Theproportionis=}\frac{210.93}{211.07}\\ =0.99\end{array}$

The interval estimate is given by $0.14$. It is a narrower value.