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Ex 3D
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Solutions
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(a)
∑fx2=1×32+1×42+1×52+1×72+1×82+1×102+1×132=432∑f=7∑fx=1×3+1×4+1×5+1×7+1×8+1×10+1×13=50Standard deviation=√∑fx2∑f−(∑fx∑f)2=√4327−(507)2=3.27014≈3.27
(b)
19,20,22,.23,24,25,27,28,30,32∑fx2=1×192+1×202+1×222+1×232+1×242+1×252+1×272+1×282+1×302+1×322=6412∑f=10∑fx=1×19+1×20+1×22+1×23+1×24+1×25+1×27+1×28+1×30+1×32=250Standard deviation=√∑fx2∑f−(∑fx∑f)2=√641210−(25010)2=4.02492≈4.02
(c)
−5,.−4,−2,0,1,4∑fx2=1×(−5)2+1×(−4)2+1×(−2)2+1×02+1×12+1×42=62∑f=6∑fx=1×(−5)+1×(−4)+1×(−2)+1×0+1×1+1×4=−6Standard deviation=√∑fx2∑f−(∑fx∑f)2=√626−(−66)2=3.05505≈3.06
(a) The table in your calculator should look like:
x | Freq |
---|---|
128 | 1 |
135 | 1 |
156 | 1 |
123 | 1 |
144 | 1 |
130 | 1 |
From the calculator, standard deviation=11.06044≈11.1
(b) The table in your calculator should look like:
x | Freq |
---|---|
0 | 1 |
1 | 1 |
25 | 1 |
14 | 1 |
2 | 1 |
16 | 1 |
22 | 1 |
4 | 1 |
From the calculator, standard deviation=9.35414≈9.35
(c) The table in your calculator should look like:
x | Freq |
---|---|
39.6 | 1 |
12 | 1 |
13.5 | 1 |
22.6 | 1 |
31.3 | 1 |
8.4 | 1 |
5.5 | 1 |
4.7 | 1 |
From the calculator, standard deviation=11.94278≈11.9
Note: Marks is the data (x) while number of students is the frequency (f)
∑fx2=5×22+7×32+6×42+4×52+9×62+3×72+6×82=1134∑f=5+7+6+4+9+3+6=40∑fx=5×2+7×3+6×4+4×5+9×6+3×7+6×8=198Standard deviation=√∑fx2∑f−(∑fx∑f)2=√113440−(19840)2=1.9615≈1.96
Note: Number of goals per match is the data (x) while number of matches is the frequency (f)
∑fx2=10×02+8×12+7×22+6×32+2×42+3×52+1×62=233∑f=10+8+7+6+2+3+1=37∑fx=10×0+8×1+7×2+6×3+2×4+3×5+1×6=69Standard deviation=√∑fx2∑f−(∑fx∑f)2=√23337−(6937)2=1.6781≈1.68
We need to use the mid-value for x: 2.5, 7.5, 12.5, 17.5, 22.5, 27.5
∑fx2=4×2.52+12×7.52+20×12.52+24×17.52+16×22.52+4×27.52=22300∑f=4+12+20+24+16+4=80∑fx=4×2.5+12×7.5+20×12.5+24×17.5+16×22.5+4×27.5=1240Standard deviation=√∑fx2∑f−(∑fx∑f)2=√2230080−(124080)2=6.2048≈6.20
We need to use the mid-value for Salary: 210, 230, 250, 270, 290
∑fx2=8×2102+23×2302+16×2502+3×2702+10×2902=3,629,200∑f=8+23+16+3+10=60∑fx=8×210+23×230+16×250+3×270+10×290=14680Standard deviation=√∑fx2∑f−(∑fx∑f)2=√3,629,20060−(1468060)2=24.99777≈25.0
(a) We need to use the mid-value for x: 35, 45, 55, 65, 75
∑fx2=16×352+25×452+35×552+14×652+10×752=291,500∑f=16+25+35+14+10=100∑fx=16×35+25×45+35×55+14×65+10×75=5270Standard deviation=√∑fx2∑f−(∑fx∑f)2=√291,500100−(5270100)2=11.7349≈11.7
(b) We need to use the mid-value for y: 72.5, 77.5, 82.5, 87.5, 92.5, 97.5, 102.5
∑fy2=4×72.52+11×77.52+15×82.52+24×87.52+18×92.52+9×97.52+3×102.52=644,025∑f=4+11+15+24+18+9+3=84∑fy=4×72.5+11×77.5+15×82.5+24×87.5+18×92.5+9×97.5+3×102.5=7330Standard deviation=√∑fy2∑f−(∑fy∑f)2=√644,02584−(733084)2=7.2335≈7.23
(i)
For Class A,∑fx=1×4+2×6+1×7+1×8+1×10+1×11+1×12=64∑f=8∑fx2=1×42+2×62+1×72+1×82+1×102+1×112+1×122=566Mean (Class A)=∑fx∑f=648=8S.D. (Class A)=√∑fx2∑f−(Mean)2=√5668−(8)2=2.5980≈2.60For Class B,∑fx=1×0+2×1+1×2+1×3+1×14+1×17+1×25=63∑f=8∑fx2=1×02+2×12+1×22+1×32+1×142+1×172+1×252=1125Mean (Class B)=638=7.875S.D. (Class B)=√∑fx2∑f−(Mean)2=√11258−(7.875)2=8.8661≈8.87
(ii)
Class A performed better on average since the mean score of Class A (8) is higher thanthe mean score of Class B (7.875).The scores obtained by students in Class A is more consistent since the standard deviationof Class A (2.60) is lower than the standard deviation of Class B (8.87).
