(a)

Number of children: \( 310 \) (M1A1, N2).

Working:
Model: \( y = -0.6T^2 + 23T + 110 \).
Given \( T = 25 \), substitute: \( y = -0.6(25)^2 + 23(25) + 110 \).
Calculate: \( 25^2 = 625 \), \( -0.6 \cdot 625 = -375 \), \( 23 \cdot 25 = 575 \).
\( y = -375 + 575 + 110 = 310 \).
So, 310 children. (M1 for substitution, A1 for answer)

[2 marks]

(b)

Regression equation: \( x = 0.0935y + 7.43 \) (M1A1A1, N3).

Working:
Data: \( (y, x) = (81, 15), (175, 27), (202, 23), (346, 35), (360, 46) \).
Perform linear regression for \( x \) on \( y \).
Calculate sums: \( \sum y = 81 + 175 + 202 + 346 + 360 = 1164 \), \( \sum x = 15 + 27 + 23 + 35 + 46 = 146 \).
\( \sum xy = 81 \cdot 15 + 175 \cdot 27 + 202 \cdot 23 + 346 \cdot 35 + 360 \cdot 46 = 1215 + 4725 + 4646 + 12110 + 16560 = 37256 \).
\( \sum y^2 = 81^2 + 175^2 + 202^2 + 346^2 + 360^2 = 6561 + 30625 + 40804 + 119716 + 129600 = 326306 \).
Slope: \( m = \frac{n \sum xy – \sum x \sum y}{n \sum y^2 – (\sum y)^2} = \frac{5 \cdot 37256 – 146 \cdot 1164}{5 \cdot 326306 – 1164^2} \approx 0.0935114 \).
Intercept: \( b = \frac{\sum x – m \sum y}{n} = \frac{146 – 0.0935114 \cdot 1164}{5} \approx 7.43053 \).
Equation: \( x = 0.0935y + 7.43 \) (3 s.f.). (M1 for regression setup, A1 for slope, A1 for intercept)

[3 marks]

(c)

Number of ice creams: \( 36 \) (M1A1, N2).

Working:
From (a): \( T = 25^\circ \text{C} \), \( y = 310 \).
Regression equation: \( x = 0.0935y + 7.43 \).
Substitute: \( x = 0.0935 \cdot 310 + 7.43 \approx 28.985 + 7.43 = 36.415 \).
Round to nearest whole number: \( 36 \) ice creams. (M1 for substitution, A1 for answer)

[2 marks]

Total [7 marks]