Chapter 9.3, Problem 89E

(a)

To determine

To explain: why randomly assigning the order in which each subject used the two knobs is important.

Explanation of Solution

Given Information:

There are two identical instruments one with right hand knob and one with left hand thread.

The table gives the times in seconds each subject took to move the indicator a fixed distance.

It is better to assign the order randomly because it will average out any effect that will make the subject do the activity better in second time.

(b)

To determine

To perform: the appropriate test at $5\%$ to investigate claim that right- handed people find it easier to use right hand threads.

Given information:

There are two identical instruments one with right hand knob and one with left hand thread.

The table gives the times in seconds each subject took to move the indicator a fixed distance.

Explanation:

Calculate the difference as shown in the table.

Subject	Right thread	Left thread	Difference
$1$	$113$	$137$	$-24$
$2$	$105$	$105$	$0$
$3$	$130$	$133$	$-3$
$4$	$101$	$108$	$-7$
$5$	$138$	$115$	$22$
$6$	$118$	$170$	$-52$
$7$	$87$	$103$	$-16$
$8$	$116$	$145$	$-29$
$9$	$75$	$78$	$-2$
$10$	$96$	$107$	$-11$
$11$	$122$	$84$	$38$
$12$	$103$	$148$	$-45$
$13$	$116$	$147$	$-31$
$14$	$107$	$87$	$-31$
$15$	$118$	$166$	$20$
$16$	$103$	$146$	$-48$
$17$	$111$	$123$	$-43$
$18$	$104$	$135$	$-12$
$19$	$111$	$112$	$-32$
$20$	$89$	$93$	$-1$
$21$	$78$	$76$	$-4$
$22$	$100$	$116$	$2$
$23$	$89$	$78$	$11$
$24$	$85$	$101$	$-16$
$25$	$88$	$123$	$-35$

The experiment is random. The size of the sample is small.

Calculate mean and standard deviation for the difference as follows.

Formula used:

  $\overline{x}=\frac{x_1+x_2+x_3+\mathrm{...}+x_n}{n}$

  $\sigma ={\displaystyle \sum _{i=1}^n\sqrt{\frac{{\left(x_i-\overline{x}\right)}^2}{n-1}}}$

Calculations:

  $\begin{array}{c}\overline{x}=\frac{-24+0-3+\mathrm{...}-35}{25}\\ =-13.36\end{array}$

  $\begin{array}{c}\sigma =\sqrt{\frac{{\left(-24-\left(-13.36\right)\right)}^2+\mathrm{...}+{\left(-35-\left(-13.36\right)\right)}^2}{25-1}}\\ =22.9236\end{array}$ .

The value of standard deviation $\sigma =22.9236$

Explanation:

The null hypothesis and the alternate hypothesis are described as follows

The null hypothesis $H_0:\mu =0$ , this implies that there is no evidence that right-handed people find right hand thread easier to use.

The alternate hypothesis $H_1:\mu <0$

Where $\mu$ denotes the mean time in seconds.

Formula used:

Describe the test statistics $t$ as follows

  $t=\frac{\overline{x}-{\mu }_0}{\frac{\sigma }{\sqrt{n}}}$

Substitute the values in $z$

Calculations:

  $t=\frac{-13.36-0}{\frac{22.9236}{\sqrt{25}}}$

  $t=-2.914$

The value of test statistics $t=-2.914$

Interpretation:

Use the test statistics to find the p- value. The p- value gives the evidence whether to accept or reject the null hypothesis.

The degree of freedom is given by $25-1=24$

Using tables, write the value at $\alpha =0.05$ level of significance.

Using the table, the probability lies between $0.0025<p<0.005$

The calculated p- value is $0.0038$

Result:

Reject null hypothesis if the level of significance is more than p- value.

Here the level of significance is $\alpha =0.05$ and $0.0025<p<0.005$ .

The p- value is $0.0025<0.0038<0.005$ .

Reject null hypothesis $H_0=\mu$ .

This implies that there is enough evidence to prove that the right handed people find right hand threads easier to use.

Explanation of Solution

Given information:

There are two identical instruments one with right hand knob and one with left hand thread.

The table gives the times in seconds each subject took to move the indicator a fixed distance.

Calculate the difference as shown in the table.

Subject	Right thread	Left thread	Difference
$1$	$113$	$137$	$-24$
$2$	$105$	$105$	$0$
$3$	$130$	$133$	$-3$
$4$	$101$	$108$	$-7$
$5$	$138$	$115$	$22$
$6$	$118$	$170$	$-52$
$7$	$87$	$103$	$-16$
$8$	$116$	$145$	$-29$
$9$	$75$	$78$	$-2$
$10$	$96$	$107$	$-11$
$11$	$122$	$84$	$38$
$12$	$103$	$148$	$-45$
$13$	$116$	$147$	$-31$
$14$	$107$	$87$	$-31$
$15$	$118$	$166$	$20$
$16$	$103$	$146$	$-48$
$17$	$111$	$123$	$-43$
$18$	$104$	$135$	$-12$
$19$	$111$	$112$	$-32$
$20$	$89$	$93$	$-1$
$21$	$78$	$76$	$-4$
$22$	$100$	$116$	$2$
$23$	$89$	$78$	$11$
$24$	$85$	$101$	$-16$
$25$	$88$	$123$	$-35$

The experiment is random. The size of the sample is small.

Calculate mean and standard deviation for the difference as follows.

Formula used:

  $\overline{x}=\frac{x_1+x_2+x_3+\mathrm{...}+x_n}{n}$

  $\sigma ={\displaystyle \sum _{i=1}^n\sqrt{\frac{{\left(x_i-\overline{x}\right)}^2}{n-1}}}$

Calculations:

  $\begin{array}{c}\overline{x}=\frac{-24+0-3+\mathrm{...}-35}{25}\\ =-13.36\end{array}$

  $\begin{array}{c}\sigma =\sqrt{\frac{{\left(-24-\left(-13.36\right)\right)}^2+\mathrm{...}+{\left(-35-\left(-13.36\right)\right)}^2}{25-1}}\\ =22.9236\end{array}$ .

The value of standard deviation $\sigma =22.9236$

The null hypothesis and the alternate hypothesis are described as follows

The null hypothesis $H_0:\mu =0$ , this implies that there is no evidence that right-handed people find right hand thread easier to use.

The alternate hypothesis $H_1:\mu <0$

Where $\mu$ denotes the mean time in seconds.

Formula used:

Describe the test statistics $t$ as follows

  $t=\frac{\overline{x}-{\mu }_0}{\frac{\sigma }{\sqrt{n}}}$

Substitute the values in $z$

Calculations:

  $t=\frac{-13.36-0}{\frac{22.9236}{\sqrt{25}}}$

  $t=-2.914$

The value of test statistics $t=-2.914$

Interpretation:

Use the test statistics to find the p- value. The p- value gives the evidence whether to accept or reject the null hypothesis.

The degree of freedom is given by $25-1=24$

Using tables, write the value at $\alpha =0.05$ level of significance.

Using the table, the probability lies between $0.0025<p<0.005$

The calculated p- value is $0.0038$

Result:

Reject null hypothesis if the level of significance is more than p- value.

Here the level of significance is $\alpha =0.05$ and $0.0025<p<0.005$ .

The p- value is $0.0025<0.0038<0.005$ .

Reject null hypothesis $H_0=\mu$ .

This implies that there is enough evidence to prove that the right handed people find right hand threads easier to use.