(i)
Total score of 6 students=10×6=60x+5+16+6+10+4=60x+41=60x=60−41x=19
(ii)
∑f=6∑fx2=1×52+1×162+1×62+1×102+1×42+1×192=794S.D.=√∑fx2∑f−(Mean)2=√7946−(10)2=5.6862≈5.69
(iii)
Priya is the best performer, since her score of 19 is much higher than the mean score of 10.
(i)
∑fx=1×23+2×15+1×8+1×13+1×28+1×6=108∑f=7∑fx2=1×232+2×152+1×82+1×132+1×282+1×62=2032Mean=∑fx∑f=1087=1537S.D.=√∑fx2∑f−(Mean)2=√20327−(1537)2=7.228≈7.23
(ii)
∑fx=1×20+2×12+1×5+1×10+1×25+1×3=87∑f=7∑fx2=1×202+2×122+1×52+1×102+1×252+1×32=1447Mean=∑fx∑f=877=1237S.D.=√∑fx2∑f−(Mean)2=√14477−(1237)2=7.228≈7.23
(iii)
On average, Kate took a shorter time to fall asleep in New Zealand as the mean time taken is lesser.The time taken to fall asleep in equally spread out in both locations as the standard deviation is the same.
(i)
For Train A,∑fx=3×2+2×3+5×4+12×5+10×6+6×7+1×8+1×9=211∑f=3+2+5+12+10+6+1+1=40∑fx2=3×22+2×32+5×42+12×52+10×62+6×72+1×82+1×92=1209Mean=∑fx∑f=21140=5.275S.D.=√∑fx2∑f−(Mean)2=√120940−(5.275)2=1.5489≈1.55For Train B,∑fx=4×2+3×3+9×4+9×5+7×6+5×7+3×8+0×9=199∑f=4+3+9+9+7+5+3+0=40∑fx2=4×22+3×32+9×42+9×52+7×62+5×72+3×82+0×92=1101Mean=∑fx∑f=19940=4.975S.D.=√∑fx2∑f−(Mean)2=√110140−(4.975)2=1.6656≈1.67
(ii)
Train A is more consistent since the standard deviation is lower.
(iii)
Train B is more punctual on the whole since the mean time by which the train arrivedafter the scheduled time was lower.
(a)(i) We need to use the mid-value for t: 21, 23, 25, 27, 29
Mean=∑fx∑f=5×21+11×23+27×25+13×27+4×2960=25
(a)(ii)
∑fx2=5×212+11×232+27×252+13×272+4×292=37740S.D.=√∑fx2∑f−(Mean)2=√3774060−(25)2=2
(b)
On the whole, the waiting time in both hospitals are similar since the mean waitingtime is the same for both hospitals. The waiting time is more consistent in Stamford Hospital as the standard deviationof the waiting time is lower for Stamford Hospital.
(a)(i) We need to use the mid-value for Temperature: 37.5, 42.5, 47.5, 52.5, 57.5, 62.5
Mean (City A)=∑fx∑f=1×37.5+4×42.5+12×47.5+23×52.5+7×57.5+3×62.550=51.5Mean (City B)=2×37.5+14×42.5+16×47.5+10×52.5+5×57.5+3×62.550=48.6
(a)(ii)
For City A,∑fx2=1×37.52+4×42.52+12×47.52+23×52.52+7×57.52+3×62.52=133962.5S.D. (City A)=√∑fx2∑f−(Mean)2=√133962.550−(51.5)2=5.19615≈5.20For City B,∑fx2=2×37.52+14×42.52+16×47.52+10×52.52+5×57.52+3×62.52=120012.5S.D. (City B)=√120012.550−(48.6)2=6.18789≈6.19
(b)
City A as the mean temperature is higher.
