Quantitative Methods for Valuation
Reading 10: Sampling and Estimation
1. A research assistant is evaluating the monthly returns on 30 different stocks over a period of 5 years. The data set she is studying is most likely an example of:
A. Longitudinal data.
B. Panel data.
C. Cross-sectional data.
2. Which of the following statements is least likely regarding the assertions of the Central Limit Theorem?
A. As long as sample size is large, the sample mean and the population mean are equal.
B. As long as sample size is large, the sample standard deviation equals the population standard deviation divided by n.
C. As long as sample size is large, the sample variance equals the population variance divided by n.
3. A sample containing the monthly returns for 75 U.S. stocks has a mean of 5% and a standard deviation of 12%. The standard error of the
sample mean:
A. Cannot be calculated given the information.
B. Approximately equals 1.39%.
C. Approximately equals 0.58%.
4. Which of the following is least likely a desirable characteristic of an estimator?
A. Unbiasedness B. Efficiency C. Effectiveness
5. Which of the following is least likely regarding a student’s t-distribution?
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A. It has a lower peak and thinner tails than the normal distribution.
B. It is defined by a single parameter, the degrees of freedom.
C. It is symmetrical.
6. In which of the following scenarios should the z-distribution be used over the t-distribution?
A. To construct confidence intervals for a normally (or approximately normally) distributed population whose variance is unknown when the sample size is small (n < 30).
B. To construct confidence intervals for a non-normally distributed population whose variance is known and the sample size is large (n
> 30).
C. To construct confidence intervals for a normally distributed population whose variance is unknown if the sample size is large (n > 30).
7. Which of the following statements is most likely?
A. A larger sample size results in a smaller standard error and reduces the width of the confidence interval.
B. A smaller sample size results in a larger standard error and reduces the width of the confidence interval.
C. A smaller sample size results in a smaller standard error and reduces the width of the confidence interval.
8. A sample of the monthly returns of ABC Stock over the last 3.5 years has a mean return of 0.5% and a standard deviation of 0.075%. The 95% confidence interval for the average monthly returns on ABC Stock is closest to?
A. 0.4766% to 0.5234%
B. 0.4773% to 0.5227%
C. 0.4850% to 0.5150%
9. A sample of the monthly returns of ABC Stock over the last 4 years has a mean return of 0.8%. The standard deviation of monthly returns over ABC’s entire history equals 0.025%. The 95% confidence interval
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for the average monthly returns on ABC Stock is closest to?
A. 0.7850% to 0.8150%
B. 0.7929% to 0.8071%
C. 0.7927% to 0.8073%
10. The sampling distribution of a statistic is most likely made up of all possible:
A. Sample statistics computed from samples of different sizes.
B. Sample statistics computed from samples of the same size.
C. Observations from the underlying population.
11. If the data set used by a researcher includes data that were not available at the time the study was conducted, the data set being studied most likely suffers from:
A. Look-ahead bias B. Survivorship bias C. Time-period bias
12. The probability (1-α) is most likely referred to as:
A. Level of significance B. Confidence interval C. Degree of confidence
13. A simple random sample is most likely:
A. One in which each member of the population has a greater than 1%
chance of being chosen
B. One in which each member of the population has an equal chance of being chosen
C. One in which each member of the population has a 1% chance of being chosen
14. Which of the following statements is most accurate?
A. Sampling error is the distribution of all possible values of a sample statistic.
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B. Sampling error equals the difference between the sample statistic and the corresponding population parameter.
C. Sampling error is the total difference between the sample statistic from one sample and the sample statistic from another sample.
15. Data collected by observing several subjects at the same point in time is most likely referred to as:
A. Cross-sectional data B. Time-series data C. Panel data
16. Longitudinal data most likely refers to?
A. Data collected over time about a single characteristic of multiple observational units
B. Data collected over time about multiple characteristics of the same observational unit
C. Data collected over time about multiple characteristics of multiple observational units
17. The standard error of a sample that has an arithmetic mean of 11, a standard deviation of 3, and 20 observations is closest to:
A. 0.9 B. 0.67 C. 0.15
18. Which of the following statements is most accurate regarding a student’s t-distribution?
A. It is symmetrical.
B. It is defined by a single parameter, the degrees of freedom, where degrees of freedom equal sample size.
C. As the degrees of freedom increase, the t-distribution becomes less peaked and the tails become fatter.
19. A sample of 100 students was selected from a school. The average points scored by these 100 students were 69. Assuming that the
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distribution of scores of all students in this school is approximately normal, and given that the population standard deviation is 31, the 90% confidence interval is closest to?
