The marks, x , of a random sample of 50 students in a test were summarised as follows: n = 50 ∑ x = 1508 ∑ x 2 = 51825 Each student's mark is scaled using the formula y = 1 . 5 x + 10 . Find estimates of the population mean and variance of the scaled marks, y .

Given data: n = 50 , Σ x = 1508 , ∑ x 2 = 51825 Formula to find an unbiased estimate of the population variance σ 2 is s 2 = 1 n - 1 Σ x 2 - ( Σ x ) 2 n . Formula to find an unbiased estimate of the population mean, μ = x ¯ = Σ x n . Thus, Est. μ = 1508 50 = 30 . 16 or 30 . 2 Est. σ 2 = 1 49 51825 - 1508 2 50 = 129 . 4636 … = 129 568 1225 or 129 We know that, E a X + b = a E X + b , Var aX + b = aVar X Now, Est. μ for scaled = 1 . 5 × 1508 50 + 10 = 55 . 24 or 55 . 2 Est. σ 2 for scaled = 1 . 5 2 × 129 . 4636 … = 291 . 2932 … or 291

The marks, x , of a random sample of 50 students in a test were summarised as follows: n = 50 ∑ x = 1508 ∑ x 2 = 51825 Each student's mark is scaled using the formula y = 1 . 5 x + 10 . Find estimates of the population mean and variance of the scaled marks, y .

Given data: n = 50 , Σ x = 1508 , ∑ x 2 = 51825 Formula to find an unbiased estimate of the population variance σ 2 is s 2 = 1 n - 1 Σ x 2 - ( Σ x ) 2 n . Formula to find an unbiased estimate of the population mean, μ = x ¯ = Σ x n . Thus, Est. μ = 1508 50 = 30 . 16 or 30 . 2 Est. σ 2 = 1 49 51825 - 1508 2 50 = 129 . 4636 … = 129 568 1225 or 129 We know that, E a X + b = a E X + b , Var aX + b = aVar X Now, Est. μ for scaled = 1 . 5 × 1508 50 + 10 = 55 . 24 or 55 . 2 Est. σ 2 for scaled = 1 . 5 2 × 129 . 4636 … = 291 . 2932 … or 291

Let n ≥ 3 . A list of numbers 0 < x 1 < x 2 < … < x n has mean μ and standard deviation σ . A new list of numbers is made as follows: y 1 = 0 , y 2 = x 2 , … . , y n - 1 = x n - 1 , y n = x 1 + x n . The mean and the standard deviation of the new list are μ ^ & σ ^ . Which of the following is necessarily true?

μ may or may not be equal to μ ^

EASY

Earn 100

The marks, $x$ , of a random sample of $50$ students in a test were summarised as follows:

$n = 50 \sum x = 1508 \sum x^{2} = 51825$

Each student's mark is scaled using the formula $y = 1.5 x + 10$ . Find estimates of the population mean and variance of the scaled marks, $y$ .

Important Questions on Estimation

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

The standard deviations of

x_{i} (i = 1, 2, \dots . 10)

and

y_{i} (i = 1, \dots 10)

are respectively

a

and

b

\bar{x}, \bar{y}

are the means of these two sets of observations respectively. If

z_{i} = (x_{i} - \bar{x}) (y_{i} - \bar{y})

and

\sum_{i = 1}^{10} z_{i} = c

, then the standard deviation of the observations

(x_{i} - y_{i}), (i = 1, 2, \dots 10)

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

The standard deviation of the set $(10, 10, 10, 10, 10)$ is

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

For a group of

100

students, the mean

{\bar{x}}_{1}

and the standard deviation

σ_{1}

of their marks were found to be

40

and

15

respectively. Later it was observed that the scores

40

and

50

were misread as

30

and

60

respectively. If the mean and the standard deviation with the corrected observations of the scores are

{\bar{x}}_{2}

and

σ_{2}

respectively, then

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

The S.D of

4, 5, 6, 7, \dots . ., 13

x

, then the S.D of

14, 15, \dots \dots 23

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

Area covered normal curve by

(μ \pm 3 σ)

is ______.

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

In a series of observations half of them equal to

a

and remaining half equal to

- a

. If the S.D. of the observations is

2

, then

| a | =

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

If the mean and the standard deviation of the data

3, 5, 7, a, b

are

5

and

2

respectively, then

a

and

b

are the roots of the equation:

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

Draw the necessary table to find the Standard Deviation for the data $20, 14, 16, 30, 21$ and $25$ .

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

In normal distribution what is the ratio of

QD : MD : SD

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

For a certain type of mobiles, the length of time between charges of the battery is normally distributed with a mean of

50

hours and a standard deviation of

15

hours. A person owns one of these mobiles and wants to know the probability that the length of time will be between

50

and

70

hours is (Given

ϕ (1.33) = 0.9082, ϕ (0) = 0.5

)

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

Let

X_{1}, X_{2}, \dots, X_{18}

be eighteen observations such that

\sum_{i = 1}^{18} (X_{i} - α) = 36

and

\sum_{i = 1}^{18} {(X_{i} - β)}^{2} = 90

, where

α

and

β

are distinct real numbers. If the standard deviation of these observations is

1

, then the value of

|α - β|

is _______.

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

The quartile deviation of a normal distribution with mean

10

and standard deviation

4

is __________

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

The Standard Deviation of the numbers

31, 32, 33 \dots \dots . 46, 47

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

The standard deviation of a variable

x

10

. Then the standard deviation of

50 + 5 x

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

Standard deviation of a collection of data is $2 \sqrt{2}$ . If each value is multiplied by $3$ , then the standard deviation of the new data is:

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

Find

SD

1, 2, 3, 4, 5, 6, 7, 8, 9

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

The standard deviation of

1

9

natural numbers is ____________

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

Let

n \geq 3

. A list of numbers

0 < x_{1} < x_{2} < \dots < x_{n}

has mean

μ

and standard deviation

σ

. A new list of numbers is made as follows:

y_{1} = 0, y_{2} = x_{2}, \dots ., y_{n - 1} = x_{n - 1}, y_{n} = x_{1} + x_{n}

. The mean and the standard deviation of the new list are

\hat{μ} & \hat{σ}

. Which of the following is necessarily true?

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

If we change the parameter (s) of a ________ distribution the Sharpe of probability curve does not change.

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Estimation>Unbiased Estimates of Population Mean and Variance

Which of the following is not true regarding the normal distribution?

The marks, x, of a random sample of 50 students in a test were summarised as follows: n=50∑x=1508∑x2=51825 Each student's mark is scaled using the formula y=1.5 x+10. Find estimates of the population mean and variance of the scaled marks, y.

Important Questions on Estimation

Learn and Prepare for any exam you want !

The marks, $x$ , of a random sample of $50$ students in a test were summarised as follows:

$n = 50 \sum x = 1508 \sum x^{2} = 51825$

Each student's mark is scaled using the formula $y = 1.5 x + 10$ . Find estimates of the population mean and variance of the scaled marks, $y$ .