Statistical Inference — practice questions

A council measures the daily water use of a random sample of 50 households in Wellington. State the population and the sample.

For the water-use study, explain the difference between a parameter and a statistic, using the median as your example.

A student says, “My sample median is 165 L, so the median for all Wellington households is exactly 165 L.” Explain why this is wrong, using the idea of sampling variability.

A sample has minimum 6, $Q_1 = 11$, median 15, $Q_3 = 21$, maximum 28. Find the IQR and the range.

For the sample above ($Q_1=11$, median 15, $Q_3=21$), describe the shape of the middle 50% and justify your answer.

A dot plot and a box plot are drawn for the same 12 values. Explain what the dot plot can show that the box plot hides, and when this matters for an inference.

Sample X has median 52; Sample Y has median 45. Both are drawn on the same scale. Write one 'I notice' statement comparing the centres in context (the variable is exam mark, %).

Sample X: $Q_1=44$, $Q_3=58$. Sample Y: $Q_1=40$, $Q_3=50$. Compare the spread and comment on what it means.

Sample X (median 52, box 44–58) and Sample Y (median 45, box 40–50) are on the same scale. Discuss the overlap and explain why the higher median alone does not prove Sample X's population scores higher.

A random sample of $n = 25$ has median 40, $Q_1 = 32$, $Q_3 = 52$. Calculate the margin $E = 1.5 \times \text{IQR}/\sqrt{n}$ and give the confidence interval.

A sample of $n = 64$ eels has median length 48 cm, $Q_1 = 40$, $Q_3 = 56$. Build the informal confidence interval and interpret it in context.

Two samples have the same median and IQR, but one has $n = 25$ and the other $n = 100$. Explain, with the margins, how the sample size changes the interval and why that matters.

Group A's interval is $(50, 60)$ and Group B's interval is $(38, 47)$. Do the intervals overlap, and can you make the call?

Sample P ($n=25$): median 60, $Q_1=50$, $Q_3=70$. Build P's interval and compare with Sample Q's interval $(58, 66)$. State your conclusion.

A study compares median incomes of two towns. The intervals overlap slightly, so no call can be made. Write a conclusion and explain what could be done to reach a call, referring to sampling variability.