C. Each unit has an equal chance of being selected
D. Researcher selects samples based on convenience
Random sampling ensures that each item in the population has an equal probability of being chosen, eliminating bias.
2. Which of the following is an example of non-random sampling?
A. Judgement sampling
B. Simple random sampling
C. Stratified random sampling
D. Systematic random sampling
Judgement sampling is a type of non-random (non-probability) sampling where the researcher selects items based on personal judgement.
3. In stratified random sampling, the population is:
A. Divided into equal groups randomly
B. Taken as a whole without division
C. Selected only from a single subgroup
D. Divided into homogeneous groups and samples drawn from each
Stratified random sampling divides the population into homogeneous strata (groups) and then draws samples from each stratum.
4. The mean of the sampling distribution of the sample mean is equal to:
A. Sample mean
B. Population mean
C. Standard deviation
D. Variance of sample
The expected value (mean) of the sampling distribution of the sample mean is equal to the population mean.
5. The standard deviation of the sampling distribution of the sample mean is called:
A. Population variance
B. Coefficient of variation
C. Standard error
D. Mean deviation
The standard deviation of the sampling distribution of the sample mean is known as the standard error.
6. If the sample size increases, the standard error of the mean:
A. Decreases
B. Increases
C. Remains constant
D. Becomes equal to variance
As sample size increases, variability among sample means reduces, hence the standard error decreases.
7. According to the Central Limit Theorem, the sampling distribution of the sample mean tends to be:
A. Always skewed
B. Exponential
C. Uniform
D. Approximately normal for large samples
Central Limit Theorem states that irrespective of the population distribution, the sampling distribution of the mean approaches normal distribution as sample size increases.
8. If a population is normally distributed with mean μ and variance σ², the distribution of sample mean (for any n) is:
A. Uniform distribution
B. Normal distribution with mean μ and variance σ²/n
C. Binomial distribution
D. Exponential distribution
For a normally distributed population, the sampling distribution of the sample mean is also normal, with mean equal to μ and variance reduced to σ²/n.
9. A population has mean 50 and standard deviation 10. If a random sample of size 25 is taken, what is the standard error of the mean?
10. Which statement about sampling from a normal population is correct?
A. The sample mean is always equal to the population mean
B. The distribution of sample means is skewed
C. Sample variance is always equal to population variance
D. The distribution of sample means remains normal regardless of sample size
When sampling from a normal population, the sampling distribution of the sample mean remains normal for any sample size n.
11. For a normal population with mean μ = 100 and σ = 20, what is the probability that the sample mean of size 4 lies within 90 to 110?
A. About 68%
B. About 50%
C. About 95%
D. About 99%
SE = σ/√n = 20/2 = 10. Range 90–110 is μ ± 1 SE. By empirical rule, probability within ±1 SE ≈ 68%.
12. Which of the following statements is TRUE regarding sampling from normal populations?
A. Sample means will not converge to population mean as n increases
B. The standard error increases with sample size
C. Larger samples give more precise estimates of population mean
D. The sample mean distribution is uniform
Larger samples reduce standard error and make the estimate of the population mean more precise.
13. When sampling from a non-normal population, which theorem ensures that the sampling distribution of the sample mean approaches normality as sample size increases?
A. Bayes’ Theorem
B. Law of Large Numbers
C. Chebyshev’s Inequality
D. Central Limit Theorem
Central Limit Theorem states that for sufficiently large sample sizes, the sampling distribution of the mean tends toward a normal distribution regardless of the population’s shape.
14. A population is skewed. If a sample of size 100 is drawn, the distribution of the sample mean will be:
A. Skewed like the population
B. Approximately normal
C. Uniform
D. Exponential
For large n, the Central Limit Theorem ensures that the sample mean distribution is approximately normal, even if the population itself is skewed.
15. For small samples (n < 30) drawn from a non-normal population, which of the following is TRUE?
A. The sampling distribution of the mean may not be normal
B. The sampling distribution will always be normal
C. Standard error is zero
D. Probability of extreme values is eliminated
For small samples from non-normal populations, the sampling distribution of the mean may not follow normal distribution, hence tests like t-test are preferred.
16. Which of the following statements best describes the effect of increasing sample size when sampling from a non-normal population?
A. Sample distribution becomes uniform
B. Sample mean diverges from population mean
C. Sampling distribution of mean becomes closer to normal
D. Variance of sample mean increases
As per Central Limit Theorem, increasing sample size makes the distribution of sample mean approximately normal, reducing skewness and kurtosis effects.
