X bar = (Σ x / n)
The value we obtain from this calculation for a specific sample is called the point estimate of
the mean.
A confidence interval is a range of estimated values within which the actual value of the
parameter will lie with a given probability of 1 - α. The term α is also called the significance
level of the test. It is also known as the confidence level.
i: Identify and describe the desirable properties of an estimate.
When we have a choice among several estimators, we want to select the one with the most
desirable statistical properties: unbiasedness, efficiency, and consistency.
An unbiased estimator is one for which the expected value of the estimator is equal to the
parameter you are trying to estimate.
An unbiased estimator is also efficient if the variance of its sampling distribution is smaller
than all the other unbiased estimators of the parameter you are trying to estimate. The
sample mean, for example, is an efficient estimator of the population mean.
A consistent estimator provided a more accurate estimate of the parameter as the sample
size increases. As the sample size increases, the standard deviation (standard error) of the
sample mean falls and the sampling distribution bunches more closely around the population
mean.
j: Calculate and interpret a confidence interval for a population mean, given a normal distribution
with a known population variance.
If the distribution of the population is normal and we know the population variance, we can
construct the confidence interval for the population mean as follows:
?
X bar Z
α/2
(σ /?
√n)
?
Example: Suppose we administer a practice exam to 100 CFA Level I candidates, and we
discover the mean score on this practice exam for all 36 of the candidates in the sample who
studied at least 10 hours a week in preparation for the exam is 80. Assume the population
standard deviation is 15. Construct a 99% confidence interval for the mean score on the
practice exam of candidates who study at least 10 hours a week.
?
80 2.575 (15 /
√36) = 80 6.4
?
The 99% confidence has a lower limit of 73.6 and an upper limit of 86.4.
k: Describe the properties of Student's t-distribution.
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