Question 1 |

A pair of six-faced dice is rolled thrice. The probability that the sum of the
outcomes in each roll equals 4 in exactly two of the three attempts is ______.
(round off to three decimal places)

0.045 | |

0.078 | |

0.018 | |

0.025 |

Question 1 Explanation:

Event, E = {(1, 3)(3, 1)(2, 2)}

n(E) = 3

n(S) = 36

p=P(E)=\frac{3}{36}=\frac{1}{12}

q=P(\bar{E})=1-\frac{1}{12}=\frac{11}{12}

P(x)=3C_2(p^2)(q^1)=3 \times \left (\frac{1}{12} \right )^2\left (\frac{11}{12} \right )=0.02

n(E) = 3

n(S) = 36

p=P(E)=\frac{3}{36}=\frac{1}{12}

q=P(\bar{E})=1-\frac{1}{12}=\frac{11}{12}

P(x)=3C_2(p^2)(q^1)=3 \times \left (\frac{1}{12} \right )^2\left (\frac{11}{12} \right )=0.02

Question 2 |

Match the following attributes of a city with the appropriate scale of
measurements.

\begin{array}{|l|l|}\hline \text{Attribute}&\text{Scale of measurement} \\ \hline \text{(P) Average temperature } (^{\circ}C)\text{ of a city} & \text{((I) Interval} \\ \hline \text{(Q) Name of a city} & \text{(II) Ordinal}\\ \hline \text{(R) Population density of a city}& \text{(III) Nominal}\\ \hline \text{(S) Ranking of a city based on ease of business}& \text{(IV) Ratio}\\ \hline \end{array}

Which one of the following combinations is correct?

\begin{array}{|l|l|}\hline \text{Attribute}&\text{Scale of measurement} \\ \hline \text{(P) Average temperature } (^{\circ}C)\text{ of a city} & \text{((I) Interval} \\ \hline \text{(Q) Name of a city} & \text{(II) Ordinal}\\ \hline \text{(R) Population density of a city}& \text{(III) Nominal}\\ \hline \text{(S) Ranking of a city based on ease of business}& \text{(IV) Ratio}\\ \hline \end{array}

Which one of the following combinations is correct?

(P)-(I), (Q)-(III), (R)-(IV), (S)-(II) | |

(P)-(II), (Q)-(I), (R)-(IV), (S)-(III) | |

(P)-(II), (Q)-(III), (R)-(IV), (S)-(I) | |

(P)-(I), (Q)-(II), (R)-(III), (S)-(IV) |

Question 2 Explanation:

Meaning of

Nominal -> a name or term

Ordinal -> in an ordered sequence

Ratio -> quantitative relation between two things

Interval -> indicates average of a range

Nominal -> a name or term

Ordinal -> in an ordered sequence

Ratio -> quantitative relation between two things

Interval -> indicates average of a range

Question 3 |

A set of observations of independent variable (x) and the corresponding
dependent variable (y) is given below.

\begin{array}{|c|c|c|c|c|} \hline x&5&2&4&3 \\ \hline y&16&10&13&12\\ \hline \end{array}

Based on the data, the coefficient a of the linear regression model

y = a + bx

is estimated as 6.1.

The coefficient b is ______________ . (round off to one decimal place)

\begin{array}{|c|c|c|c|c|} \hline x&5&2&4&3 \\ \hline y&16&10&13&12\\ \hline \end{array}

Based on the data, the coefficient a of the linear regression model

y = a + bx

is estimated as 6.1.

