Statistics · NDA previous year questions

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2-2bx+c^2=0$ where $b,c$ are positive real numbers. Let $A$ be the arithmetic mean of $\alpha$ and $\beta$; and $G$ be the geometric mean of $\alpha$ and $\beta$. What are the roots of the quadratic equation $x^2-(b+c)x+bc=0$ ?

1. A.$A,G$
2. B.$2A,G$
3. C.$A,2G$
4. D.$2A,2G$

If $p,g_1,g_2$ and $q$ are in GP and $m$ is the arithmetic mean of $p$ and $q$, then $\frac{g_1^2}{g_2}+\frac{g_2^2}{g_1}$ is equal to

1. A.$m$
2. B.$2m$
3. C.1
4. D.$\frac{1}{2}$

Passage: The frequency distribution of marks obtained by 100 students in a certain subject is given below : $$\begin{array}{lllll}\text{Marks}& 0\text{-}10& 10\text{-}20& 20\text{-}30& 30\text{-}40\\ \text{No. of students}& 10& 20& f& 40\end{array}$$ Question: What is the standard deviation ?

1. A.$8\cdot 5$
2. B.9
3. C.$9\cdot 5$
4. D.10

Passage: The frequency distribution of marks obtained by 100 students in a certain subject is given below : $$\begin{array}{lllll}\text{Marks}& 0\text{-}10& 10\text{-}20& 20\text{-}30& 30\text{-}40\\ \text{No. of students}& 10& 20& f& 40\end{array}$$ Question: What is the mean deviation about the arithmetic mean ?

1. A.$5\cdot 5$
2. B.6
3. C.$6\cdot 5$
4. D.8

The standard deviation of 100 observations is 10. If 5 is multiplied to each of the observations, then what is the new standard deviation ?

1. A.20
2. B.25
3. C.40
4. D.50

If the mean of 20 observations, namely $x_1,x_2,x_3,\dots ,x_{20}$ is $1\cdot 414$, then what is the value of ${\sum }_{i=1}^{20}100(2x_i+4)$ ?

1. A.24168
2. B.20828
3. C.15248
4. D.13656

Let two lines of regression be $x+y+11=0$ and $2x+3y+4=0$ for some data. What is the value of correlation coefficient between $x$ and $y$ ?

1. A.$-\sqrt{\frac{2}{3}}$
2. B.$-\sqrt{\frac{1}{6}}$
3. C.$\sqrt{\frac{2}{3}}$
4. D.$\sqrt{\frac{1}{6}}$

Passage: The equation of the sphere $S$ is $x^2+y^2+z^2-4x-6y-12z+k=0$. Question: What is the radius of the sphere passing through origin and concentric with the sphere $S$ ?

1. A.$\frac{7}{2}$
2. B.5
3. C.7
4. D.Cannot be determined due to insufficient data

For a Binomial distribution with mean 6 and standard deviation $\sqrt{2}$, what is the value of $P(X=0)$ ?

1. A.${\left(\frac{1}{3}\right)}^9$
2. B.${\left(\frac{2}{3}\right)}^9$
3. C.$\frac{1}{3}{\left(\frac{2}{3}\right)}^8$
4. D.$\frac{2}{3}{\left(\frac{1}{3}\right)}^8$

What is the harmonic mean of the numbers C(10, 3), C(10, 4), C(10, 5), C(10, 6) and C(10, 7) ?

1. A.3150/19
2. B.4000/19
3. C.252
4. D.225

The standard deviation of 100 observations is 10. If 20 is added to each observation, then what will be the new standard deviation?

1. A.10
2. B.15
3. C.20
4. D.25

The frequency distribution of the marks obtained by students in a Science examination is given below: $$\begin{array}{lllll}\text{Marks}& 5\text{-}15& 15\text{-}25& 25\text{-}35& 35\text{-}45\\ \text{Number of students}& 20& 30& 30& 20\end{array}$$ What is the arithmetic mean?

1. A.20
2. B.25
3. C.30
4. D.35

If P(A) = 0.3 , P(B) = 0.4 and P( A |B)=0.5, then what is the value P( B |A) ?

1. A.0.325
2. B.0.333
3. C.0.375
4. D.0.667

If P(A) = 1/3 , P(B) = 1/2 and $P(A\cap B)=1/4$, then what is the value of $P(\overline{A}\cup B)$ ?

