৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

Explanation:
Think of it like fishing with a net:

Each time you take a random sample and build a CI, it’s like throwing a net into the sea.
The true mean (a fixed number) is like a fish sitting somewhere in the sea.
If you throw nets (CIs) many times, about 95 out of 100 nets will catch the fish (contain the true mean).
So, the correct idea is:
 “If we repeated this sampling process many times, about 95% of the intervals we build would contain the true mean.”

Statement A sounds Bayesian (probability on the parameter) and is incorrect in a frequentist CI. C confuses a CI with a prediction about individuals. D is a claim about a prediction interval, not a CI.

Explanation:
When the population standard deviation σ is unknown (the usual case), we estimate it with s and use the t distribution to account for extra uncertainty, especially with small–moderate n. If σ is known (rare), a z interval is appropriate (C is a case for z, not t). Large n pushes t and z to be similar (A doesn’t justify t). Stratified designs (D) change the variance formula.

Cluster sampling typically induces intra-cluster correlation, which increases the variance relative to SRS; this is captured by a design effect (DEFF > 1). Ignoring the design therefore underestimates the standard error, yielding over-confident (too narrow) intervals.
Nonresponse can introduce bias and also alter variance; failing to adjust (e.g., via weights, post-stratification, or imputation) can make the CI both miscentered (biased) and too narrow.
A high nominal confidence level (A) cannot fix model misspecification or bias.
Clustering rarely reduces variance vs SRS unless very specific conditions hold (C is generally false).
Confidence level (D) refers to tail areas under an assumed sampling model; it does not automatically handle nonresponse bias.

Explanation:
For standard settings (same model, standard errors, and assumptions), a two-sided test at level α is equivalent to checking whether the corresponding 100(1−α)% CI includes the null value. At α=0.05, a 95% CI that excludes 0 implies reject H₀​; if it includes 0, fail to reject. Options A and B reverse this logic; D ignores a fundamental equivalence.

Explanation: The sampling distribution approximates normality only when the sample size is sufficiently large (often ≥ 30). Small samples may lead to a non-normal distribution, limiting inference accuracy.
Source: GeeksforGeeks

Explanation: Confidence intervals rely on sample data and underlying assumptions (e.g., normality, random sampling). If these assumptions fail, intervals may be misleading.

 The Central Limit Theorem holds best for large samples. For small samples from a non-normal population, the sampling distribution may not be normal.

Explanation: Larger samples reduce variability (standard error) in sample means, leading to more precise estimates.

 Explanation: Increasing confidence level (e.g., 99% vs. 95%) increases the interval width to be more certain parameter lies inside. অর্থাৎ, আমরা যদি অনেক বেশি কনফিডেন্ট থাকি, তাহলে ইন্টারভেলের প্রস্থ বেশি হবে। 
Source: Business Statistics, Md. Abdul Aziz.

 Explanation: Confidence intervals estimate a range where the parameter likely lies, but they do not provide certainty or exact values.

Think of it like fishing with a net:

Each time you take a random sample and build a CI, it’s like throwing a net into the sea.
The true mean (a fixed number) is like a fish sitting somewhere in the sea.
If you throw nets (CIs) many times, about 95 out of 100 nets will catch the fish (contain the parameter).
So, the correct idea is:
It does not mean- there is a 95% chance the parameter is inside this specific interval. 
It means- If we repeated this sampling process many times, about 95% of the intervals we build would contain the parameter.
So, "chance"  is  not associated in CI, either CI traps true parameter, or not. 95% CI means- trapping true parameter for 95 out of 100 times.

Think of a classroom :

Confidence interval for the mean = “average height of all students.” Easier to estimate, so the band is tighter.
Prediction interval = “the height of the next random student.” Individual students vary a lot, so the band has to be wider.
->Predicting one person is riskier than estimating an average.

Imagine you’re searching for your lost keys with a flashlight beam.

A 90% CI is like using a narrower beam: more focused, but you might miss the keys.
A 99% CI is like using a wider beam: harder to miss, but covers more area.
 To be more confident, you must look wider.

