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)
3. What is the sample variance of the original dataset? (Correct up to 2 decimal place accuracy)
Solution:
Sample variance,
$$
s^2 = \frac{\sum(x_i - \bar{x})^2}{N-1}
$$
Let $\sum x_i^2 = A$ for noted dataset and for the original dataset be $B$.
$$
B = A - p^2 + x^2
$$
where,
$$
A = \left(\frac{s^2 + N m^2}{N-1}\right) \times (N-1)
$$$$
\text{Sample variance for the original dataset} = \frac{B}{N-1} - \frac{(T')^2}{N(N-1)}
$$
4. Let the data $x_1, x_2, …, x_n$ represent the retail prices in rupees of a certain commodity in n randomly selected shops in a particular city. What will be the sample variance in the retail prices, if c rupees is added to all the retail prices? (Correct up to 2 decimal place accuracy)
Solution:
If $c$ rupees is added to all retail prices, new prices $y_i = x_i + c$.
No observations outside these bounds. Hence, no outliers1.
8. In a deck, there are cards numbered 1 to n such that the number of cards of a given number is the same as the number on the card. Which of the following statement(s) is/are true about the mean and mode of the numbers on this deck of card?
a. Mode is n.
b. Mean is $\frac{2n + 1}{3}$.
c. Mode is n − 1.
d. Mean is n.
e. Mean is $\frac{n + 1}{2}$.
f. Mode is not defined for this data.
Answer: a, b
Solution:
Number (xi), Frequency (fi): 1:1, 2:2, …, n:n
Mode = n
Total observations = $1 + 2 + … + n = \frac{n(n+1)}{2}$
9. Figure 3.1.G shows a stem and leaf plot of the ratings (out of 100) of an actor’s performance in different movies. What is the Inter Quartile Range (IQR) (Correct up to 1 decimal point accuracy)?
