The complete data set, measured in minutes, is \(10,13,13,16,19,22,22,25,31\). Treat these nine observations as a population.
Stem
Leaf
1
0 3 3 6 9
2
2 2 5
3
1
Key: 2 | 7 represents 27 minutes.
Find the mean in minutes. Explain your reasoning. [2]
Find the population variance in square minutes. Explain your reasoning. [2]
Find the median in minutes. Explain your reasoning. [2]
Solution
Hint
Use population variance when describing the complete listed data set. Add all observations and divide by nine.
Worked solution
(a)
\(\bar x=19\)
Answer: \(19\).
Answer: \(19\)
(b)
\(\operatorname{Var}(X)=\sum x^2/9-\bar x^2=40\)
Answer: \(40\).
Answer: \(40\)
(c)
\(x_{(5)}=19\)
Answer: \(19\).
Answer: \(19\)
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.