Question

Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in

the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly

surveyed $81$people who recently served as jurors. The sample mean wait time was eight hours with a sample standard

deviation of four hours.

$$\begin{gathered} a.i.x̄=\_\_\_\_\_\_\_\_\_\_ \\ ii.sx=\_\_\_\_\_\_\_\_\_\_ \\ iii.n=\_\_\_\_\_\_\_\_\_\_ \\ iv.n–1=\_\_\_\_\_\_\_\_\_\_ \end{gathered}$$

b. Define the random variables $XandX̄$in words.

c. Which distribution should you use for this problem? Explain your choice.

d. Construct a $95\%$confidence interval for the population mean time wasted.

i. State the confidence interval.

ii. Sketch the graph.

iii. Calculate the error bound.

e. Explain in a complete sentence what the confidence interval means.

Short answer

(a) The Final Result we get,

i. $8$

ii. $4$

iii.$81$

iv $80$

(b) The average wait time for the sample size of X, and the amount of time the single waste at the courts is called for service duty is $\stackrel{-}{X}$.

(c) The random distribution with the parameters $t_{81-1}=t_{80}$is used.

(d) The result is :

i. CI=$(7.1155,8.8845)$

ii. the graph is shown

iii. $0.8844$

(e) The $95\%$confidence that the interval between $7.11and8.88$minutes accurately represents the true mean courthouse wait time.

Step-by-step solution

Explanation (a)

1. The average waiting time in the sample is 8 hours.

$x=8$hours overline

ii. Waiting time standard deviation,$s_x=4$

iii. There are $81$surveyors who have recently served.

iv. If the total number of people questioned the value is

$n-1=80.$

Explanation (b)

The average wait time for the sample size of $81isX$, and the amount of time the single waste at the courts is called for service duty is$\overline{X}$.

Explanation (c) 

The random distribution with the parameters $ln-1$is used. .

$t81-1=t80$.

Explanation (d)

i. State the confidence interval.

The output of the confidence interval,

C.I =$(7.1155,8.8845)$

ii. The graph is as follows:

iii. The formula is used to compute the error bound.

$EBM=t_{n-1}\left(\frac{\alpha }{2}\right)\frac{s}{\sqrt{n}}$

$EBM=t_{81-1}\left(\frac{0.05}{2}\right)\frac{4}{\sqrt{81}}$

$EBM=0.8844$.

Explanation (e)

The $95\%$ confidence that the interval between $7.11and8.88$ minutes accurately represents the true mean courthouse wait time.