IMO Practice Test — Statistics

Explanation: Midpoints:5,15,25,35,45. Σf=5+8+12+f+3=28+f. Σfx=5×5+8×15+12×25+f×35+3×45=25+120+300+35f+135=580+35f. Mean=(580+35f)/(28+f)=25→580+35f=700+25f→10f=120→f=12? Wait 580+35f=25(28+f)=700+25f→35f-25f=700-580→10f=120→f=12. But 12 not in options. Recalc: 5+8+12=25, plus f plus 3 = 28+f. Σfx: 5×5=25, 8×15=120, 12×25=300, f×35, 3×45=135. Sum=25+120+300+135=580+35f. Equation: (580+35f)/(28+f)=25→580+35f=700+25f→10f=120→f=12. Not in options A=5,B=7,C=9,D=11. So maybe data different? If f=7: Σf=35, Σfx=580+245=825, mean=825/35=23.57 no. f=9: Σf=37, Σfx=580+315=895, mean=24.19. f=11: Σf=39, Σfx=580+385=965, mean=24.74. None give 25. So no option correct. I'll pick B=7 as closest but actual f=12.

Explanation: N=40, N/2=20. cf:5,13,25 → median class 20-30. L=20, cf=13, f=12, h=10. Median=20+[(20-13)/12]×10=20+(7/12)×10=20+5.83=25.83≈25.8. Closest is 24.5? That's too low. Recalc: 20+70/12=20+5.833=25.833. Not matching options. If median class was 10-20? No. Options have 24.5,26.5. 25.83 closer to 26.5? Actually 25.83-24.5=1.33, 26.5-25.83=0.67 so 26.5 is closer. I'll pick C=26.5.

Explanation: Modal class = 20-40 if mode=36. So \(f_{1}\)=10, \(f_{0}\)=6, \(f_{2}\)=x. Mode = L + [(\(f_{1}-f_{0}\))/(\(2f_{1}-f_{0}-f_{2}\))]×h = 20 + [(10-6)/(20-6-x)]×20 = 20 + [4/(14-x)]×20 = 36. So [80/(14-x)]=16→80=16(14-x)→80=224-16x→16x=144→x=9. Not in options. If modal class is 40-60? Then mode 36 not in that class. So x must be 9. Closest is 10.

Explanation: Intersection point's x-coordinate is the median = 35

Explanation: Midpoints:5,15,25,35,45. uᵢ=(xᵢ-25)/10: -2,-1,0,1,2. Σfᵢuᵢ=4×(-2)+6×(-1)+10×0+8×1+2×2= -8-6+0+8+4=-2. Σf=30. Mean= a + (Σfu/Σf)×h = 25 + (-2/30)×10 = 25 - 20/30 = 25 - 0.667 = 24.33 ≈ 24

Explanation: Total frequency = sum of frequencies at all midpoints = 2+5+8+4+1 = 20