1. Which of the following best defines an estimator?
A. A specific numerical value calculated from sample data
B. The process of drawing inferences about population
C. A statistical rule or formula used to estimate a population parameter
D. A measure of error in prediction
An estimator is a statistical rule or formula that provides an estimate of a population parameter.
For example, the sample mean (x̄) is an estimator of the population mean (μ).
2. Which of the following is an example of an estimate?
A. Population mean μ
B. Formula for sample mean
C. Formula for sample variance
D. The value of x̄ = 50 computed from a given sample
An estimate is the actual numerical value obtained from the sample data using an estimator.
For example, if sample mean (x̄) = 50, this is the estimate of the population mean.
3. Which of the following is NOT a desirable property of a good estimator?
A. High bias
B. Consistency
C. Efficiency
D. Unbiasedness
A good estimator should be unbiased, consistent, efficient, and sufficient.
High bias is undesirable since it systematically over or underestimates the parameter.
4. If the sample mean (x̄) is used to estimate the population mean (μ), then x̄ is:
A. A parameter
B. An estimator and also an estimate when computed
C. Always equal to μ
D. A biased estimator
The sample mean (x̄) is an estimator of the population mean μ.
Once we calculate its numerical value from data, it becomes an estimate.
Hence, it plays both roles.
5. A statistic is called unbiased if:
A. Its variance is zero
B. Its expected value is greater than the parameter
C. Its expected value equals the population parameter
D. It is always equal to the sample value
An estimator is unbiased if its expected value equals the population parameter it is estimating.
For example, the expected value of the sample mean is equal to the population mean.
6. Suppose the true population mean (μ) is 100. A sample of 25 observations gives a mean (x̄) of 98.
Here, 98 is considered as:
A. Population parameter
B. Estimator
C. Standard error
D. Estimate of μ
The sample mean value (x̄ = 98) is an estimate of the true population mean μ = 100.
The formula (Σx/n) is the estimator, but the computed value (98) is the estimate.
7. Which of the following best describes a point estimate?
A. A single value calculated from sample data to estimate a population parameter
B. A range of values that may contain the parameter
C. A fixed known population value
D. The level of confidence chosen for estimation
A point estimate is a single numerical value obtained from sample data that is used to estimate the population parameter.
For example, the sample mean (x̄ = 52) is a point estimate of the population mean.
8. An interval estimate provides:
A. An exact value equal to the population parameter
B. The variance of the population
C. A range of values within which the parameter is expected to lie
D. A measure of sampling bias
An interval estimate gives a range of values (confidence interval) around the point estimate,
within which the population parameter is likely to lie with a certain probability (confidence level).
9. Which of the following is true about confidence intervals?
A. Wider intervals imply higher precision
B. Narrower intervals imply lower accuracy
C. They always contain the true population parameter
D. Higher confidence level usually leads to a wider interval
Increasing the confidence level (say from 90% to 95%) makes the interval wider
because we want to be more certain that the parameter lies within the range.
10. A sample of 100 observations gives a mean of 80 and standard deviation of 10.
The 95% confidence interval for the population mean is approximately:
A. (75, 85)
B. (78.04, 81.96)
C. (70, 90)
D. (76, 84)
For 95% confidence, Z ≈ 1.96.
Standard Error = σ/√n = 10/√100 = 1.
Margin of Error = 1.96 × 1 = 1.96.
Confidence Interval = 80 ± 1.96 → (78.04, 81.96).
11. Which statement correctly differentiates point and interval estimates?
A. Point estimate gives a single value, interval estimate gives a range of values
B. Point estimate is always unbiased, interval estimate is biased
C. Point estimate uses population data, interval uses sample data
D. Both are always equal
A point estimate is a single value (like x̄ = 50),
while an interval estimate gives a range (like 48 to 52) within which the true parameter is likely to lie.
12. If the confidence level is increased from 90% to 99%, what happens to the interval estimate?
A. The interval becomes narrower
B. The point estimate changes
C. The interval becomes wider
D. It remains the same
A higher confidence level means we want to be more certain,
so we must allow a wider interval to capture the population parameter with higher probability.
13. A confidence interval is constructed around a point estimate in order to:
A. Reduce the sampling error to zero
B. Provide a range of values that is likely to include the population parameter
C. Increase the sample size automatically
D. Eliminate bias in the estimator
A confidence interval provides a range of values around a point estimate within which the true population parameter is likely to lie, with a chosen probability (confidence level).
14. If the standard error decreases, the confidence interval:
A. Becomes wider
B. Shifts upward
C. Remains the same
D. Becomes narrower
Smaller standard error means the sample statistic is more precise, which results in a narrower confidence interval.
15. For a 95% confidence interval, the corresponding Z-value (standard normal) is approximately:
A. 1.28
B. 1.65
C. 1.96
D. 2.58
For a 95% confidence interval, the Z-critical value is approximately ±1.96 under the standard normal distribution.
