Statistics & Applications

Inferential

Medium

applied

If a sample has 'n' observation $x_1$ , $x_2$ ............. $x_n$ with 'm' constraints on these values then degree of freedom of sample statistic is:

$\frac{n - m}{2}$

$\frac{n + m}{2}$

$n + m$

$n - m$

✅ Correct Option: 4

The degree of freedom represents the number of values that are free to vary after accounting for constraints.

When a sample has $n$ observations: $x_1, x_2, ..., x_n$ with $m$ constraints on these values, each constraint restricts one value from varying freely.

Consider an example with 5 numbers where their sum must equal 20 (1 constraint).

The first 4 numbers can be chosen freely: 2, 3, 4, 5

The 5th number must be 6 to satisfy the constraint (sum = 20)

Only 4 numbers are free to vary = 5 - 1 = 4 degrees of freedom

Each constraint reduces the freedom by 1 since it "locks" one value that would otherwise be free to vary.

With $n$ observations and $m$ constraints:

Degree of Freedom $= n - m$

Examples:

10 values with no restrictions: $10 - 0 = 10$ degrees of freedom

10 values with mean fixed: $10 - 1 = 9$ degrees of freedom

10 values with mean and variance fixed: $10 - 2 = 8$ degrees of freedom

Therefore, the degree of freedom of the sample statistic is $n - m$ .

2025: 30 May Shift 1

Match List-I with List-II

List-I	List-II
(A) The chance of observing a specific outcome in an experiment	(I) Parameter
(B) A method used to estimate the population parameters	(II) Estimation
(C) A value that describes an entire population	(III) Statistic
(D) A measurable characteristic of a sample	(IV) Probability distribution

Choose the correct answer from the options given below:

2025: 14 May Shift 2

The simple random sample consists of six observations: 5, 8, 10, 7, 10, 14. The point estimate of the population mean is:

2026: 25 May Shift 1

The value of t-statistic for the following data as Population mean is 51, Sample mean is 54, Sample size is 10 and sample variance is 36, is:

2026: 30 May Shift 1

In reference to the Inferential Statistics, which of the following statements are correct?

A. The number of degrees of freedom of a statistic is the number of independent varieties used to compute the statistic.

B. A confidence interval is a range that could be expected to contain the population parameter of interest.

C. In a hypothesis test, the significance level is the probability of making wrong decision when the null hypothesis is true.

D. The confidence level of the sampling distribution of a statistic is known as its standard error.

Choose the correct answer from the options given below:

2026: 15 May Shift 1

The central limit theorem states that the sampling distribution of the sample mean approaches the 'X' distribution, as the sample size gets larger, no matter what the shape of the population is, then 'X' is

2025: 15 May Shift 1

A 95 % confidence interval states that the population mean is greater than 152 and less than 160. If $\sigma = 15$ and $Z_{0.025} = 1.96$ , then what sample size was used in the study?

If a sample has 'n' observation x1x_1x1​, x2x_2x2​ ............. xnx_nxn​ with 'm' constraints on these values then degree of freedom of sample statistic is:

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If a sample has 'n' observation $x_1$ , $x_2$ ............. $x_n$ with 'm' constraints on these values then degree of freedom of sample statistic is: