Can we trust the engagement survey?

Palmstone Group (a fictional services company, 1,400 employees in six departments) runs an annual engagement survey. The executive committee set a target: at least 50% of employees “proud to work here” (agree or strongly agree with item q1). The draft report says 50.5% — target met, and the CEO plans to announce it at the town hall.

The HR director is uneasy: only 56% of staff responded, and response rates differed a lot by department. She asks you to check, before the announcement:

1. how representative the respondents are, and whether any responses are low quality;
2. how the Likert items should be summarised;
3. what the headline would be if each department counted in proportion to its real size;
4. what the CEO can honestly say.

The data. engagement_survey has one row per respondent (department, tenure band, work mode, six 1–5 Likert items, completion time, date); engagement_population has each department's headcount and number of responses. Both are synthetic: built for teaching, describing no real company.

Stakeholders. The CEO (wants a positive message), the HR director (owns the survey's credibility), and department heads (whose teams may be under-represented).