The marketing department of a renowned technology firm is determined to scrutinize the impact of its most recent social media marketing initiative on the level of user engagement. With this objective in mind, the team embarks on a quantitative analysis, aiming to draw meaningful insights into how effectively their advertising efforts have translated into changes in the interactions among their target audience on social media platforms. In pursuit of this goal, the department selects a random sample comprising $8$ users from their broader user base. This selection is carefully designed to represent a wide array of user types to ensure the reliability and applicability of the findings. The data collection process involves meticulously recording the average number of daily interactions for each user within this group. These interactions encompass a variety of engagement metrics, such as likes, comments, shares, and other forms of active response to content.

The period for data collection is strategically set to cover two distinct phases: one week immediately BEFORE the launch of the advertising campaign and one week immediately AFTER the campaign begins. This timeframe is chosen to capture any immediate variations in user engagement levels attributable to the campaign, providing a before$-$and$-$after snapshot of its effectiveness.

The primary question for analysis is:

How does the launch of the company's latest social media advertising campaign influence user engagement, as measured by the average daily interactions of a randomly selected sample of users?

The data $($in average interactions per day$)$ is as follows:

$\backslash$table$[[$Interactions per day,$],[$Before campaign,$56,63,59,68,55,66,60,58,60\mathrm{.}6,4\mathrm{.}7],[,,,,,,,,,,],[$After campaign,$72,67,65,75,59,74,61,66,67\mathrm{.}4,5\mathrm{.}9],[,,,,,,,,,],[16,4,6,7,4,8,1,8,6\mathrm{.}8,4\mathrm{.}4]]$

Given this information, perform the following:

$($a$)$ Construct a $99\%$ confidence interval for the true mean difference in user engagement before and after the advertising campaign. $(2$ marks$)$