Question

19) Five people who were convicted of speeding were ordered by the court to attend a workshop. A special device put into their cars kept records of their speeds for two weeks before and after the workshop. The maximum speeds for each person during the two weeks before and the two weeks after the workshop follow. Using a .05 significance level, should we conclude that people are likely to drive more slowly after such a workshop? Use the 5-step hypothesis testing procedure. Participants Before After LB 65.00 60.00 1.5. JK 70.00 65.00 RC 60.00 56.00

I need help with step 2. I dont know how to do the table!

Answer 1

19)

Suppose, random variable X and Y denote maximum speed before and after workshop respectively. Also random variable D denotes difference in maximum speeds as D=X-Y.

We are given data in pairs corresponding to each sample unit and so we have to perform paired t-test.

We have to test for null hypothesis _{$H_0:\bar d=0$}

against the alternative hypothesis
_{H :d > 0}

Participants	Before (x)	After (y)	d=x-y	d-M = d-24/5 = d-4.8	(d-M)2
LB	65	60	5	0.2	0.04
JK	70	65	5	0.2	0.04
RC	60	56	4	-0.8	0.64
RT	70	66	4	-0.8	0.64
JM	68	62	6	1.2	1.44
Total	-	-	24	-	2.80

Here, number of observation $n=5$

Mean was calculated as

M =d = id : 4 = 4.8 d) = 4.8

Sample standard deviation of difference (d) is given by

$s_d=\frac{1}{n-1}\sum_{i=1}^{n}(d_i-\bar d)^2=\frac{1}{4}\sum_{i=1}^{5}(d_i-\bar d)^2=\frac{2.8}{4}=0.7$

Our test statistic is given by

$t=\frac{\bar d}{s_d/\sqrt n}$

$\therefore t_{calculated}=\frac{4.8}{0.7/\sqrt 5}=15.33304$

Level of significance $\alpha = 0.05$

Degrees of freedom $v=n-1=5-1=4$

In case of right tailed (greater than type alternative hypothesis) test critical value is given by

_{$t_{critical}=t_{\alpha, v}=t_{0.05,4}=2.131847$} [As P(t₄>2.131847) = 0.05]

In this type of (right tailed) test we reject null hypothesis if _{$t_{calculated}>t_{critical}$} .

Here we observe that _{$t_{calculated}=15.33304>2.131847=t_{critical}$} .

So, we reject our null hypothesis.

Hence, based on the given data we conclude that people are likely to drive more slowly after this type of workshop.

A key note- In the given images of question and attempted answer, I noticed that in Research hypothesis it is mentioned that lower driving speed than those who don't attend the workshop. However, it must not be so. There is no comparison with other people. the comparison is between maximum speed before and after workshop for each individual attending the workshop (as clearly mentioned in the given problem statement). Thus mistake was done in constructing null hypothesis also.

However, I would like to rectify and restate Research and Null hypothesis (in terms of language) as follows.

Null hypothesis- Those who attended the workshop has no change in driving speed between before and after attending the workshop.

Alternative hypothesis- Those who attended the workshop has lower driving speed after attending the workshop than before the workshop.