A. 19.7 to 43.32 B. 63.9 to 74.1 C. 58.64 to 74.08
20. Ashley selected a sample of 20 stocks and calculated their mean return over a 3-year period to be 4.25%. Given that the sample standard deviation is 0.3% and assuming that the population is normally distributed, the 90% confidence interval is closest to:
A. 4.13% to 4.37%
B. 4.22% to 4.44%
C. 4.14% to 4.36%
21. Brian selected a sample of 20 stocks and calculated their mean return for the year to be 6.3%. Given that the population standard deviation is 1.2% and that the population is normally distributed, the 90%
confidence interval is closest to:
A. 5.86% to 6.74%
B. 5.78% to 6.83%
C. 5.84% to 6.76%
22. Given that the sample size is less than 30, the population variance is known, and the population is not normally distributed, which of the following is most likely?
A. The t-stat is the appropriate test statistic.
B. The z-stat is the appropriate test statistic.
C. Neither the z nor the t-stats can be used.
23. Compared to simple random sampling, stratified random sampling most likely:
A. Increases the size of the sample.
B. Improves the representativeness of the sample.
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C. Increases the sampling error.
24. A data set is examined extensively until a relationship that appears to be statistically significant is discovered. This most likely indicates that the study might suffer from:
A. Data mining bias B. Sample selection bias C. Look-ahead bias
25. Which of the following will least likely result in a wider confidence interval?
A. Use of the z-distribution over the t-distribution B. Use of a smaller sample size
C. Use of a lower significance level
transmitted without publisher's prior permission. Violators will be prosecuted.
Reading 10: Sampling and Estimation
1. A research assistant is evaluating the monthly returns on 30 different stocks over a period of 5 years. The data set she is studying is most likely an example of:
A. Longitudinal data.
B. Panel data.
C. Cross-sectional data.
Answer: B
Panel data refers to data collected over time (5 years) about a single characteristic (monthly returns) of multiple observational units (30 different stocks).
2. Which of the following statements is least likely regarding the assertions of the Central Limit Theorem?
A. As long as sample size is large, the sample mean and the population mean are equal.
B. As long as sample size is large, the sample standard deviation equals the population standard deviation divided by n.
C. As long as sample size is large, the sample variance equals the population variance divided by n.
Answer: B
As long as sample size is large, the sample standard deviation equals the population standard deviation divided by the square root of n.
3. A sample containing the monthly returns for 75 U.S. stocks has a mean of 5% and a standard deviation of 12%. The standard error of the
sample mean:
A. Cannot be calculated given the information.
B. Approximately equals 1.39%.
C. Approximately equals 0.58%.
transmitted without publisher's prior permission. Violators will be prosecuted.
Answer: B
Standard error = 12/8.66 = 1.39
4. Which of the following is least likely a desirable characteristic of an estimator?
A. Unbiasedness B. Efficiency C. Effectiveness Answer: C
Effectiveness is not one of the desirable characteristics of an estimator.
5. Which of the following is least likely regarding a student’s t-distribution?
A. It has a lower peak and thinner tails than the normal distribution.
B. It is defined by a single parameter, the degrees of freedom.
C. It is symmetrical.
Answer: A
The t-distribution does have a lower peak, but has fatter or thicker tails compared to a normal distribution.
6. In which of the following scenarios should the z-distribution be used over the t-distribution?
A. To construct confidence intervals for a normally (or approximately normally) distributed population whose variance is unknown when the sample size is small (n < 30).
B. To construct confidence intervals for a non-normally distributed population whose variance is known and the sample size is large (n
> 30).
