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1.

Find the variance of the first 10 natural numbers.(a) 7.25(b) 7(c) 8.25(d) 8The question was asked in examination.Question is taken from Statistics in chapter Statistics of Mathematics – Class 11

Answer» RIGHT answer is (c) 8.25

To ELABORATE: Variance = \(\FRAC{1}{10}\) [1^2 + 2^2 +…+ 10^2] – \(\frac{1}{20}\)[1 + 2 +…. 10]^2 = 38.5 – 30.25 = 8.25.
Discussion
2.

The mean of 5 observations is 3 and variance is 2. If three of the five observations are 1, 3, 5, find the other two.(a) 2, 6(b) 3, 3(c) 1, 5(d) 2, 4I had been asked this question in unit test.My query is from Statistics in section Statistics of Mathematics – Class 11

Answer»

Correct ANSWER is (d) 2, 4

To EXPLAIN I would say: \(\frac{1}{5}\) (Σ xi) = 3 = 1 + 3 + 5 + x4 + X5

⇒ x4 + x5 = 15 ———— (i)

Now, \(\frac{1}{5}\) Σ (xi)^2 – \([\frac{1}{5}\) Σ xi]^2 = 4

⇒ (x4)^2 + (x5)^2 = 30 —————-(ii)

Solving equation i and ii, x4 = 2 and x5 = 4.
Discussion
3.

If the S.D. is a set of observations is 4 and if each observation is divided by 4, find the S.D. of the new observations.(a) 4(b) 3(c) 2(d) 1I had been asked this question in an interview.This is a very interesting question from Statistics in section Statistics of Mathematics – Class 11

Answer»

The CORRECT option is (d) 1

For explanation I WOULD say: σ1 = 4 and d = (X – 0)/4

And σ1 = |4|σd

⇒ σd = 4/4 = 1.
Discussion
4.

What is the geometric mean of 5,5^2, ….,5^n?(a) 5^n/2(b) 5^(n+1)/2(c) 5^n(n+1)/2(d) 5^nThis question was addressed to me at a job interview.This intriguing question comes from Statistics in section Statistics of Mathematics – Class 11

Answer» RIGHT choice is (B) 5^(n+1)/2

Explanation: Geometric Mean = (5 X 5^2 x …… x 5^n)^1/n = [5^(1+2+…+n)]^1/n = [5^n(n+1)/2]^1/n = 5^(n+1)/2.
Discussion
5.

Find the mean deviation from mean of the observations: a, a+d, …., (a+2nd).(a) n(n + 1)d^2/3(b) n(n + 1)d^2/2(c) a + n(n + 1)d^2/2(d) n(n + 1)d/(2n + 1)I have been asked this question in an interview for job.This interesting question is from Statistics in division Statistics of Mathematics – Class 11

Answer» RIGHT answer is (d) n(n + 1)d/(2N + 1)

To elaborate: Mean = \(\FRAC{1}{n}\) Σ XI = \(\frac{1}{2n+1}\) [a + (a + d) + … + (a + 2nd)] = a + nd

⇒ Mean Deviation = \(\frac{1}{2n+1}\) [2 × d × (1 + 2 + … + n)] = [n (n + 1) (d)]/(2n + 1).
Discussion
6.

If the standard deviation of a data is 820 and mean of the data is 50, find the coefficient of variation.(a) 16.4(b) 164(c) 1640(d) 1.64I got this question by my school teacher while I was bunking the class.Question is taken from Statistics in division Statistics of Mathematics – Class 11

Answer»

Right answer is (a) 16.4

The EXPLANATION is: COEFFICIENT of Variance = (Standard Deviation/Mean) × 100

⇒ Coefficient of Variance = (820/50) × 100 = 16.4.
Discussion
7.

The change in which of following terms does not affect the standard deviation?(a) Origin(b) Scale(c) Origin and scale(d) Neither origin nor scaleI have been asked this question in an interview for job.My doubt stems from Statistics in division Statistics of Mathematics – Class 11

Answer»

The CORRECT answer is (a) ORIGIN

The best explanation: Change in origin does not AFFECT the standard DEVIATION, whereas standard deviation is affected by scale.
Discussion
8.

If the standard deviation of the numbers 2, 4, 5 & 6 is a constant a, then find the standard deviation of the numbers 4, 6, 7 & 8.(a) a + 2(b) 2a(c) 4a(d) aI have been asked this question in my homework.This intriguing question originated from Statistics topic in division Statistics of Mathematics – Class 11

Answer»

Right option is (d) a

The explanation is: The standard DEVIATION is INDEPENDENT of the CHANGE of origin. So, the standard deviation for the new set will REMAIN the same.
Discussion
9.

Assuming the variance of four numbers w, x, y, and z as 9. Find the variance of 5w, 5x, 5y and 5z.(a) 225(b) 5/9(c) 9/5(d) 54The question was posed to me during an interview.Enquiry is from Statistics topic in division Statistics of Mathematics – Class 11

Answer» CORRECT OPTION is (a) 225

Explanation: (σx)^2 = h^2(σu)^2, if U = (X – h)/a

Now, h = (1/5). ⇒ Variance, (σu)^2 = 9 × 25 = 225
Discussion
10.

A fisherman is weighing each of 50 fishes. Their mean weight worked out is 50 gm and a standard deviation of 2.5 gm. Later it was found that the measuring scale was misaligned and always under reported every fish weight by 2.5 gm. Find the mean and standard deviation of fishes.(a) 52.5,2.5(b) 30,5(c) 50,5(d) 48.5,2.5The question was asked during an interview for a job.My doubt stems from Statistics in chapter Statistics of Mathematics – Class 11

Answer» CORRECT ANSWER is (a) 52.5,2.5

For explanation I would say: SINCE mean(X + B) = meanX + b and Var(X + b) = VarX, so we get correct mean as 50 + 2.5 = 52.5 GM and S.D. is 2.5 gm.
Discussion
11.

