Q.89 - The Practice of Statistics for AP

Question

The design of controls and instruments affects how easily people can use them. A student project investigated this effect by asking $25$ right-handed students to turn a knob (with their right hands) that moved an indicator. There were two identical instruments, one with a right-hand thread (the knob turns clockwise) and the other with a left-hand thread (the knob must be turned counterclockwise). Each of the $25$ students used both instruments in a random order. The following table gives the times in seconds each subject took to move the indicator a fixed distance:

(a) Explain why it was important to randomly assign the order in which each subject used the two knobs. 

(b) The project designers hoped to show that right-handed people find right-hand threads easier to use. Carry out a significance test at the $5\%$ significance level to investigate this claim 

Step-by-Step Solution

Answer

(a)It is good to appoint the request randomly on the grounds that it will average out any effect that will cause the subject to improve in second time.

(b) There is enough evidence to prove that the right handed people find right hand threads easier to use. 

Part (a) Step 1: Given information

Given in the question that, The ease of use of controls and instruments is influenced by their design. A student project studied this effect by having 25 right-handed students operate a knob that moved an indicator using their right hands. One had a right-hand thread (the knob turned clockwise) while the other had a left-hand thread (the knob turned counterclockwise) (the knob must be turned counterclockwise). In a random order, each of the 25 students played both instruments. The following table shows how long each participant took to move the indicator a specified distance in seconds.

Part (a) Step 2: Explanation

Two identical instruments are available, one with a right-hand knob and the other with a left-hand thread. 

The table shows how long each participant took to move the indication a reasonable distance in seconds. 

It is preferable to assign the request at random because this will average out any effects that cause the subject to improve the second time around.

Part (b) Step 1: Given information

Right-handed persons find right-hand threads easier to use, according to the project's designers. To investigate this allegation, we must do a 5% significance test at the significance level.

Part (b) Step 2: Explanation

Two identical instruments are available, one with a right-hand knob and the other with a left-hand thread. The table shows how long each participant took to move the indicator a set distance in seconds. Calculate the difference using the table as a guide. 

Subject	Right hand	Left hand	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

Part (b) Step 3: Calculate mean and standard deviation

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

Calculate mean and standard deviation for the difference as follows.

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

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

$\overline{x}=\frac{-24+0-3+\dots -35}{25}$

$=-13.36$

$\sigma =\sqrt{\frac{(-24-(-13.36){)}^2+\dots +(-35-(-13.36){)}^2}{25-1}}$

$=22.9236$

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$\mathit{\mu }$denotes the mean time in seconds.

Part (b) Step 4: find the p-value

Describe the test statistics $t$ as follows

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

Substitute the values in $z$.

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

$t=-2.914$

The  test statistics value will be: $t=-2.914$

To determine the $p-$value, just use test statistics. The p-value indicates whether the null hypothesis should be accepted or rejected. 

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 $0.0038$

Null hypothesis is rejected if the significance value exceeds the $p$-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 shows that there is enough data to demonstrate that right-handed people would prefer typing with their right hand.