C. To construct confidence intervals for a normally distributed population whose variance is unknown if the sample size is large (n > 30).
Answer: B
When the population variance is known, the z-distribution should
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ideally be used.
7. Which of the following statements is most likely?
A. A larger sample size results in a smaller standard error and reduces the width of the confidence interval.
B. A smaller sample size results in a larger standard error and reduces the width of the confidence interval.
C. A smaller sample size results in a smaller standard error and reduces the width of the confidence interval.
Answer: A
A larger sample size results in a smaller standard error and reduces the width of the confidence interval. A smaller sample size results in a larger standard error and increases the width of the confidence
interval.
8. A sample of the monthly returns of ABC Stock over the last 3.5 years has a mean return of 0.5% and a standard deviation of 0.075%. The 95% confidence interval for the average monthly returns on ABC Stock is closest to?
A. 0.4766% to 0.5234%
B. 0.4773% to 0.5227%
C. 0.4850% to 0.5150%
Answer: A
n = 42; df = 41; t-critical value (5% sig. level 2-tailed) = 2.0211;
standard error = 0.01157. Therefore, the confidence interval is given as:
0.5 ± 2.0211 × 0.01157
Use the t-distribution as the population variance is not known and the sample size is large.
9. A sample of the monthly returns of ABC Stock over the last 4 years has a mean return of 0.8%. The standard deviation of monthly returns over ABC’s entire history equals 0.025%. The 95% confidence interval for the average monthly returns on ABC Stock is closest to?
transmitted without publisher's prior permission. Violators will be prosecuted.
A. 0.7850% to 0.8150%
B. 0.7929% to 0.8071%
C. 0.7927% to 0.8073%
Answer: B
n = 48; z-critical value (5% sig. level 2-tailed) = 1.96; standard error = 0.00361 Therefore, the confidence interval is given as:
0.8 ± 1.96 × 0.00361
Use the z-distribution as the population variance is known.
10. The sampling distribution of a statistic is most likely made up of all possible:
A. Sample statistics computed from samples of different sizes.
B. Sample statistics computed from samples of the same size.
C. Observations from the underlying population.
Answer: B
The sampling distribution of a statistic is made up of all possible sample statistics computed from samples of the same size.
11. If the data set used by a researcher includes data that were not available at the time the study was conducted, the data set being studied most likely suffers from:
A. Look-ahead bias B. Survivorship bias C. Time-period bias Answer: A
Look-ahead bias arises when a study uses information that was not available on the test date.
12. The probability (1-α) is most likely referred to as:
A. Level of significance B. Confidence interval C. Degree of confidence
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Answer: C
The probability (1 − α) is referred to as the confidence level while α refers to the significance level.
13. A simple random sample is most likely:
A. One in which each member of the population has a greater than 1%
chance of being chosen
B. One in which each member of the population has an equal chance of being chosen
C. One in which each member of the population has a 1% chance of being chosen
Answer: B
14. Which of the following statements is most accurate?
A. Sampling error is the distribution of all possible values of a sample statistic.
B. Sampling error equals the difference between the sample statistic and the corresponding population parameter.
C. Sampling error is the total difference between the sample statistic from one sample and the sample statistic from another sample.
Answer: B
15. Data collected by observing several subjects at the same point in time is most likely referred to as:
A. Cross-sectional data B. Time-series data C. Panel data
Answer: A
Time-series data consists of observations measured at uniform
intervals over a period of time. Panel data refers to data collected over time about a single characteristic of multiple observational units.
16. Longitudinal data most likely refers to?
transmitted without publisher's prior permission. Violators will be prosecuted.
A. Data collected over time about a single characteristic of multiple observational units
B. Data collected over time about multiple characteristics of the same observational unit
C. Data collected over time about multiple characteristics of multiple observational units
Answer: B
Longitudinal data refers to data collected over time about multiple characteristics of the same observational unit.
17. The standard error of a sample that has an arithmetic mean of 11, a standard deviation of 3, and 20 observations is closest to:
A. 0.9 B. 0.67 C. 0.15 Answer: B
Standard error = 3 / 200.5 = 0.67.
