وَأَحْصَىٰ كُلَّ شَيْءٍ عَدَدًا
"And He has enumerated everything in numbers." (Allah perfectly counts and knows the exact measure of all creations).
Q1. The types of statistical inferences are:
A. Estimation of parameters
B. Testing of hypotheses
C. Central tendency
D. (a) & (b) ✓ Correct Answer
Explanation: Statistical inference has two branches: estimating
parameters and testing hypotheses — so both (a) and (b).
Q2. The conclusions made about the parameters of a
population from sample data:
A. Statistical
Inferences ✓ Correct Answer
B. Census
C. True values
D. None
Explanation: Drawing conclusions about population parameters from
sample data is exactly what statistical inference means.
Q3. Technique of obtaining estimates from sample
data:
A. Estimation ✓ Correct Answer
B. Testing of hypotheses
C. Central tendency
D. (a) & (b)
Explanation: The process of obtaining numerical estimates of
parameters from sample data is called estimation.
Q4. Types of statistical estimation:
A. Point estimation
B. Interval estimation
C. Central tendency
D. (a) & (b) ✓ Correct Answer
Explanation: Statistical estimation has two types: point estimation
and interval estimation.
Q5. It provides a single value as an estimate:
A. Testing of hypothesis
B. Interval estimation
C. Point estimation ✓ Correct Answer
D. a and b
Explanation: Point estimation gives one single numerical value as the
estimate of a parameter.
Q6. An estimator is always:
A. A statistic
B. Function of sample of
observations
C. Random variable
D. All ✓ Correct Answer
Explanation: An estimator is a statistic, is computed as a function of
sample observations, and varies from sample to sample, so it is a random
variable — all true.
Q7. The specific value of an estimator:
A. Estimation
B. Estimate ✓ Correct Answer
C. Statistic
D. All
Explanation: Once computed from actual sample data, the specific
numerical value of an estimator is called an estimate.
Q8. A point estimator is a random variable whereas an
estimate is:
A. Also a random variable
B. Constant ✓ Correct Answer
C. Statistic
D. None
Explanation: The estimator varies sample to sample (random variable),
but once evaluated on a particular sample it becomes a fixed constant (the
estimate).
Q9. Population parameters are estimated from:
A. Sample data ✓ Correct Answer
B. Population data
C. Census data
D. None
Explanation: Estimation, by definition, uses sample data to infer
about population parameters.
Q10. Expected value of a statistic is equal to the
population parameter:
A. Unbiasedness ✓ Correct Answer
B. Biasedness
C. Inconsistentency
D. a and b
Explanation: When E(statistic) = parameter, the statistic is called an
unbiased estimator — this property is unbiasedness.
Q11. A point estimator is said to be unbiased if the
expected value of estimator is:
A. Not equal to parameter
B. Equal to parameter ✓ Correct Answer
C. Greater than parameter
D. Less than parameter
Explanation: Unbiasedness means E(estimator) exactly equals the true
parameter value.
Q12. The difference of expected value of a statistic
from the parameter:
A. Bias ✓ Correct Answer
B. Accuracy
C. Sampling error
D. None
Explanation: By definition, Bias = E(statistic) - parameter.
Q13. If E(T) = θ then estimator T is:
A. Unbiased ✓ Correct Answer
B. Biased
C. Positively unbiased
D. None
Explanation: When the expected value of T equals θ exactly, T is an
unbiased estimator.
Q14. An estimator 'T' is unbiased for θ if:
A. E(T) = θ ✓ Correct Answer
B. E(T) < θ
C. E(T) > θ
D. None
Explanation: The defining condition for unbiasedness is E(T) = θ.
Q15. It has least variance among the class of
unbiased estimators:
A. Best estimator ✓ Correct Answer
B. Biased estimator
C. Worst estimator
D. a and b
Explanation: An unbiased estimator with the smallest possible variance
is called the best (most efficient) estimator.
Q16. Estimator will tend to give estimates nearer to
the parameter as sample size increases:
A. Unbiased
B. Biased
C. Consistent ✓ Correct Answer
D. None
Explanation: This describes a consistent estimator — as n grows, its
estimates converge closer to the true parameter.
Q17. The unbiased point estimator of population
variance is:
A. P = X/n
B. Ś² = Σ(x-x̄)²/(n-1) ✓ Correct Answer
C. X̄ = Σx/n
D. None
Explanation: The sample variance formula with (n-1) in the
denominator, Ś² = Σ(x-x̄)²/(n-1), is the unbiased estimator of population
variance.
Q18. The unbiased point estimator of population
proportion:
A. P = X/n ✓ Correct Answer
B. Ś² = Σ(x-x̄)²/(n-1)
C. X̄ = Σx/n
D. None
Explanation: The sample proportion P = X/n is the unbiased estimator
of the population proportion.
Q19. The unbiased point estimator of population mean:
A. P = X/n
B. Ś² = Σ(x-x̄)²/(n-1)
C. X̄ = Σx/n ✓ Correct Answer
D. None
Explanation: The sample mean X̄ = Σx/n is the unbiased estimator of
the population mean.
Q20. A point estimate is a specific value of:
A. Estimator
B. Statistic
C. Parameter
D. a and b ✓ Correct Answer
Explanation: A point estimate is a specific value taken by an
estimator, and since an estimator is a statistic, both (a) and (b) apply.
