The Instagram Library Poll
For three years the library has closed at 9 pm. A student group wants it open until 11 pm during the April exam fortnight and runs a poll on the college's Instagram account: "Should the library stay open till 11 pm?" The poll is live from 10 am to 2 pm on a Wednesday and draws 240 responses; 156 say yes. The account has 3,000 followers, roughly 4,800 students are enrolled, and the library committee will decide on the basis of the poll. The librarian objects: "Two-thirds of students want it, so we have to fund it." Extending hours would cost ₹18,000 for the fortnight, mainly for a security guard and electricity, and the library budget has ₹12,000 unspent.
1. Build a 95% confidence interval for the proportion supporting the extension and state precisely what the interval means, and what it does not mean.
Read the worked answer
p̂ = 156/240 = 0.65.
Standard error = √(p̂(1−p̂)/n) = √(0.65 × 0.35 / 240) = √0.000948 ≈ 0.0308.
Margin of error = 1.96 × 0.0308 ≈ 0.0603, so the 95% interval is 0.590 to 0.710, i.e. 59% to 71%.
Interpretation: if we repeated this sampling procedure many times, about 95% of the intervals built this way would contain the true proportion of the population being sampled. The interval does not mean there is a 95% chance that the true value sits between 59% and 71%, and it says nothing about whether that population is the student body - it is the population of Instagram followers who chose to respond.
Note also that the interval comfortably contains 0.50 and comfortably contains 0.667. On these data the claim "two-thirds want it" is not established, and neither is the opposite.
2. Identify two distinct sources of bias in this poll and give a concrete fix for each.
Read the worked answer
1. Self-selection (voluntary response). 240 responses from 3,000 followers is an 8% response rate, and the people who answer a poll about library hours are the people with strong views or free time at that moment. The estimate is not a random sample of anything. Fix: draw a random sample from the enrolment register, contact selected students individually, and follow up non-responders at least once.
2. Coverage and timing. Only Instagram followers could be counted, so students who avoid social media - plausibly heavy library users - are structurally excluded. The 10 am–2 pm Wednesday window also skews towards students whose timetables are free then. Fix: stratify by year and programme, run the survey across a full week including evening hours, and include a paper or email route using college IDs.
A third, less obvious issue: the question is leading ("stay open till 11 pm" offers no trade-off), so even a clean sample would overstate support once students are told it costs ₹18,000 against a ₹12,000 budget.
3. The group wants the margin of error at ±4 percentage points with 95% confidence. What sample size is needed, and what does that imply for the committee's decision?
Read the worked answer
Using the conservative worst case p = 0.5:
n = (1.96)² × 0.25 / (0.04)² = 3.8416 × 0.25 / 0.0016 = 0.9604 / 0.0016 = 600.25, so 601 students.
(Using p̂ = 0.65 instead gives n = 3.8416 × 0.2275 / 0.0016 ≈ 546, but the safe figure to plan with is about 600.)
Implication: the current poll is roughly 40% of the required size, and quadrupling the response count means going back to the field with a proper sampling frame, not simply leaving the poll open longer - a bigger self-selected sample reduces random error while leaving the bias untouched.
For the committee: the poll as run cannot justify the librarian's "two-thirds" claim. The defensible move is a short properly sampled survey before committing the ₹18,000, or a conditional approval that funds the guard only if a valid survey shows majority support.