Sampling And Estimation (Reading 10)

Learning Outcome Statements (LOS)

a

Define simple random sampling and a sampling distribution:

b

Explain sampling error:

c     

Distinguish between simple random and stratified random sampling:

d    

Distinguish between time-series and stratified cross-sectional data:     

e  

Explain the central limit theorem and its importance:

f 

Calculate and interpret the standard error of the sample mean:

g

Identify and describe desirable properties of an estimator:

h

Distinguish between a point estimate and a confidence interval estimate of a population parameter:

i

Describe the properties of Student’s t-distribution and calculate and interpret its degrees of freedom:

j

Calculate and interpret a confidence interval for a population mean, given a normal distribution with 1) a known population variance, 2) an unknown population variance, or 3) an unknown variance and a large sample size:

k

Describe the issues regarding selection of the appropriate sample size, data-mining bias, sample selection bias, look-ahead bias, and time-period bias.

Formulas:

Exercise Problems:

1.      A mutual fund manager wants to create a fund based on a high-grade corporate bond index. She first distinguishes between utility bonds and individual bonds; she then, for each segment, defines maturity intervals of less than 5 years, 5 to 10 years, and greater than 10 years. For each segment and maturity level, she classifies the bonds as callable or non-callable. She then selects bonds from each of the subpopulations she has created. For the manager’s sample, which of the following best describes the sampling approach?

A.    systematic

B.     simple random

C.     stratified random

 Ans: C; in stratified random sampling, the population is divided into subpopulation based on one or more classification criteria. Simple random samples are then drawn from each stratum in sizes proportional to the relative size of each stratum in the population. These samples are then pooled to form a stratified random sample.

In this problem, the manager first divide the population based on several criteria, and then select from each of the subpopulation. So it’s stratified random sampling.

A is incorrect; systematic sampling is always used when the population size cannot be identified. With systematic sampling, we select every kth member until we have a sample of desired size.

B is incorrect; a simple random sample is a subset of a large population created in such a way that each element of the population has an equal probability of being selected to the subset. The procedure of drawing a sample to satisfy the definition of a simple random sample is called simple random sampling.

2.      An analyst collects data relating to five commonly used measures of use of debt and interest coverage for a randomly chosen sample of 300 firms. The data comes from those firms’ fiscal year 2011 annual reports. This data is best characterized as:   

A.    Time-series data.

B.     Longitudinal data.

C.     Cross-sectional data.

Ans: C; since with the cross-sectional data, the observations in the sample represent a characteristic of individuals, groups, geographical regions, or companies at a single point in time.

In this problem, the analyst chooses 300 firms in a single point in time, so it’s cross-sectional data.

A is incorrect; time-series is a sequence of returns collected at discrete and equally spaced intervals of times.

B is incorrect; longitudinal data consist of observations on characteristic of the same observational unit through time.

3.      If the distribution of the population from which the samples are drawn is positive skewed, and given that the sample size is large, the sampling distribution of the sample means is most likely:

A.    Approximately normally distributed

B.     To have a variance equal to that of the entire population

C.     To have a mean smaller than the mean of the entire population

Ans: A; according to the Central Limit Theorem, when the sample size n is large, we can assume the sample mean is normal distributed whatever the distribution of the population.

B is incorrect; the variance should be the population variance divided by n.

C is incorrect; the mean is equal to the population mean..

4.      The Central Limit Theorem is best described as stating that the sampling distribution of the sample mean will be approximately normal for large-size sample:

A.  if the population distribution is normal 

B.  if the population distribution is symmetric

C. for populations described by any probability distribution.

Ans: C; according to the Central Limit Theorem, when the sample size n is large, we can assume the sample mean is normal distributed whatever the distribution of the population.

5.      An According to the Central Limit Theorem, a sampling distribution of the sample mean will be approximately normal only if the:

A.    Sample size is large

B.     Underlying distribution is normally distributed

C.     Variance of the underlying distribution is known.

Ans: A:  according to the Central Limit Theorem, when the sample size n is large, we can assume the sample mean is normal distributed whatever the distribution of the population.

6.      The following sample of 10 items is selected from a population. The population variance is unknown.

The standard error of the sample mean is closest to:

A.  2.8

B.  2.7

C.  9.0

Ans: A;

7.      A sample of 25 observations has a mean of 8 and a standard deviation of 15. The standard error of the sample mean is closest to:

A.    1.60

B.     3.00

C.     3.06      

Ans: B;

8.      An analyst gathered the following information about a stock index:

If the analyst takes a sample of 36 companies from the index, the standard error of the sample mean is closest to:

A.  $88,889

B.  $400,000

C.  $533,333

Ans: C;

9.      An analyst gathers the following information about a sample:

The standard error of the sample mean is closest to:

A.  0.47

B.  0.64

C.  0.80

Ans: C;

10.  In generating an estimate of a population parameter, a large sample size is most likely to improve the estimator’s:

A.  Efficiency

B.  Consistency

C.  Unbiasedness

 Ans: B; a consistent estimator is one for which the probability of estimates close to the value of the population parameter increases as sample size increase. So a large size is most likely to improve the estimator’s efficiency.

A is incorrect; an unbiased estimator is efficient if no other unbiased estimator of the same parameter has a sampling distribution with smaller variance.

C is incorrect; an unbiased estimator is one whose expected value equals the parameter it is intended to estimate.

11.  When an analyst is unsure of the underlying population distribution, which of the following is least likely to increase the reliability of parameter estimates?

A.  Increase in the sample size

B.  Use of point estimates rather than confidence intervals

C.  Use of the t-distribution rather than the normal distribution to establish confidence intervals

Ans: B; because of sampling error, the point estimate is not likely to equal the population parameter. So a confidence interval gives a more reliable estimate of the parameter.

A is incorrect; according the formula, sampling error will decrease with the increase of sample size. So increase in the sample size could increase the reliability of parameter estimates.

C is incorrect; when the population variance, using a reliability factor based on the t-distribution is essential for a small sample size. Even when we have a large sample, the t-distribution provides more-conservative confidence intervals.

12.  Compared to the normal distribution, the Student’s t-distribution most likely:

A.  Has fatter tails

B.  Is more peaked

C.  Has greater degrees of freedom

Ans: A; the tails of t-distribution will be thinner with the increase of degree of freedom. And t-distribution will approach normal distribution when the degree of freedom is very large. So compared to normal distribution, t-distribution always has fatter tails.

B is incorrect; normal distribution is more peaked than t-distribution.

C is incorrect; normal distribution has greater degree of freedom.

13.  All else held constant, the width of a confidence interval is most likely to be smaller if the sample size is:

A.  Larger and the degree of confidence is lower

B.  Larger and the degree of confidence is higher

C.  Smaller and the degree of confidence is lower

Ans: A; consider the formula of confidence interval:

The width of interval is smaller when:

1.       n is larger which means sample size is larger

2.       z is smaller which means degree of confidence is lower

14.  Use the following values from Student’s t-distribution to establish a 95% confidence interval for the population mean given a sample size of 10, a sample mean of 6.25, and a sample standard deviation of 12. Assume that the population from which the sample is drawn is normally distributed and the population variance is not known.

The 95% confidence interval is closest to:

A.    A lower bound of -2.33 and an upper bound of 14.83

B.     A lower bound of -2.20 and an upper bound of 14.70

C.     A lower bound of -0.71 and an upper bound of 13.20

Ans: A; consider the formula of confidence interval:

Where degree of freedom is sample size minus 1, which is 9. So two-tail t value is 2.262.

So the lower bound is -2.33 and the higher bound is 14.83