16. A sample of 64 has mean 120 and standard deviation 16.
Find the 95% confidence interval for the population mean.
A. (116.08, 123.92)
B. (114, 126)
C. (118, 122)
D. (115, 125)
SE = σ/√n = 16/√64 = 16/8 = 2.
Margin of Error = 1.96 × 2 = 3.92.
CI = 120 ± 3.92 → (116.08, 123.92).
17. Increasing the sample size while keeping confidence level constant will generally:
A. Widen the confidence interval
B. Narrow the confidence interval
C. Not affect the interval
D. Change the Z-value
Larger sample size reduces the standard error, which results in a narrower confidence interval.
18. For a 99% confidence interval, which Z-critical value is generally used?
A. 1.65
B. 1.96
C. 2.00
D. 2.58
For a 99% confidence level, the Z-critical value is approximately ±2.58 in the standard normal distribution.
19. When estimating the mean from a large sample (n ≥ 30), the sampling distribution of the mean is assumed to follow:
A. Poisson distribution
B. Exponential distribution
C. Approximately normal distribution
D. Chi-square distribution
By the Central Limit Theorem, the sampling distribution of the sample mean approaches normal distribution as sample size increases (n ≥ 30 is a common rule of thumb).
20. A large sample of 400 observations has mean 50 and standard deviation 8.
What is the 95% confidence interval for the population mean?
A. (49.22, 50.78)
B. (48.5, 51.5)
C. (47, 53)
D. (49, 51)
SE = σ/√n = 8/√400 = 8/20 = 0.4.
Margin of Error = 1.96 × 0.4 = 0.78.
CI = 50 ± 0.78 → (49.22, 50.78).
21. In large samples, the population standard deviation (σ) is usually:
A. Ignored completely
B. Approximated by the sample standard deviation (s)
C. Equal to the sample mean
D. Not required for interval estimation
In practice, the true population σ is unknown. For large samples, it is reasonably approximated by the sample standard deviation (s).
22. A large sample of 1000 customers reports an average deposit of ₹25,000 with a standard deviation of ₹2,000.
Find the 99% confidence interval for the population mean.
A. (24,950, 25,050)
B. (24,960, 25,040)
C. (24,900, 25,100)
D. (24,947.6, 25,052.4)
SE = σ/√n = 2000/√1000 ≈ 63.25.
For 99% CI, Z = 2.58.
Margin of Error = 2.58 × 63.25 ≈ 163.2.
CI = 25,000 ± 163.2 → (24,947.6, 25,052.4).
23. Which of the following factors will NOT directly affect the width of a confidence interval for large samples?
A. Sample size
B. Variability (σ or s)
C. Population size (when very large)
D. Confidence level
For large samples, the width depends on sample size, variability, and confidence level.
Very large population size does not materially affect the interval.
24. A bank wants to estimate the mean loan amount disbursed. From a sample of 400 accounts, the mean is ₹1,20,000 and s = ₹20,000.
Construct the 95% confidence interval.
A. (1,18,040, 1,21,960)
B. (1,17,000, 1,23,000)
C. (1,18,500, 1,21,500)
D. (1,16,000, 1,24,000)
SE = s/√n = 20,000/√400 = 20,000/20 = 1,000.
Margin of Error = 1.96 × 1,000 = 1,960.
CI = 1,20,000 ± 1,960 → (1,18,040, 1,21,960).
25. The confidence interval for population proportion in large samples is based on:
A. Student’s t-distribution
B. Standard normal distribution (Z)
C. Chi-square distribution
D. Poisson distribution
For large n, the sampling distribution of sample proportion (p̂) is approximately normal, hence Z distribution is used.
26. A survey of 400 customers finds that 120 use mobile banking.
Find the 95% confidence interval for the true proportion.
27. As sample size increases, the confidence interval for a population proportion becomes:
A. Narrower
B. Wider
C. Remains the same
D. Depends only on p̂
Larger n reduces the standard error (SE), which makes the confidence interval narrower, improving precision.
28. Out of 900 account holders, 450 are women. Estimate the 99% confidence interval for the proportion of women account holders.
A. (0.46, 0.54)
B. (0.47, 0.53)
C. (0.48, 0.52)
D. (0.463, 0.537)
p̂ = 450/900 = 0.50.
SE = √(0.5×0.5/900) = √(0.25/900) = √0.000278 ≈ 0.01667.
For 99% CI, Z = 2.58.
Margin = 2.58 × 0.01667 ≈ 0.043.
CI = 0.50 ± 0.043 → (0.463, 0.537).
29. A bank survey finds that 60% of customers prefer digital banking.
If 2,500 customers were surveyed, what is the 95% confidence interval for this proportion?
