Questions and Solutions 🧠 1. Identify the sample and population. Question: The education minister wants to know the status of campus placements of B.Tech students in different engineering institutes of India. An analyst did a survey on the randomly selected four IITs of India and analysed the status of campus placements. Based on the information given, answer the question. Options: (a) The sample consists of all the engineering institutes of India and the population consists of randomly selected four IITs of India. (b) The sample consists of all the IITs of India and the population consists of all the engineering institutes of India. (c) The sample consists of all IITs of India and the population consists of randomly selected four IITs of India. (d) The sample consists of four randomly selected IITs of India and the population consists of all the engineering institutes of India.

1. Which of the following statements is/are incorrect? Options: (a) To represent the share of a particular category, bar chart is the most appropriate graphical representation. (b) The multiplication of the total number of observations and relative frequency of a particular observation should be equal to the frequency of that observation. (c) Mean can be defined for a categorical variable. (d) Mode of a categorical variable is the widest slice in a pie chart.

1. The numbers a, b, c, d have frequencies (x + 6), (x + 2), (x − 3) and x respectively. If their mean is m, find the value of x. (Enter the value as next highest integer) Solution: $$ \frac{a(x + 6) + b(x + 2) + c(x − 3) + dx}{(x + 6) + (x + 2) + (x − 3) + x} = m $$$$ \frac{ax + 6a + bx + 2b + cx − 3c + dx}{4x + 5} = m $$$$ ax + bx + cx + dx + 6a + 2b − 3c = m(4x + 5) = (4m)x + 5m $$$$ (a + b + c + d − 4m)x = 5m − 6a − 2b + 3c $$$$ x = \frac{5m − 6a − 2b + 3c}{a + b + c + d − 4m} $$Suppose, we substitute values of a, b, c, d and m as 2, 7, 9, 17 and 6.88 respectively,

Questions 1–6: Sales Data Analysis Context: The phone brands OnePlus, Vivo, and Oppo are owned by BBK Electronics. Table 4.1.G represents sales (in Lakhs) of OnePlus and BBK Electronics by different dealers in Chennai and Punjab in 2010. Dealer’s Location OnePlus BBK Electronics Chennai a b Punjab c d Chennai e f Punjab g h Chennai i j Punjab k l Chennai m n 1. What is the population standard deviation of sales of OnePlus? Solution: Let $ m_x $ and $ \sigma_x $ be the mean and population standard deviation of sales of OnePlus, respectively.

1. Covering Registers with Coloured Papers Question: Vinod has $ n $ registers and $ m $ cover papers of different colours. In how many ways can he cover all the registers with cover papers? Answer: $ m \times (m - 1) \times (m - 2) \times \dots \times (m - n + 1) $ Solution: For the 1st register: $ m $ ways. For the 2nd register: $ m-1 $ ways. … For the $ n $th register: $ m-n+1 $ ways. Total: $ m \times (m - 1) \times \dots \times (m - n + 1) $.

1. Five-Digit Numbers Divisible by 4 Question: How many 5-digit numbers can be formed from the numbers 0, 2, 4, 5, 7 and 9 (without repetition), such that it is divisible by 4? Options: a. 120 b. 144 c. 132 d. 104 Answer: b. 144 Solution: A number is divisible by 4 if its last two digits form a number divisible by 4. Possible last two digits: 04, 20, 24, 40, 52, 72, 92.

Questions and Solutions 1. Arranging Boys and Girls with Exactly 4 Boys Between Two Girls Question: $ m $ boys and 2 girls are to be placed next to each other in the school ground for morning assembly. What is the probability that there are exactly 4 boys between the 2 girls? Options: a. $\frac{2 m-5}{m+2 P_{2}}$ b. $\frac{2 m-6}{m+2 P_{2}}$ c. $\frac{2 m-6}{m+3 P_{2}}$ d. $\frac{2 m-4}{m+2 P_{2}}$

1. Probability of the k-th Record Being the Last Five-Wicket Haul Question: Zaheer Khan has taken $ m $ five-wicket hauls in his last $ n $ matches. His match records are selected at random, one by one, and analyzed. If none of the match records is analyzed more than once, then what is the probability that the $ k^{th} $ one analyzed is his last five-wicket haul match?

A discrete random variable $ X $ can take the values $ 1,2,3,···,n $. For these values the cumulative distribution function is defined by - $$ F(x) = P(X < x) = \frac{x^2 + k}{m}; \quad x = 1, 2, 3, \ldots, n $$ Find the value of $ k $.

There are $2^n$ numbered cards in a deck, among which ${}^n C_i$ cards bear the number $i$ ($i = 0,1,2,\ldots,n$). From the deck, $m$ cards are drawn with replacement. What is the expectation of the sum of their numbers? (Enter the answer correct to one decimal accuracy.)

A match predictor claims that he can predict the result of a match correctly $x\%$ of the time. It is agreed that his claim will be accepted if he correctly predicts the results of at least $m$ of $n$ matches. What is the probability that his claim gets rejected?

Let a random variable be uniformly distributed over $[a, b]$ with expectation $e$ and variance $v$ respectively. Find the value of $ab$.

Questions (1)–(6): Based on Table 4.1.G Dealer’s Location OnePlus BBK Electronics Chennai 2 | 2 | | Punjab |