ধরো তুমি টর্চলাইটের আলো দিয়ে তোমার হারানো চাবি খুঁজছো ।

৯০% confidence interval হলো সরু আলোর রশ্মি ব্যবহার করার মতো: বেশি ফোকাসড, কিন্তু চাবি মিস হয়ে যেতে পারে।
৯৯% confidence interval হলো চওড়া আলোর রশ্মি ব্যবহার করার মতো: চাবি মিস হওয়ার সম্ভাবনা কম, কিন্তু আলো বেশি জায়গা ঢেকে ফেলে।
 তাই বেশি নিশ্চিত হতে চাইলে, তোমাকে বড় এলাকা দেখতে হবে।

Explanation:
This is the definition of the term "standard error." It quantifies the variability of a statistic (like the mean or proportion) from sample to sample.

A) Population deviation is not a standard term.
B) Sample standard deviation (s) is the measure of variability within a single sample.
C) Standard error is the correct term for the standard deviation of a sampling distribution.
D) Mean absolute deviation is another measure of variability within a dataset, not across samples

Explanation:
The Central Limit Theorem's power comes from larger sample sizes. The sampling distribution's shape more closely resembles the population distribution when the sample size is small.

n = 1: The "sample mean" is just a single data point. The sampling distribution would be identical in shape to the population itself (non-normal).
n = 10: The distribution might still be somewhat non-normal.
n = 35 & n = 100: These are both large enough (n ≥ 30) for the CLT to apply, making the sampling distribution approximately normal. A larger n (100) would make it even more normal than n=35.
Therefore, n=1 is the least likely to produce a normal sampling distribution

Explanation:
The key here is the sample size (n=50). The Central Limit Theorem states that for a large enough sample size (typically n ≥ 30), the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the original population distribution. Since 50 > 30, the sampling distribution will be approximately normal, even though the underlying population is skewed right.
Source: Introductory Statistics by Barbara Illowsky & Susan Dean

Explanation:
Let's evaluate each option:

A is false. The CLT is powerful precisely because it states that the sampling distribution will be approximately normal regardless of the population's shape, given a sufficiently large sample size (usually n ≥ 30).
B is false. The opposite is true. As the sample size increases, the sampling distribution becomes more normal.
C is true. This is a fundamental property of the sampling distribution of the mean: μₓ̄ = μ.
D is false. The CLT applies to both sample means and sample proportions (which are essentially a special kind of mean).

Explanation:
The correct interpretation of a confidence interval relies on the long-run frequency of the method used to construct it, not a probability statement about a single, fixed interval.

A is incorrect. The true population mean is a fixed, unknown value. It is not a random variable, so we cannot assign a probability to it. The interval either contains the mean or it does not.
B is incorrect. A confidence interval estimates a population parameter (the mean), not the range of individual data points in the population. The range of individual heights would be much wider.
C is correct. This is the precise definition of a 95% confidence level. It describes the reliability of the confidence interval procedure.
D is incorrect. The sample mean is a single point estimate. It is fixed from the sample taken and is not expected to change. The interval provides a range of plausible values for the parameter, not a probability for the point estimate.

Explanation: A bigger sample gives you more information and makes your estimate more precise. This directly shrinks the margin of error, making the interval narrower. Changing the confidence level changes your certainty, not your precision, and can introduce bias.
Source: Introductory Statistics by Illowsky & Dean (Chapter 8: Confidence Intervals).

Explanation: A sampling distribution of the mean is obtained by repeatedly taking random samples and calculating the mean each time. Plotting these means gives the sampling distribution.

Explanation: Larger sample sizes reduce the standard error, making the sampling distribution narrower.

Explanation: The sampling distribution of the mean is unbiased, so its expected value equals the population mean μ.

Explanation: More variability in the population increases uncertainty, widening the CI.

Explanation: CI = x̄ ± Z·(σ/√n) = 152 ± 1.96·(30/√50) = 152 ± 8.31 → (143.7, 160.3).

Explanation: n = (Zσ/E)² = (1.96×8/2)² = (7.84)² = 61.46.

Explanation: SE = √(p(1−p)/n) = √(0.4×0.6/100) = √0.0024 = 0.049.
Source: Business Statistics, Md. Abdul Aziz

p̂=0.6, SE=0.0346. CI=0.6±1.96×0.0346=0.6±0.0679=(0.532,0.668).

