AS & A Level · Practice questions · Unbiased estimates and confidence intervals

Estimate, then give the margin.

Six original Cambridge-style Paper 6 questions on unbiased estimates and confidence intervals, with full worked solutions.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Estimate, standard error, \(z\), interval, interpret.

Estimates: \(\bar{x} = \dfrac{\sum x}{n}\), \(s^{2} = \dfrac{1}{n-1}\left(\sum x^{2} - \dfrac{(\sum x)^{2}}{n}\right)\).
Coded data: work with \(x - a\), then add \(a\) back to the mean; the variance is unchanged.
Mean: \(\bar{x} \pm z\dfrac{\sigma}{\sqrt{n}}\); use \(s\) when \(n\) is large and \(\sigma\) is unknown.
Proportion: \(\hat{p} \pm z\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\), large \(n\) only.
Interpret: a stated value inside the interval is plausible; outside, it is doubtful.

Recall

[1 marks]

Explain what is meant by saying that \(\bar{x}\) is an unbiased estimate of \(\mu\).

Individual sample means vary, but on average (over all possible samples) they equal \(\mu\): \(\text{E}(\bar{X}) = \mu\). ✓

Calculation

[3 marks]

The times, in minutes, taken by 8 randomly chosen students to solve a puzzle were

4.2 3.8 5.1 4.6 4.9 3.5 4.4 5.0

Find unbiased estimates of the population mean and variance.

\(\sum x = 35.5\), \(\sum x^{2} = 159.87\). ✓
\(\bar{x} = \dfrac{35.5}{8} = 4.4375\) (4.44 to 3 s.f.). ✓
\(s^{2} = \dfrac{1}{7}\left(159.87 - \dfrac{35.5^{2}}{8}\right) = \dfrac{1}{7}(159.87 - 157.53125) = 0.334\) (3 s.f.). ✓

Calculation

[3 marks]

For a random sample of 60 values of \(x\), \(\sum (x - 50) = 84\) and \(\sum (x - 50)^{2} = 1260\). Find unbiased estimates of the population mean and variance.

Mean of \(x - 50\): \(\dfrac{84}{60} = 1.4\), so \(\bar{x} = 51.4\). ✓
\(s^{2} = \dfrac{1}{59}\left(1260 - \dfrac{84^{2}}{60}\right) = \dfrac{1}{59}(1260 - 117.6) = \dfrac{1142.4}{59}\). ✓
\(s^{2} = 19.4\) (3 s.f.); subtracting 50 does not change the variance. ✓

Calculation

[3 marks]

The diameters of bolts are \(N(\mu, 0.8^{2})\) mm. A random sample of 15 bolts has mean diameter 24.32 mm. Find a 98% confidence interval for \(\mu\).

\(z = 2.326\) (1% in each tail). ✓
Margin: \(2.326 \times \dfrac{0.8}{\sqrt{15}} = 0.480\). ✓
Interval: \((23.8, 24.8)\) mm (3 s.f.). ✓

Analysis

[5 marks]

In a random sample of 400 adults, 132 said they cycle to work.

(a) Find a 95% confidence interval for the proportion of all adults who cycle to work.

(b) A council claims that 30% of adults cycle to work. Comment on this claim.

(c) Using the same estimate of the proportion, find the smallest sample size for which the width of the 95% interval is at most 0.05.

(a) \(\hat{p} = 0.33\); margin \(1.96\sqrt{\dfrac{0.33 \times 0.67}{400}} = 1.96 \times 0.023\,51 = 0.0461\). Interval \((0.284, 0.376)\). ✓
(b) 0.30 lies inside the interval, so the data do not contradict the council's claim. ✓
(c) Width \(= 2 \times 1.96\sqrt{\dfrac{0.2211}{n}} \le 0.05\), so \(n \ge \left(\dfrac{3.92}{0.05}\right)^{2} \times 0.2211 = 1359.004\). ✓
Smallest \(n = 1360\). ✓

Analysis

[6 marks]

A random sample of 100 calls to a help desk had durations \(x\) minutes with \(\sum x = 4250\) and \(\sum x^{2} = 184\,000\).

(a) Find unbiased estimates of the population mean and variance.

(b) Find a 95% confidence interval for the mean call duration, and explain why it is valid even though call durations are not normally distributed.

(a) \(\bar{x} = 42.5\); \(s^{2} = \dfrac{1}{99}\left(184\,000 - \dfrac{4250^{2}}{100}\right) = \dfrac{3375}{99} = 34.1\) (3 s.f.). ✓
(b) Margin \(1.96\sqrt{\dfrac{34.09}{100}} = 1.144\); interval \((41.4, 43.6)\) minutes. ✓
\(n = 100\) is large, so by the Central Limit Theorem \(\bar{X}\) is approximately normal whatever the distribution of durations. ✓
(c) Each interval contains \(\mu\) with probability 0.95, independently: \(0.95^{10} = 0.599\) (3 s.f.). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.