৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size grows large.

Sampling distributions are concerned with how sample statistics (like the sample mean or sample proportion) behave across many samples.

A sampling distribution is the probability distribution of a statistic (like sample mean) obtained from multiple samples of the same size drawn from a population.

The expected value of the sample mean equals the population mean (unbiased estimator).

Source: Business Statistics (Md. Abdul Aziz)  

The formula for the standard error of the sample mean is population standard deviation divided by the square root of the sample size.

Source: Live MCQ class

Sampling distributions show how a statistic varies across samples, which differs from the population distribution.

If the population is normal, the sample mean’s distribution is normal regardless of sample size.

Source: Business Statistics, MP Gupta, SP Gupta 

The primary reason to study the sampling distribution of a statistic—is that sampling distributions help to understand the variability of the statistic from sample to sample.

This purpose is essential because it allows researchers to estimate the sampling error and make inferences about the population based on sample data.

Sampling distributions provide a framework for understanding how sample statistics vary and the precision with which population parameters are estimated.

Source: Statistics by Jim.

The variability of the sampling distribution of a statistic, such as the sample mean, is quantified by the standard error. The standard error (SE) is the standard deviation of the sampling distribution.

The formula for the standard error of the sample mean is:
SE=σ/√n
Here, σ is the population standard deviation and n is the sample size.

As the sample size n increases, the denominator √n increases, causing the standard error to decrease.
A smaller standard error means the sampling distribution is more clustered (less spread out) around the population mean. 

Thus, larger sample sizes lead to narrower sampling distributions, improving the precision of the estimate of the population parameter.

Source: Khan Academy Statistics Library

When population variance is unknown and the sample size is small, the sampling distribution of the sample mean follows a t-distribution.

Source: Business Statistics (MK Roy) 

Without a large sample size, the Central Limit Theorem does not guarantee normality of the sampling distribution for non-normal populations.

The Central Limit Theorem (CLT) states that, regardless of the population distribution shape, the sampling distribution of the sample mean tends toward a normal distribution as nn becomes large (usually n>30 is considered sufficient).

Since the sample size is 50 (large), the sampling distribution of the sample mean will be approximately normal, even though the population is right-skewed.

Standard error is defined as SE=σ/√n
Given σ=12, n=4,
SE=12/√4=12/2=6 (True for A)

As sample size increases, SE decreases (B is true).

SE measures variability of sample means between samples (C is true).

SE depends only on population standard deviation and sample size, not on the shape of the distribution (D is false).

Hence, D is false.

The standard deviation of the sampling distribution of the sample mean (called the standard error) is:

SE = σ/√n = 15/√25 = 15/5 = 3

This reflects that the sample mean varies less than individual data points.

Source: GeeksforGeeks.

The Central Limit Theorem states that regardless of the population distribution shape, the distribution of the sample mean approaches a normal distribution as the sample size becomes large.

Source: Statistics By Jim

The standard error decreases as the sample size increases because the variability of the sample mean reduces with larger samples.

The formula for standard error is SE=σ/√n, so increasing n decreases SE.

Source: Live MCQ lecture slide 

A sampling distribution is the distribution of a statistic (such as the sample mean) that is calculated from many different samples drawn from the same population.

It shows how that statistic varies from sample to sample

When a sample of size nn is drawn from a normal population, the statistic:
(n−1)×S²/σ²
follows a chi-square distribution with (n−1) degrees of freedom.
Hence, the sampling distribution of the sample variance is a scaled chi-square distribution.

The Law of Large Numbers states that as the number of trials becomes very large, the observed proportion of heads will get closer and closer to the true probability of 0.5 for a fair coin.

The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, no matter the original population distribution.

According to the Central Limit Theorem, when the sample size is large enough, the sampling distribution of the sample mean approximates a normal distribution regardless of the shape of the population distribution.

Source: "Introduction to Statistical Theory" by Paul H. Garthwaite, J. Jennifer Hand, and Henry P. Wynn

A parameter is a fixed numerical characteristic of a population, usually unknown and constant.

