2005 CFA level 1 Junwei Notes 03

Emily is analyzing the monthly returns on the common stock of Mathews Inc. over the past 30 years. The mean monthly return over this period is 2% with a standard deviation of 5%. Compute and interpret the standard error of the sample mean for a sample of 3, 30, or 300.

“Sampling and Estimation”
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The Standard Error of the Sample Mean (Cont’d)
In most practical applications we need to use this formula because the population standard deviation is almost never available. The sample standard deviation (s) is estimated using the formula below:
A population has a mean µ and a variance σ2. Samples of size n are selected from this population. The sample size n is sufficiently large.
Level 1 Session 3 Quantitative Methods
Use the sample mean to infer the population mean . Construct confidence intervals for the population mean based on the normal distribution.
Level 1 Session 3 Quantitative Methods
The standard error of the sample mean refers to the standard deviation of the distribution of the sample means. If the population standard deviation (σ) is known, the standard error (σx) of the sample mean is computed as below:
•
LOS 1.B.f
The standard deviation of the sampling distribution equals σ/ n.
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409ຫໍສະໝຸດ “Sampling and Estimation”
LOS 1.B.g
If the population standard deviation (σ) is unknown, the standard error of the sample mean can be estimated using the sample standard deviation (s).
Note that the central limit theorem does not prescribe that the underlying population must be normally distributed. Therefore, the central limit theorem can be applied on a population with any probability distribution.
•
LOS 1.B.f
Sample size is 3:
σx =
σ n σ n σ n
= = =
5% 3 5% 30 5% 300
= 2.89% = 0.91% = 0.29%
(To be continued) 414
•
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Sample size is 30: σx = Sample size is 300: σx =
•

“Sampling and Estimation”
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Calculate the Standard Error of the Sample Mean: The Mathews Example (Cont’d)
Level 1 Session 3 Quantitative Methods
Solution:
Note that the population standard deviation is known in this example. The standard error of the sample mean is computed as:
Junwei®
Illustrated CFA® Study Notes 2005 Level 1 Book 2
Quantitative Methods
Thomas Wang, CFA
Book
2
“Sampling and Estimation”
What’s Next?
Sampling
√
Level 1 Session 3 Quantitative Methods
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408
“Sampling and Estimation”
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The Central Limit Theorem
Assume that:
LOS 1.B.f
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As the sample size (n) gets larger, the variability of the sample means gets smaller, causing the standard error of the sample mean to decrease.
Solution (Cont’d):
Here is how to interpret the standard error of the sample mean: • If we take all possible samples of size 3 from the monthly returns on Mathews stock over the past 30 years and construct a sampling distribution of the sample mean, the mean is 2% and the standard error is 2.89%. • If we increase the sample size to 30, the mean remains 2%, while the standard error falls to 0.91%. • If we increase the sample size to 300, the mean is still 2%, but the standard error falls further to 0.29%. Note that the standard error of the sample mean decreases as the sample size gets larger. • Also note that a sample size of 30 is large enough to apply the central limit theorem.
Level 1 Session 3 Quantitative Methods
σx =
σ n
Where: σx = The standard error of the sample mean σ = the population standard deviation n = the sample size
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The Central Limit Theorem: A Graphical Illustration
Probability Underlying Population Probability Probability
Level 1 Session 3 Quantitative Methods
Where: s = The sample standard deviation
i=1
Level 1 Session 3 Quantitative Methods
s=
i=1
∑ (xi –
n–1
n
x
)2
X = The sample mean =
∑ Xi
n
n
Xi = An observation in the sample Xi – X = Xi’s deviation from the sample mean n = the sample size
The mean of the sampling distribution equals to µ, the population mean. The variance of the sampling distribution equals σ2/n, the population variance divided by the sample size .