What is the median and standard deviation of a distribution are 50 and 5 respectively, if each item is increased by 4.(a) Median will increase and S.D. will increase(b) Both will remain same(c) median will go up by 2 but S.D. will remain same(d) median will increase and S.D. will decreaseThis question was addressed to me in an international level competition.Asked question is from Statistics topic in chapter Statistics of Mathematics – Class 11

Answer»

Correct answer is (C) median will go up by 2 but S.D. will remain same

For EXPLANATION: Median will CHANGE if the observations are changes but standard deviation is UNAFFECTED by the origin. So, here the median will go up by 4 and S.D. will remain same.
Discussion
12.

In a class there are 20 juniors, 15 seniors and 5 graduate students. If the junior averaged 65 in the midterm exam, the senior averaged 70 and the graduate students averaged 91, then what is the mean of the centre class approximately?(a) 71(b) 74(c) 70(d) 72The question was posed to me in semester exam.I'm obligated to ask this question of Statistics topic in chapter Statistics of Mathematics – Class 11

Answer»

Correct option is (c) 70

To elaborate: COMBINED MEAN = (Σ xini)/(Σ NI) = (20 × 65 + 15 × 70 + 5 × 91)/(20 + 15 + 5) = 70.
Discussion
13.

If standard deviation of a data is 40 and the coefficient of variation is 25600, then find the mean.(a) 64(b) 6.4(c) 640(d) 0.64The question was posed to me during an internship interview.The origin of the question is Statistics in portion Statistics of Mathematics – Class 11

Answer»

The CORRECT choice is (B) 6.4

The EXPLANATION: COEFFICIENT of Variance = (Standard Deviation/Mean) × 100

⇒ Mean = 25600/4000 = 6.4.
Discussion
14.

The mean deviation of an ungrouped data is 150. If each observation is increased by 3.5%, then what is the new mean deviation?(a) 153.5(b) 3.5(c) 155.25(d) 150I had been asked this question in homework.My doubt stems from Statistics in section Statistics of Mathematics – Class 11

Answer»

Right CHOICE is (c) 155.25

Explanation: If X1, x2, …, xn are the OBSERVATIONS, then the new observations are (1.035) x1, (1.035) x2, ……, (1.035) xn.

Therefore, the new mean is (1.035) X̄

Now, Mean deviation = \(\frac{1}{n}\)[Σi = n |XI – mean|]

⇒ New mean deviation = \(\frac{1}{n}\)[Σi = n|(1.035)xI – (1.035) x̄|] = (1.035) × \(\frac{1}{n}\)[Σi = n |xI – mean|] = 1.035 x 150 = 155.25.
Discussion
15.

If the coefficient of variation is 100 the mean of the data is 25, then find the standard deviation.(a) 5(b) 10(c) 15(d) 25This question was posed to me by my school principal while I was bunking the class.The doubt is from Statistics in chapter Statistics of Mathematics – Class 11

Answer» RIGHT option is (d) 25

Explanation: Coefficient of Variance = (STANDARD DEVIATION/Mean) × 100

⇒ Standard Deviation = 25.
Discussion
16.

If the standard deviation of a data is 0.012. Find the variance.(a) 0.144(b) 0.00144(c) 0.000144(d) 0.0000144This question was posed to me by my school teacher while I was bunking the class.I would like to ask this question from Statistics topic in portion Statistics of Mathematics – Class 11

Answer»

Correct ANSWER is (c) 0.000144

The EXPLANATION is: S.D. = √VARIANCE

⇒ (0.012)^2 = Variance

⇒ Variance = 0.000144
Discussion
17.

The mean of two samples of the sizes 250 and 320 were found to be 20,12 respectively. Their standard deviations were 2 & 5, respectively. Find the variance of combined sample of size 650.(a) 10.23(b) 32.5(c) 65(d) 26.17I got this question in an internship interview.This interesting question is from Statistics topic in portion Statistics of Mathematics – Class 11

Answer»

The correct answer is (d) 26.17

Explanation: COMBINED Mean = X = (n1x1 + n2x2)/(n1 + N2) = (250 × 20 + 320 × 12)/650 = 13.6

Let d1 = x1 – X = 20-13.6 = 6.4, d2 = x2 – X = 12-13.6 = -1.6

Variance = [250(6.4 + 40.96) + 320(13.6 + 2.56)]/650 = 26.17.
Discussion
18.

The mean and Standard deviation of a sample were found to be 9.5 and 2.5, respectively. Later, an additional observation 15 was added to the original data. Find the S.D. of the 11 observation.(a) 2.6(b) 2.8(c) 2.86(d) 3.24This question was posed to me in exam.My question is from Statistics topic in division Statistics of Mathematics – Class 11

Answer» RIGHT answer is (c) 2.86

The EXPLANATION: \(\frac{1}{n}\) (Σ xi) = 9.5,

\(\frac{1}{n}\) (Σ [(xi)^2 – (X)^2] = 6.25, Σ xi = 95 and \(\frac{1}{10}\) Σ[(X)^2] = 96.5

Corrected MEAN = (95 + 15)/11 = 10

Corrected Variance = [(1/11) x (965 + 225)] – 100 = 90/11

⇒ Standard deviation = √Variance = √(90/11) = 2.86.
Discussion