Question $4(10$ marks$)-$ this module is statistics for data science

A company wants to determine if a new training program has a significant effect on employee

productivity. The company measures the productivity of a sample of employees before and after

the training program. The productivity is measured by the number of tasks completed per week.

Data:

$$ Sample size $($n$)$: $10$ employees

$$ Productivity before training: $[40,42,45,43,44,41,39,46,38,47]$

$$ Productivity after training: $[45,44,46,47,49,43,41,48,42,50]$

Hypotheses:

$$ H$0$: There is no difference in productivity before and after the training program $($mean

difference $=0).$

$$ H$1$: There is a difference in productivity before and after the training program $($mean

difference $!=0).$

Carry out a paired t$-$test to determine the effectiveness of the training program in enhancing

productivity. Draw the relevant conclusions using the p$-$value from the test using alpha value of

$0\mathrm{.}05.$