Solutions: Statistics - Test 3

Q1. (b) Median

Cumulative frequency tables are primarily used to find the Median.

Q2. (b) 12

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
$n=10$ (even). Median = Average of 5th and 6th terms = $\frac{11+13}{2} = \frac{24}{2} = 12$.

Q3. (c) cumulative frequency of the class preceding the median class

In the median formula, $cf$ is the cumulative frequency of the class preceding the median class.

Q4. (a) Both A and R are true and R is the correct explanation of A.

Data: 5, 7, 9, 11, 15, 17, 19. $n=7$ (odd).
Median = $(\frac{7+1}{2})^{th} = 4^{th}$ term = 11.

Q5. Arrange in ascending order: 20, 22, 23, 25, 26, 28, 31, 32, 34, 35.

$n=10$ (even). Median = Average of 5th and 6th terms.
5th term = 26, 6th term = 28.
Median = $\frac{26+28}{2} = 27$.

Q6. Data: 6, 7, $x-2$, $x$, 17, 20. $n=6$.

Median = Average of 3rd and 4th terms = $\frac{(x-2) + x}{2} = 16$.
$2x - 2 = 32 \Rightarrow 2x = 34 \Rightarrow x = 17$.

Q7. Total frequency $N = 4+4+8+10+12 = 38$.

$N/2 = 19$.
Cumulative Frequencies: 4, 8, 16, 26, 38.
The cumulative frequency just greater than 19 is 26, which corresponds to the class 30-40.
Median Class is 30-40.

Q8. Calculate Median.

$N = 5+15+25+20+7 = 72$. $N/2 = 36$.
CF: 5, 20, 45, 65, 72.
Median Class: 20-30 (CF > 36 is 45).
$l=20, cf=20, f=25, h=10$.
Median = $20 + \frac{36-20}{25} \times 10 = 20 + \frac{16}{25} \times 10 = 20 + \frac{160}{25} = 20 + 6.4 = 26.4$.

Q9. Empirical Formula: 3 Median = Mode + 2 Mean.

Given Median = 24, Mode = 29, Mean = 21.5.
LHS = $3 \times 24 = 72$.
RHS = $29 + 2(21.5) = 29 + 43 = 72$.
Since LHS = RHS, the relationship holds. No variable $p$ was in the question context provided, but assuming verification was the goal. If $p$ was intended to be one of the values, say Median = $p$, then $3p = 72 \Rightarrow p=24$.

Q10. Median = 525. $N=100$.

CF Table:
0-100: 2
100-200: 7
200-300: $7+x$
300-400: $19+x$
400-500: $36+x$
500-600: $56+x$
600-700: $56+x+y$
700-800: $65+x+y$
800-900: $72+x+y$
900-1000: $76+x+y$
Total $76+x+y = 100 \Rightarrow x+y = 24$.
Median is 525, so Median Class is 500-600.
$l=500, f=20, cf=36+x, h=100, N/2=50$.
$525 = 500 + \frac{50 - (36+x)}{20} \times 100$
$25 = (14-x) \times 5 \Rightarrow 5 = 14-x \Rightarrow x = 9$.
$y = 24 - 9 = 15$.
Values: $x=9, y=15$.

(i) CF: 5, 22, 40, 48, 50.

(ii) $N=50, N/2=25$. CF just greater than 25 is 40.
Median Class is 50-55.

(iii) $l=50, cf=22, f=18, h=5$.
Median = $50 + \frac{25-22}{18} \times 5 = 50 + \frac{3}{18} \times 5 = 50 + \frac{15}{18} = 50 + 0.83 = 50.83$ kg.