Lecture 4 Sampling and Estimation. Dr Peter Wheale. Sampling. To make inferences about the parameters of a population, we will use a sample A simple random sample is one where every population member has an equal chance of being selected
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Lecture 4Sampling and Estimation
Dr Peter Wheale
x = 25+34+19+54+17 / 5
= 29.8%
In the above calculations the population size, N, is 12, and the sample size, n, is 5.
All interval and ratio data sets have an arithmetic mean.
1. Create subgroups from population based on important characteristics, e.g. identify bonds according to: callable, ratings, maturity, coupon
2. Select samples from each subgroup in proportion to the size of the subgroup
Used to construct bond portfolios to match a bond index or to construct a sample that has certain characteristics in common with the underlying population
Time-series data
e.g. Monthly prices for IBM stock for 5 years
Cross-sectional data
e.g. Returns on all health care stocks last month
construct confidence intervals for population means based on sample means
Standard error of sample mean is the standard deviation of the distribution of sample means.
Example: The mean P/E for a sample of 41 firms is 19.0, and the standard deviation of the population is 6.6. What is the standard error of the sample mean?
Interpretation: For samples of size n = 41, the distribution of the sample means would have a mean of 19.0 and a standard error of 1.03.
Example: The mean P/E for a sample of 41 firms is 19.0, the standard error of the sample mean is 1.03, and the population is normal
Point estimate of mean is 19.0
90% confidence interval is 19 +/- 1.65 (1.03)
17.3 < mean < 20.7
95% confidence interval is 19 +/- 1.96 (1.03)
17.0 < mean < 21.0
Confidence interval: a range of values around an expected outcome within which we expect the actual outcome to occur some specified percent of the time.
1. Unbiased- expected value equal to parameter
2. Efficient- sampling distribution has smallest variance of all unbiased estimators
3. Consistent– larger sample → better estimator
Standard error of estimate decreases with larger sample
size
Properties of Student’s t-Distribution
The figure below shows the shape of the t-distribution with different degrees of freedom.