Q21. The best point estimator of population mean is:
A. X̄ ✓ Correct Answer
B. P = X/n
C. Ś² = Σ(x-x̄)²/(n-1)
D. None of these
Explanation: The sample mean X̄ is both unbiased and has the least
variance, making it the best estimator of the population mean.
Q22. The best point estimator of population
proportion is:
A. X̄
B. P = X/n ✓ Correct Answer
C. Ś² = Σ(x-x̄)²/(n-1)
D. None of these
Explanation: The sample proportion P = X/n is the best (unbiased,
minimum variance) estimator of the population proportion.
Q23. The best point estimator of population variance
is:
A. Ś² = Σ(x-x̄)²/(n-1) ✓ Correct Answer
B. X̄
C. P = X/n
D. None of these
Explanation: Ś² = Σ(x-x̄)²/(n-1) is the best estimator of the
population variance.
Q24. The technique of constructing interval from
sample data:
A. Interval estimation ✓ Correct Answer
B. Testing of hypothesis
C. Point estimation
D. None
Explanation: Building a range (interval) of plausible values for a
parameter from sample data is called interval estimation.
Q25. As the confidence coefficient gets nearer 100%,
the confidence interval rapidly:
A. Widens ✓ Correct Answer
B. Contracts
C. Shortens
D. None
Explanation: Higher confidence requires capturing more of the
distribution, which widens the interval rapidly as it approaches 100%.
Q26. The choice of method used in constructing the
confidence interval for μ depends upon:
A. Normality
B. Information of σ²
C. Sample size
D. All ✓ Correct Answer
Explanation: The appropriate method depends on whether the population
is normal, whether σ² is known, and the sample size — all matter.
Q27. The finite population correction factor can be
ignored if:
A. Sample size is less than 5% of
population size
B. Sampling is done with
replacement from finite population
C. Sampling is done without
replacement from an infinite population
D. All ✓ Correct Answer
Explanation: All three conditions mean the finite population
correction has negligible effect and can be safely ignored.
Q28. Which of the following is a necessary condition
for using a t-distribution?
A. Small sample size
B. Unknown σ²
C. a & b ✓ Correct Answer
D. Infinite population
Explanation: The t-distribution is used when the sample size is small and
the population variance σ² is unknown — both conditions together.
Q29. The interval estimate of a population mean with
large sample size and known standard deviation is:
A. x̄ ± z₁₋ₐ/₂ · σ/√n ✓ Correct Answer
B. x̄ ± z₁₋ₐ/₂ · Ś/√n
C. x̄ ± tᵥ;₁₋ₐ/₂ · σ/√n
D. x̄ ± tᵥ;₁₋ₐ/₂ · Ś/√n
Explanation: With a large sample and known σ, the z-based interval
using the true standard deviation is used: x̄ ± z₁₋ₐ/₂ · σ/√n.
Q30. By increasing 'n', the length of the confidence
interval for μ:
A. Increases
B. Decreases ✓ Correct Answer
C. No effect
D. None
Explanation: A larger sample size n shrinks the standard error (σ/√n),
so the confidence interval becomes shorter.
Q31. By decreasing 'n', the length of the confidence
interval for μ:
A. Increases ✓ Correct Answer
B. Decreases
C. No effect
D. None
Explanation: A smaller sample size increases the standard error,
widening the confidence interval.
Q32. The probability the confidence interval doesn't
contain the parameter:
A. 1 - α
B. 1 + α
C. α ✓ Correct Answer
D. -α
Explanation: The significance level α is exactly the probability that
the confidence interval fails to capture the true parameter.
Q33. The probability the confidence interval contains
the parameter:
A. 1 - α ✓ Correct Answer
B. 1 + α
C. α
D. -α
Explanation: The confidence coefficient 1 - α is the probability that
the interval does contain the true parameter.
Q34. By increasing '1 - α', the length of the
confidence interval for μ:
A. Increases ✓ Correct Answer
B. Decreases
C. No effect
D. None
Explanation: A higher confidence level (1-α) requires a wider interval
to be more certain of capturing the parameter.
Q35. By decreasing '1 - α', the length of the
confidence interval for μ:
A. Increases
B. Decreases ✓ Correct Answer
C. No effect
D. None
Explanation: Lowering the confidence level allows for a narrower
(shorter) interval.
Q36. By increasing 'Ś²', the length of the confidence
interval for μ:
A. Increases ✓ Correct Answer
B. Decreases
C. No effect
D. None
Explanation: A larger sample variance Ś² increases the standard error,
which widens the confidence interval.
Q37. By decreasing 'Ś²', the length of the confidence
interval for μ:
A. Increases
B. Decreases ✓ Correct Answer
C. No effect
D. None
Explanation: A smaller sample variance reduces the standard error,
shortening the confidence interval.
Q38. By increasing 'α', the length of the confidence
interval for μ:
A. Increases
B. Decreases ✓ Correct Answer
C. No effect
D. None
Explanation: Increasing α means lowering the confidence level (1-α),
which narrows the confidence interval.
Q39. By decreasing 'α', the length of the confidence
interval for μ:
A. Increases ✓ Correct Answer
B. Decreases
C. No effect
D. None
Explanation: Decreasing α raises the confidence level (1-α), which
widens the confidence interval.
Q40. By increasing 'x̄', the length of the confidence
interval for μ:
A. Increases
B. Decreases
C. No effect ✓ Correct Answer
D. None
Explanation: x̄ only shifts the center of the interval; the width
depends on the margin of error, not on x̄, so there is no effect.
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