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

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.

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.

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.

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.

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 localid="1653413723568" $p-$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.