Level 1 CFA® Exam:
Interval Estimation of Population Mean
As we pointed out in the previous lesson, in the case of interval estimation we need to find a range of values that we expect to include the parameter with a particular degree of confidence. This range of values is called the confidence interval.
For a \(100\times(1-\alpha)\%\) confidence interval:
- the lower confidence limit is equal to point estimate minus reliability factor times standard error, and
- the upper confidence limit is equal to point estimate plus reliability factor times standard error.
Where:
- Point estimate is a point estimate of the parameter, for example it may be a value of the sample mean,
- Reliability factor is a number based on the assumed distribution and the degree of confidence. We will look up the reliability factor in tables.
- Standard error equals population standard deviation divided by the square root of the size of the sample, or standard error equals sample standard deviation divided by the square root of the size of the sample.
As you can see the width of the confidence interval depends on the value of:
- the reliability factor, and
- the standard error.
What is more the value of the reliability factor depends on:
- distribution, and
- degree of confidence.
Additionally the value of the standard error depends on:
- population standard deviation or sample standard deviation, and
- sample size.
Having said some things about the confidence interval we can move on to the reliability factor. The value of the reliability factor depends on the distribution.
On the other hand, the distribution we choose depends on 3 factors:
- the size of the sample,
- whether the population distribution is normal, and
- whether we know the population mean.
Jus to remind you, a sample size is small if it is lower than 30, and it is large if it is at least 30.
In the table, you can see the distributions used in constructing confidence intervals and finding reliability factors.
| Sampling from: | Statistics for small sample | Statistics for large sample |
|---|---|---|
| Normal distribution with known variance | z-statistic | z-statistic |
| Normal distribution with unknown variance | t- statistic | t-statistic or z-alternative |
| Nonnormal distribution with known variance | no statistics available | z-statistic |
| Nonnormal distribution with unknown variance | no statistics available | t-statistic or z-alternative |
But what are the z-statistic, z-alterative and t-statistic anyway?
Z-statistic is based on normal distribution and is used when we know the population variance.
Z-alternative is also based on normal distribution, but we use it when the population variance is unknown (hence we use the sample standard deviation in the formula for the reliability factor). For both z-statistic and z-alternative, we look up the value of the reliability factor in the tables for normal distribution. We can use z-alternative only if the sample is large, because from the central limit theorem we know that for large sample the sample mean is approximately normally distributed.
T-statistic is based on Student’s t-distribution.
Let's assume the students of a school to be our population. We'd like to know what is the average of their grades. We know the variance for all students, which is 1.2, and the grade average for the sample of 50 students, which is 3.9. What is the 90% confidence interval for the population grade average?
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In the exam, it's worth remembering the values for the most popular degrees of confidence:
- for a 90% confidence interval: \(z_{0.05}\) is 1.65.
- for a 95% confidence interval: \(z_{0.025}\) is 1.96.
- for a 99% confidence interval: \(z_{0.005}\) is 2.58.
The higher the degrees of confidence, the wider the confidence interval.
We've drawn a sample of 35 companies listed on the exchange in Kula Hula country. We want to construct the 95% confidence interval for the mean return of shares. Unfortunately, we don't know the variance of the population. However, we do know that the sample mean return is 6% with a standard deviation of 8%, and the returns are approximately normally distributed. Use t-distribution.
(...)
We've drawn a sample of 35 companies listed on the exchange in Kula Hula country. We want to construct the 95% confidence interval for the mean return of shares. Unfortunately, we don't know the variance of the population. However, we do know that the sample mean return is 6% with a standard deviation of 8%, and the returns are approximately normally distributed. Use z-alternative.
(...)
If we cannot use the formulas we discussed in this lesson, for example, to estimate the sample mean, standard error, etc. or we are trying to find more complicated estimators, we can use the so-called resampling which is based on resampling new samples from a given sample. There are two resampling methods at our disposal: bootstrap and jackknife.
For both bootstrap and jackknife, we start by drawing a sample from the population.
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- The width of the confidence interval depends on the value of the reliability factor and the standard error.
- The value of the reliability factor depends on distribution and degree of confidence.
- The value of the standard error depends on population standard deviation or sample standard deviation and
sample size. - The distribution we choose depends on 3 factors: the size of the sample, whether the population distribution is normal, and whether we know the population mean.
- Z-statistic is based on normal distribution and is used when we know the population variance.
- Z-alternative is also based on normal distribution, but we use it when the population variance is unknown.
- T-statistic is based on Student’s t-distribution.
- Resampling is based on resampling new samples from a given sample.
- There are two resampling methods at our disposal: bootstrap and jackknife.