The coefficient b is ______________ . (round off to one decimal place)

6.1 | |

1.9 | |

2.2 | |

3.6 |

Question 3 Explanation:

We know that, normal equation for fitting of straight lines are

\begin{aligned} \Sigma y&=na+b\Sigma x\\ \Sigma xy&=a\Sigma x+b\Sigma x^2\\ n&=4\\ 51&=4a+b(14)\\ 188&=a(14)+b(54) \end{aligned}

After solving, a=6.1 and b=1.9

\begin{aligned} \Sigma y&=na+b\Sigma x\\ \Sigma xy&=a\Sigma x+b\Sigma x^2\\ n&=4\\ 51&=4a+b(14)\\ 188&=a(14)+b(54) \end{aligned}

After solving, a=6.1 and b=1.9

Question 4 |

The shape of the cumulative distribution function of Gaussian distribution is

Horizontal line | |

Straight line at 45 degree angle | |

Bell-shaped | |

S-shaped |

Question 4 Explanation:

PDF:f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-(x-\mu )^2/(2\sigma ^2)}

CDF:F(x)=\frac{1}{2}\left [ 1+eff\left ( \frac{x-\mu }{\sigma \sqrt{2}} \right ) \right ]

Question 5 |

A fair (unbiased) coin is tossed 15 times. The probability of getting exactly 8 Heads
(round off to three decimal places), is ________.

0.523 | |

0.421 | |

0.196 | |

0.223 |

Question 5 Explanation:

P(H)=\frac{1}{2}

P(T)=\frac{1}{2}

Probability of getting exactly 8 heads out of 15 trial ={^{15}C_8}\left [ \frac{1}{2} \right ]^8 \times \left [ \frac{1}{2} \right ]^{15-8}=0.196

P(T)=\frac{1}{2}

Probability of getting exactly 8 heads out of 15 trial ={^{15}C_8}\left [ \frac{1}{2} \right ]^8 \times \left [ \frac{1}{2} \right ]^{15-8}=0.196

Question 6 |

The probability density function of a continuous random variable distributed uniformly between x and y (for y \gt x) is

\frac{1}{x-y} | |

\frac{1}{y-x} | |

x-y | |

y-x |

Question 6 Explanation:

Probability density function of uniform distribution is f(x)=\frac{1}{y-x}

Question 7 |

Probability (up to one decimal place) of consecutively picking 3 red balls without replacement from a box containing 5 red balls and 1 white ball is ______

0 | |

0.25 | |

0.5 | |

1 |

Question 7 Explanation:

Probability, \bar{P}=\frac{5}{6}\times \frac{4}{5}\times \frac{3}{4}=\frac{1}{2}=0.5

Question 8 |

A probability distribution with right skew is shown in the figure.

The correct statement for the probability distribution is

The correct statement for the probability distribution is

Mean is equal to mode | |

Mean is greater than median but less than mode | |

Mean is greater than median and mode | |

Mode is greater than median |

Question 8 Explanation:

t_{L}\lt t_{mean}=Curve is skew to right.

Mode \lt mean

i.e., Mean \gt median and mode

Mean is greater than the mode and the median.This is common for a distribution that is skewed to the right [i.e., bunched up toward the left and a 'tail' stretching toward the right].

Question 9 |

The graph of a function f(x) is shown in the figure.

For f(x) to be a valid probability density function, the value of h is

For f(x) to be a valid probability density function, the value of h is

0.33 | |

0.66 | |

1 | |

3 |

Question 9 Explanation:

\begin{aligned} \int_{0}^{3}f\left ( x \right )dx &=1 \\ \int_{0}^{1}f\left ( x \right )dx+\int_{1}^{2}f\left ( x \right )dx+\int_{2}^{3}f\left ( x \right )dx &=1 \\ \frac{h}{2}+\frac{2h}{2+\frac{3h}{2}} &=1 \\ 6h &=2 \\ \Rightarrow\; \; h &=\frac{1}{3} \end{aligned}

Question 10 |

A two-faced fair coin has its faces designated as head (H) and tail (T). This coin is tossed three times in succession to record the following outcomes: H, H, H. If the coin is tossed one more time, the probability (up to one decimal place) of obtaining H again, given the previous realizations of H, H and H, would be______

0.25 | |

0.5 | |

0.75 | |

0.2 |

There are 10 questions to complete.