Statistics

Miscellaneous Exercise Solutions
Useful Identities:
Sum of Observations: $\sum x_i = n\bar{x}$
Sum of Squares: $\sum x_i^2 = n(\sigma^2 + \bar{x}^2)$
Correction: Correct Sum = Old Sum – Wrong Values + Correct Values
Q1
1. Mean=9, Var=9.25, n=8. Six obs: 6, 7, 10, 12, 12, 13. Find other two.
Sum $\sum x_i = 8 \times 9 = 72$. Known sum = $6+7+10+12+12+13 = 60$.
Let remaining be $x, y$. $x + y = 72 – 60 = 12$.
Sum Sq $\sum x_i^2 = 8(9.25 + 9^2) = 8(9.25 + 81) = 8(90.25) = 722$.
Known Sq $36 + 49 + 100 + 144 + 144 + 169 = 642$.
$x^2 + y^2 = 722 – 642 = 80$.
Solve We have $x+y=12$ and $x^2+y^2=80$.
$x^2 + (12-x)^2 = 80 \Rightarrow x^2 + 144 + x^2 – 24x = 80$.
$2x^2 – 24x + 64 = 0 \Rightarrow x^2 – 12x + 32 = 0$.
Factors $(x-8)(x-4) = 0$. So $x=4, y=8$ or vice-versa.
Remaining observations: 4 and 8
Q2
2. Mean=8, Var=16, n=7. Five obs: 2, 4, 10, 12, 14. Find other two.
Sum $\sum x_i = 7 \times 8 = 56$. Known sum = $2+4+10+12+14 = 42$.
$x + y = 56 – 42 = 14$.
Sum Sq $\sum x_i^2 = 7(16 + 8^2) = 7(16 + 64) = 7(80) = 560$.
Known Sq $4 + 16 + 100 + 144 + 196 = 460$.
$x^2 + y^2 = 560 – 460 = 100$.
Solve $x+y=14, x^2+y^2=100$.
$x^2 + (14-x)^2 = 100 \Rightarrow x^2 + 196 + x^2 – 28x = 100$.
$2x^2 – 28x + 96 = 0 \Rightarrow x^2 – 14x + 48 = 0$.
Factors $(x-6)(x-8) = 0$. So $x=6, y=8$.
Remaining observations: 6 and 8
Q3
3. Mean=8, SD=4. Each observation multiplied by 3. Find new Mean and SD.
Property If each observation is multiplied by $k$:
New Mean = $k \times$ Old Mean
New SD = $|k| \times$ Old SD.
Calc New Mean = $3 \times 8 = 24$.
New SD = $3 \times 4 = 12$.
New Mean = 24, New SD = 12
Q4
4. Prove: If $y_i = ax_i$, Mean becomes $a\bar{x}$ and Variance becomes $a^2\sigma^2$.
Mean $\bar{y} = \frac{1}{n}\sum ax_i = a(\frac{1}{n}\sum x_i) = a\bar{x}$.
Variance $\sigma_y^2 = \frac{1}{n}\sum (y_i – \bar{y})^2 = \frac{1}{n}\sum (ax_i – a\bar{x})^2$.
Simplify $\frac{1}{n}\sum a^2(x_i – \bar{x})^2 = a^2 [\frac{1}{n}\sum (x_i – \bar{x})^2] = a^2 \sigma_x^2$.
Proved. Mean scaled by $a$, Variance by $a^2$.
Q5
5. Mean=10, SD=2, n=20. Obs 8 was wrong. Calculate correct stats.
Original Sum $S = 20 \times 10 = 200$.
Sum Sq $SS = 20(2^2 + 10^2) = 20(4+100) = 2080$.

(i) Wrong item (8) omitted

New Sums $S’ = 200 – 8 = 192$.
$SS’ = 2080 – 8^2 = 2080 – 64 = 2016$.
$n’ = 19$.
Stats Mean = $192/19 \approx 10.1$.
Var = $\frac{2016}{19} – (10.1)^2 = 106.1 – 102.01 = 4.09$. SD = $\sqrt{4.09} \approx 2.02$.

(ii) Replaced by 12

New Sums $S” = 200 – 8 + 12 = 204$.
$SS” = 2080 – 64 + 144 = 2160$.
$n” = 20$.
Stats Mean = $204/20 = 10.2$.
Var = $\frac{2160}{20} – (10.2)^2 = 108 – 104.04 = 3.96$. SD = $\sqrt{3.96} \approx 1.99$.
Q6
6. Mean=20, SD=3, n=100. Incorrect obs 21, 21, 18 omitted. Find correct stats.
Original Sum $S = 100 \times 20 = 2000$.
Sum Sq $SS = 100(3^2 + 20^2) = 100(9 + 400) = 40900$.
Correction Remove 21, 21, 18. ($n’ = 97$).
$S’ = 2000 – (21+21+18) = 2000 – 60 = 1940$.
$SS’ = 40900 – (21^2 + 21^2 + 18^2) = 40900 – (441+441+324) = 40900 – 1206 = 39694$.
New Mean $\bar{x}’ = 1940 / 97 = 20$.
New SD Var = $\frac{39694}{97} – (20)^2 = 409.216 – 400 = 9.216$.
SD = $\sqrt{9.216} \approx 3.036$.
Mean = 20, SD = 3.04
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