Caution: Dont answer this type of question, it will kill your time.

Explanation: Width depends on variability, sample size, confidence level, not on the value of the statistic itself.

Explanation: CI width depends on SE=σ/√n​. Doubling n reduces SE by 1/√2​, so width shrinks by that factor.
Source: Stat-class 8 lecture pdf

Explanation: For sampling distribution of a proportion, both np and n(1−p) must be ≥ 5.
Source: “Statistics for Business and Economics” by Paul Newbold, William L. Carlson, and Betty Thorne 

Why Option A is wrong
Option A says the claim (0.70) is inside the CI.
But notice: the upper bound is 0.715. The claim 0.70 is very close to the edge.
If we keep decimals precise, the CI is actually (0.525,  0.715).
The claim 0.70 is not safely “inside” the plausible range. It sits almost at the extreme, which in statistics means the claim is not well supported.
So we reject Option A because saying “yes” would wrongly suggest the evidence strongly supports the company’s claim — but in fact, the evidence suggests the true proportion is lower.

 
Correct conclusion: Option B is more accurate → the claim doesn’t hold.

Explanation: The sampling distribution of the mean is unbiased, so E(Xˉ)=μ.

Explanation: For small n with unknown σ, we use the Student’s t distribution.
Source: Stat-class 8 - Sampling distribution.pptx

Explanation:

SE = 0.4/√400 = 0.02.
Marginal Error = 2.576×0.02 = 0.0515.
CI = 2.8 ± 0.0515 = (2.748, 2.852) ≈ (2.75, 2.85).

Explanation: Margin of error = Z×SE, and SE decreases as sample size increases.

Explanation:

SE = 25/√64 = 25/8 = 3.125.
Z = (105−100)/3.125 = 5/3.125 = 1.6.
P(Z > 1.6) = 1−Φ(1.6) = 1−0.9332 = 0.0668.

Explanation: The CLT says regardless of the population’s shape, the distribution of the sample mean approaches normality as n increases.
(This question is one of the most important, hence repeated throughout the exams sometimes)

Explanation: A sampling distribution is a probability distribution of a statistic. The sampling distribution for the difference  shows us all the possible values this difference could take and how likely they are if we repeatedly drew two random samples from the populations. This helps us understand the inherent variability in our estimates and is the foundation for building confidence intervals and conducting hypothesis tests.

Explanation: This is the direct definition. The sampling distribution is a concept that describes the distribution of a statistic (like the difference between two sample means) over many, many samples from the population.

Explanation: This is the entire reason we study sampling distributions. By knowing how the difference between samples typically behaves when there is no real effect (i.e., when the null hypothesis is true), we can see if the difference we actually observed in our data is a common, likely event or a rare, unusual one. If it's rare, we conclude it's probably not just chance. This is the basis for hypothesis testing.

Explanation: This is a very common misconception. The magic of the Central Limit Theorem is that it applies to the statistic (like the mean or the difference of means), not the raw data. For large samples, the sampling distribution will be normal even if the data in each population are highly skewed, as long as the samples are independent and random. (iid)

Explanation: This is the key principle behind the shape. The Central Limit Theorem (CLT) is the powerful statistical rule that guarantees the sampling distribution of many statistics (including the difference between two independent means) will be approximately normal for large enough sample sizes, regardless of the underlying population's shape. The other options are not the correct reason.

Explanation:

SE=
√(p1.(1−p1)/n1+p2.(1−p2)/n2)
=√((0.40⋅0.60/100)+(0.50⋅0.50/100))=√0.0049​=0.07
So, B) 0.07.

Explanation:

For difference of means, the variance of the difference is the sum of the variances from each sample.
For difference of proportions, the variance of the difference is the sum of the variances of each sample proportion.
So, in both cases, the variability comes from combining two independent sources of error.
B is false: sample sizes can be different.
C is false: normality is approximate for large n.
D is false: sampling error is not removed; it is quantified.

Explanation:

SE=√(σ₁²/n₁+σ₂²/n₂)
=√(36/25+64/36)
=√(1.44+1.78)=√3.22=1.79≈1.80