Hint

Use population variance when describing the complete listed data set. Add all observations and divide by nine.

Worked solution

(a)

\(\bar x=23\)

Answer: \(23\).

Answer: \(23\)

(b)

\(\operatorname{Var}(X)=\sum x^2/9-\bar x^2=\frac{640}{9}\)

Answer: \(\frac{640}{9}\).

Answer: \(\frac{640}{9}\)

(c)

\(x_{(5)}=23\)

Answer: \(23\).

Answer: \(23\)

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.

(a) · 2 marks

• 1 mark: Add all observations and divide by nine.
• 1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: Do not divide descriptive population variance by n−1.

(b) · 2 marks

• 1 mark: Calculate the second moment and subtract the square of the mean.
• 1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: A change in location does not by itself change spread.

(c) · 2 marks

• 1 mark: Locate the fifth ordered observation.
• 1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: Do not divide descriptive population variance by n−1.

Total: 6 marks.