(c)
City A as the standard deviation is lower.
Mean=∑fx∑f=1×10+1×6+1×18+1×x+1×15+1×y6=10+6+18+x+15+y6=49+x+y69=49+x+y66(9)=49+x+y54=49+x+y54−49=x+y5=x+yy=5−x000 --- (1)∑fx2=1×102+1×62+1×182+1×x2+1×152+1×y2=100+36+324+x2+225+y2=685+x2+y2S.D.=√∑fx2∑f−(Mean)2=√685+x2+y26−(9)2=√685+x2+y26−816=√685+x2+y26−8162=685+x2+y26−8136+81=685+x2+y26117=685+x2+y266(117)=685+x2+y2702=685+x2+y2702−685=x2+y217=x2+y2000 --- (2)Substitute (1) into (2),17=x2+(5−x)217=x2+52−2(5)(x)+x217=x2+25−10x+x20=2x2−10x+25−170=2x2−10x+80=x2−5x+40=(x−1)(x−4)x=1 or x=4Substitute into (1),y=5−1 or y=5−4y=4000000y=1∴
(i)
\begin{align*} \text{Original mean} & = 5 \\ \\ \text{Total (for } n \text{ numbers)} & = 5 \times n \\ & = 5n \\ \\ \text{New mean} & = 5 \\ \\ \text{Total (for } n + 2 \text{ numbers)} & = 5 \times (n + 2) \\ & = 5(n + 2) \\ & = 5n + 10 \\ \\ \text{Sum of two additional numbers added} & = 5n + 10 - 5n \\ & = 10 \\ \\ \therefore & \phantom{0} \text{Sets } A \text{ and } C \end{align*}
(ii)
\begin{align} & \text{Set C} \\ \\ & \text{Since the mean is 5 and the original standard deviation is 1.8, majority of the data is around 3.2-6.8.} \\ \\ & \text{Since the values from Set C {4, 6} is within 3.2-6.8, the new standard deviation will be close to 1.8.} \end{align}
(i)
\begin{align*} \bar{x} & = {\sum fx \over \sum f} \\ & = {\sum fx \over 100} \\ \\ \bar{y} & = {\sum fy \over \sum f} \\ & = {\sum fy \over 100} \\ \\ {\bar{x} + \bar{y} \over 2} & = { {\sum fx \over 100} + {\sum fy \over 100} \over 2} \\ & = { {\sum fx + \sum fy \over 100} \over 2} \\ & = { \sum fx + \sum fy \over 200} \\ \\ \\ \text{Yes, } & {\bar{x} + \bar{y} \over 2} \text{ can be used} \end{align*}
(ii)
\begin{align} & \text{No. The combined standard deviation depends on the combined mean mass of both schools.} \end{align}
(iii)
\begin{align*} \sum fx & = 5 \times 45 + 36 \times 50 + 28 \times 55 + 22 \times 60 + 7 \times 65 + 2 \times 70 \\ & = 5480 \\ \\ \sum fy & = 7 \times 40 + 21 \times 45 + 24 \times 50 + 6 \times 55 + 3 \times 60 + 26 \times 65 + 8 \times 70 + 1 \times 75 + 4 \times 80 \\ & = 5580 \\ \\ \text{Mean} & = {5480 + 5580 \over 100 + 100} \\ & = 55.3 \\ \\ \\ \sum fx^2 & = 5 \times 45^2 + 36 \times 50^2 + 28 \times 55^2 + 22 \times 60^2 + 7 \times 65^2 + 2 \times 70^2 \\ & = 303400 \\ \\ \sum fy^2 & = 7 \times 40^2 + 21 \times 45^2 + 24 \times 50^2 + 6 \times 55^2 + 3 \times 60^2 + 26 \times 65^2 + 8 \times 70^2 + 1 \times 75^2 + 4 \times 80^2 \\ & = 322950 \\ \\ \text{S.D.} & = \sqrt{ {303400 + 322950 \over 100 + 100} - (55.3)^2 } \\ & = 8.5825 \\ & \approx 8.58 \end{align*}