1. A.7/12
2. B.2/3
3. C.3/4
4. D.11/12

Consider the following statements: I. Mean and variance have the same unit of measurement. II. Mean deviation and standard deviation have the same unit of measurement. Which of the statements given above is/are correct?

1. A.I only
2. B.II only
3. C.Both I and II
4. D.Neither I nor II

Consider the following for the four (04) items that follow: The frequency distribution of height of students of a class is given below: $$\begin{array}{cc}\text{Height (in cm}& \text{Number of Students}\\ 160\text{-}162& 12\\ 162\text{-}164& 15\\ 164\text{-}166& 24\\ 166\text{-}168& 13\end{array}$$ What is the total number of students whose height is less than or equal to 165 cm?

1. A.15
2. B.39
3. C.51
4. D.None of the above

Consider the following for the four (04) items that follow: The frequency distribution of height of students of a class is given below: $$\begin{array}{cc}\text{Height (in cm)}& \text{Number of Students}\\ 160\text{-}162& 12\\ 162\text{-}164& 15\\ 164\text{-}166& 24\\ 166\text{-}168& 13\end{array}$$ What is the median height of the class?

1. A.162.41 cm
2. B.163.41 cm
3. C.164.41 cm
4. D.165.41 cm

Consider the following for the four (04) items that follow: The frequency distribution of height of students of a class is given below: $$\begin{array}{cc}\text{Height (in cm}& \text{Number of Students}\\ 160\text{-}162& 12\\ 162\text{-}164& 15\\ 164\text{-}166& 24\\ 166\text{-}168& 13\end{array}$$ The height which occurs most frequently in the class is

1. A.163.5 cm
2. B.163.9 cm
3. C.164.5 cm
4. D.164.9 cm

Consider the following for the four (04) items that follow: The frequency distribution of height of students of a class is given below: $$\begin{array}{cc}\text{Height (in cm}& \text{Number of Students}\\ 160\text{-}162& 12\\ 162\text{-}164& 15\\ 164\text{-}166& 24\\ 166\text{-}168& 13\end{array}$$ The most appropriate graphical representation of the given frequency distribution is

1. A.bar chart
2. B.percentage bar chart
3. C.histogram
4. D.pie chart

Consider the following for the two (02) items that follow: The sum and the sum of squares of the observations corresponding to length X (in cm) and weight Y (in gm) of 50 tropical tubers are given as $\Sigma X=200,\Sigma Y=250,\Sigma X^2=900$ and $\Sigma Y^2=1400$ Which one of the following is correct?

1. A.Variance (X) > Variance (Y)
2. B.Variance (X) < Variance (Y)
3. C.Variance (X) = Variance (Y)
4. D.Cannot be determined from the given data

Consider the following for the two (02) items that follow: The sum and the sum of squares of the observations corresponding to length X (in cm) and weight Y (in gm) of 50 tropical tubers are given as $\Sigma X=200,\Sigma Y=250,\Sigma X^2=900$ and $\Sigma Y^2=1400$ Which one of the following statements is correct?

1. A.Coefficient of variation of X is strictly more than coefficient of variation of Y.
2. B.Coefficient of variation of X is strictly less than coefficient of variation of Y
3. C.Coefficient of variation of X is same as coefficient of variation of Y.
4. D.Coefficient of variation cannot be determined from the given data.

The arithmetic mean of 100 observations is 50. If 5 is subtracted from each observation and then divided by 20, then what is the new arithmetic mean?

1. A.2.25
2. B.3.5
3. C.4.25
4. D.5.5

The standard deviation of 100 observations is 10. If 5 is added to each observation and then divided by 20, then what will be the new standard deviation?

1. A.0.25
2. B.0.5
3. C.0.75
4. D.1.00

If $P(A)=1/3$, $P(B)=1/2$ and $P(A\cap B)=1/4$, then what is the value of $P(B|A^C)$?

1. A.1/8
2. B.3/8
3. C.5/8
4. D.7/8

If $P(A)=\frac{1}{3},P(B)=\frac{1}{2}$ and $P(A\cap B)=\frac{1}{4}$, then what is the value of $P(A^C\cap B^C)$?

1. A.1/4
2. B.5/12
3. C.7/12
4. D.11/12