18. Which of the following statements is most accurate regarding a student’s t-distribution?
A. It is symmetrical.
B. It is defined by a single parameter, the degrees of freedom, where degrees of freedom equal sample size.
C. As the degrees of freedom increase, the t-distribution becomes less peaked and the tails become fatter.
Answer: A
The t-distribution is defined by a single parameter, the degrees of freedom, where degrees of freedom equal (n − 1).
As the degrees of freedom increase, the t-distribution curve becomes more peaked and the tails become thinner.
19. A sample of 100 students was selected from a school. The average points scored by these 100 students were 69. Assuming that the
transmitted without publisher's prior permission. Violators will be prosecuted.
distribution of scores of all students in this school is approximately normal, and given that the population standard deviation is 31, the 90% confidence interval is closest to?
A. 19.7 to 43.32 B. 63.9 to 74.1 C. 58.64 to 74.08 Answer: B
The population variance is known and the sample size is greater than 30, so we will use the z-values. The confidence interval is:
69 − 1.645 (31 / √l00) to 69 + 1.645 (31 / √l00) = 63.9 to 74.1.
20. Ashley selected a sample of 20 stocks and calculated their mean return over a 3-year period to be 4.25%. Given that the sample standard deviation is 0.3% and assuming that the population is normally distributed, the 90% confidence interval is closest to:
A. 4.13% to 4.37%
B. 4.22% to 4.44%
C. 4.14% to 4.36%
Answer: A
Since the population variance is not known, but the population is assumed to be normally distributed, and the sample size is small we must use the t-distribution.
Standard error = 0.3 / (20)0.5 = 0.06708 Degrees of freedom = 20 − 1 = 19
For a 90% confidence interval, we need 5% in either tail. The relevant t-score with 19 degrees of freedom is 1.7291.
Confidence interval = 4.25% ± (1.7291 * 0.067%) = 4.13% to 4.37%
21. Brian selected a sample of 20 stocks and calculated their mean return for the year to be 6.3%. Given that the population standard deviation is 1.2% and that the population is normally distributed, the 90%
confidence interval is closest to:
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A. 5.86% to 6.74%
B. 5.78% to 6.83%
C. 5.84% to 6.76%
Answer: A
Since the population variance is known, the population is normally distributed, and the sample size is small, we must use the
z-distribution.
Standard error = 1.2 / (20)0.5 = 0.2683
For a 90% confidence interval, we need 5% in either tail. The relevant z-score is 1.645.
Confidence interval = 6.3% ± (1.645 * 0.2683%) = 5.86% to 6.74%.
22. Given that the sample size is less than 30, the population variance is known, and the population is not normally distributed, which of the following is most likely?
A. The t-stat is the appropriate test statistic.
B. The z-stat is the appropriate test statistic.
C. Neither the z nor the t-stats can be used.
Answer: C
When studying a population that does not follow a normal
distribution, neither the z nor the t-stat can be used if the sample size is less than 30.
23. Compared to simple random sampling, stratified random sampling most likely:
A. Increases the size of the sample.
B. Improves the representativeness of the sample.
C. Increases the sampling error.
Answer: B
Stratified random sampling improves the representativeness of the sample and reduces sampling error.
transmitted without publisher's prior permission. Violators will be prosecuted.
24. A data set is examined extensively until a relationship that appears to be statistically significant is discovered. This most likely indicates that the study might suffer from:
A. Data mining bias B. Sample selection bias C. Look-ahead bias Answer: A
Data mining bias is the practice of developing a model by extensively searching through a data set for statistically significant relationships until a pattern “that works” is discovered.
25. Which of the following will least likely result in a wider confidence interval?
A. Use of the z-distribution over the t-distribution B. Use of a smaller sample size
C. Use of a lower significance level Answer: A
Use of the z-distribution over the t-distribution results in a narrower confidence interval.
A smaller sample size increases standard error and increase the width of the confidence interval.
A lower significance basically means a higher confidence level, which implies a wider interval.
transmitted without publisher's prior permission. Violators will be prosecuted.