A statistic is a numerical value calculated from a sample, which varies from sample to sample.

Source: Whatagraph

The standard deviation of the sampling distribution of the sample mean, also called the standard error, is σ/√n.

As the sample size n increases, the standard error decreases, which means sample means become less spread out.

Source: Live MCQ lecture slide 

The mean of the sampling distribution of the sample mean is equal to the population mean μ.

This means the sample mean is an unbiased estimator of the population mean.

The sampling distribution of the sample mean is the probability distribution of all possible sample means from samples of a particular size drawn from the population.

It shows how the sample mean varies from sample to sample.

Source: Live MCQ class 

A parameter is a fixed numerical summary describing a population characteristic (B).

Unlike a statistic, it is generally unknown and constant, not calculated from a sample.

A statistic is a value computed from sample data (B), used to estimate population parameters.

It varies from sample to sample.

Sampling distribution of the sample mean shows how sample means fluctuate in repeated sampling, helping us understand variability and make inferences (B).

Sampling cannot provide exact population parameters (C); this is a limitation but phrased as something sampling cannot do, so it is actually a limitation.

The question asks for what is not a limitation; (C) is actually a limitation, so the correct answer is the option that is true in respect to sampling but not a limitation (Check carefully).

The best correct answer here is (C) as it implies sampling achieves exactness, which it does not—that is a limitation, so the correct answer is (C) because this is not a limitation (it is an impossibility, hence not a property of sampling itself).

To clarify: sampling limitation includes error, bias, uncertainty; exactness is impossible, so "ability" is not a limitation but an impossibility.

By the Central Limit Theorem, the sampling distribution of the sample mean approaches a normal distribution as sample size increases (C).

Knowing the sampling distribution helps to make probabilistic inferences (confidence intervals, hypothesis tests) about the population mean (B).

The Central Limit Theorem allows approximation of the sampling distribution of the mean to be normally distributed regardless of the population shape when sample size is large, facilitating inference (B).

Source: Investopedia

Sampling error causes sample statistics to vary randomly from sample to sample, which is a key limitation (B).

Source: Live MCQ class 

Averaging tends to reduce variability since extreme values are smoothed out in sample means (B).

Source: Statistics by Jim

Statistics are calculated from samples to estimate parameters, which describe populations. So (C) is correct.

Source: Live MCQ class lecture 

 Explanation:

Central Limit Theorem: For large n, sample mean distribution is approximately normal, regardless of population shape.

Sample variance is an estimator, not always equal.

It only approximates the population variance as n grows large.

We don’t usually observe the population → so we use sample statistics.

Sampling distributions help us connect sample → population, with probabilities.

SE = 50/√100 
      = 50/10
      = 5

Mean of sampling distribution = μ = 80.

দেওয়া আছে:
μ = 170 সেমি
σ = 12 সেমি
n = 36

Standard Error (SE)
SE = σ / √n = 12 / 6 = 2

P(|X̄ - μ| ≤ 2)

⇒ Z = (X̄ - μ) / SE
⇒ P(|Z| ≤ 2/2) = P(|Z| ≤ 1)

P(|Z| ≤ 1) = 2Φ(1) - 1 = 2(0.8413) - 1 = 0.6826 ≈ 0.68

Probability distribution is about a random variable X (like one person’s height, or one coin toss).

Sampling distribution is about a statistic (like average height of a sample, or proportion of heads in many samples).

Example: Toss a coin → Probability distribution: chance of head or tail. Sampling distribution: if you toss the coin 50 times repeatedly, what distribution the proportion of heads follows.

The Gamma distribution is a continuous probability distribution that is often used to model waiting times. Specifically, it generalizes the exponential distribution.

The exponential distribution models the waiting time until the first event in a Poisson process.
The Gamma distribution models the waiting time until the α-th event in a Poisson process (where α > 0 is called the shape parameter).
This makes the Gamma distribution a perfect model in scenarios where:

Events happen independently and at a constant average rate.
You're interested in the aggregate waiting time for multiple events, not just the first one.