17. A bank analyst is studying loan default amounts from a highly skewed population. To apply Z-test reliably, which approach should be followed?
A. Use very small samples
B. Assume population is symmetric
C. Use convenience sampling
D. Take a large sample size to approximate normality
For skewed populations, larger sample sizes make the sample mean distribution approximately normal, allowing Z-test to be applied reliably.
18. The Central Limit Theorem (CLT) is important in statistics because:
A. It ensures all populations are normally distributed
B. It states variance always decreases with sample size
C. It allows normal distribution to be used as an approximation for sample means
D. It makes population mean equal to sample mean
CLT shows that the sampling distribution of the sample mean approximates a normal distribution as the sample size increases, even if the population is not normal.
19. According to the CLT, the sampling distribution of the sample mean approaches normality when:
A. Sample size is less than 5
B. Sample size is sufficiently large
C. Population is uniform
D. Variance is always zero
The CLT states that for large enough sample sizes (commonly n ≥ 30), the distribution of the sample mean becomes approximately normal.
20. A population has mean 200 and standard deviation 50. If a sample of size 100 is drawn, what is the approximate distribution of the sample mean?
A. Uniform with mean 200 and variance 25
B. Exponential with mean 200
C. Skewed with mean 200
D. Normal with mean 200 and standard error 5
By CLT, sample mean ~ N(μ, σ/√n). Here, SE = 50/√100 = 5. Hence, N(200, 5).
21. Which of the following is NOT a condition for applying CLT?
A. Population must always be normally distributed
B. Samples must be independent
C. Sample size should be sufficiently large
D. Variance of the population should be finite
CLT does not require the population to be normal. It only needs large independent samples and finite variance.
22. A bank is analyzing ATM withdrawal amounts which follow a skewed distribution. If 64 transactions are randomly sampled, which statement is correct as per CLT?
A. The sample mean will remain skewed
B. The distribution of the sample mean will be approximately normal
C. Variance of the sample mean becomes zero
D. The population becomes normal
Even if the population is skewed, with n = 64, the sample mean distribution will approximate normality by CLT.
23. The finite population multiplier (FPC) is applied when:
A. Sample size is very small compared to population
B. Population is infinite
C. Sampling is done with replacement
D. Sampling is without replacement from a finite population
The finite population multiplier adjusts the standard error when sampling without replacement from a finite population.
24. The formula for the finite population correction factor (FPC) is:
A. √(N / (N + n))
B. √((N – n) / (N – 1))
C. √(n / N)
D. √((N – 1) / (N – n))
Finite Population Correction Factor (FPC) = √((N – n) / (N – 1)), where N = population size, n = sample size.
25. Why is the finite population correction factor applied?
A. To increase the standard error
B. To ignore population variance
C. To reduce the standard error when the sample size is a large fraction of the population
D. To change mean of the distribution
When n/N is large, variability reduces. The FPC adjusts standard error downward to reflect this reduced variability.
26. A population has N = 200 items, and a sample of n = 50 is drawn without replacement. What is the finite population multiplier (FPC)?
27. If the population size is very large compared to the sample size, the FPC value will be:
A. Very small
B. Equal to zero
C. Negative
D. Close to 1
For large N relative to n, (N – n) / (N – 1) ≈ 1, so FPC ≈ 1. Hence, adjustment becomes negligible.
28. Point estimation in statistics refers to:
A. Interval within which population parameter lies
B. A single value estimate of a population parameter
C. Sampling error in population
D. Standard deviation of a sample
A point estimate gives one numerical value (e.g., sample mean) as the best guess of a population parameter.
29. Which of the following is NOT a desirable property of a good estimator?
A. Unbiasedness
B. Consistency
C. Efficiency
D. Biasedness
A good estimator should be unbiased, consistent, and efficient. Biasedness is undesirable.
30. The width of a confidence interval depends on:
A. Only sample size
B. Only population variance
C. Sample size, population variability, and confidence level
D. Only confidence level
Confidence interval width is influenced by sample size (n), population variability (σ), and confidence level (Z value).
31. For a 95% confidence interval, the Z-value used is approximately:
A. 1.96
B. 2.58
C. 1.64
D. 3.29
For a 95% confidence level in a normal distribution, the critical Z-value is 1.96.
32. A sample of 100 customers shows an average balance of ₹50,000 with a standard deviation of ₹10,000. What is the 95% confidence interval for the mean?
