IB DP Math MAI SL : IB Style Mock Exams – Set 5 Paper 1
Question
The table below summarizes the monthly energy usage, \(x\), measured in kilowatt hours (kWh), for a sample of 150 residential properties in the town of Helvetia:
| Energy usage (\(x\) kWh) | Frequency |
|---|---|
| \(500 \le x < 700\) | 38 |
| \(700 \le x < 900\) | 45 |
| \(900 \le x < 1100\) | 25 |
| \(1100 \le x < 1300\) | 18 |
| \(1300 \le x < 1500\) | 16 |
| \(1500 \le x < 1700\) | 8 |
(a) Utilizing the mid-interval values for this distribution, calculate an estimate for:
(i) the mean monthly consumption
(ii) the standard deviation
(i) the mean monthly consumption
(ii) the standard deviation
(b) In the neighboring district of Eureka, the standard deviation of monthly energy usage is recorded as 95 kWh. Compare this value to your result from part (a)(ii) and interpret what this indicates about the energy usage patterns in Eureka relative to Helvetia.
▶️ Answer/Explanation
(a)(i)
Midpoints: 600, 800, 1000, 1200, 1400, 1600.
\( \text{Mean} \approx \frac{600 \times 38 + 800 \times 45 + 1000 \times 25 + 1200 \times 18 + 1400 \times 16 + 1600 \times 8}{150} \)
\( = \frac{22800 + 36000 + 25000 + 21600 + 22400 + 12800}{150} \)
\( = \frac{140600}{150} \approx 937.33 \)
\(\boxed{937 \ \text{kWh}}\)
(a)(ii)
Using GDC or formula for grouped data standard deviation:
\( \sigma \approx 299.45 \)
\(\boxed{299 \ \text{kWh}}\)
(b)
Standard deviation measures spread. Eureka’s \(\sigma = 95\) kWh is much smaller than Helvetia’s \(\sigma \approx 299\) kWh, so monthly energy consumption in Eureka is less spread out / more consistent / less variable than in Helvetia.
\(\boxed{\text{Eureka’s consumption is less variable than Helvetia’s.}}\)