# Understanding Estimators and Confidence Intervals | CFA Level I Quantitative Methods

In this lesson, we’ll dive into the properties of estimators, types of estimators, and how to compute confidence intervals. Let’s get started!

## Desirable Properties of Estimators: Consistent, Unbiased, and Efficient

Before we delve into the types of estimators, let’s understand the desirable properties of an estimator using the acronym CUE:

• Consistent
• Unbiased
• Efficient

### Consistent Estimators

A consistent estimator is one in which the accuracy of the estimate increases as the sample size increases. The sample mean is a consistent estimator of the population mean.

### Unbiased Estimators

An unbiased estimator is one in which the expected value of the estimator is equal to the parameter to be estimated. For example, if we are estimating the mean of a population, the estimator is considered unbiased if the expected value of the sample mean is equal to the population mean.

### Efficient Estimators

An unbiased estimator is also efficient if no other unbiased estimator has a sampling distribution with smaller variance. The sample mean is both an unbiased and efficient estimator of the population mean, as there is no other unbiased estimator that has a sampling distribution with a smaller variance.

## Types of Estimators: Point Estimate and Confidence Interval

There are two main types of estimators:

1. Point estimate – a single value used to estimate population parameters.
2. Confidence interval – a range of values in which the population parameter is expected to fall within, with a prescribed level of confidence.

Now, let’s discuss how to compute confidence intervals using the following structure:

Confidence Interval = Point estimate ± (Reliability factor × Standard error)

## Applying Z-Statistics to Compute Confidence Intervals

Remember that the Z-table and Z-statistics are only valid for normal distributions. Thanks to the Central Limit Theorem, we can assume that the sampling distribution is normal when we have a large sample size (n ≥ 30), regardless of the underlying population distribution. This means we can apply Z-statistics when constructing confidence intervals.

### EXAMPLE

Veron Foo wants to study the mean monthly return of a stock for the past 20 years, which has a known standard deviation of 7.18%. She picks a sample of 50 random observations from the past, and found that the sample mean is 1.47% with a standard deviation of 5.23%. What is the 95% confidence interval for the mean monthly return of the stock?

Here’s the solution:

1. Since the sample size (n) is 50, we can approximate the sampling distribution to normal.
2. Apply the Z-statistic for a 95% confidence interval: ±1.96 standard deviations.
3. Use the population standard deviation (7.18%) divided by the square root of the sample size (50) to get the standard error of the mean: 1.02%.
4. Calculate the 95% confidence interval for the mean return: 1.47% ± (1.96 × 1.02%) = -0.53% to 3.47%.

We can interpret this as being 95% confident that the mean monthly return falls between -0.53% and 3.47%.

### Computing Confidence Intervals with Different Confidence Levels

If we want to find an 80% confidence interval instead, we need to look up the Z-table for the corresponding Z-value. In this case, the level of significance is 20%, and the degree of confidence is 80%.

From the Z-table, we find the Z-value corresponding to a 90% area under the curve (80% confidence plus 10% on the right tail) to be 1.28.

Now, we can calculate the 80% confidence interval for the mean return: 1.47% ± (1.28 × 1.02%) = 0.16% to 2.78%. This implies that we are 80% confident that the mean monthly return falls between 0.16% and 2.78%.

### Using T-Statistic When Sample Size is Less Than 30

In cases where the sample size is less than 30, we cannot assume that the sampling distribution is normal. As a result, we cannot use Z-statistics to solve for the confidence interval. However, if the underlying distribution is normal, we can use the Student’s t-statistic instead.

## Understanding Student’s t-Distribution

Student’s t-distribution is an alternative to the z-distribution, used when the z-statistic isn’t appropriate. It’s less peaked and has fatter tails than the normal distribution. The deviation from normality is determined by one single parameter – the degrees of freedom (n-1).

As the degrees of freedom increase, the t-distribution approaches the normal distribution.

## When to Use the z-statistic or t-statistic

1. Is the population normally distributed?
2. Is the population variance known?
3. Is the sample size at least 30?

If the population is normal, the sampling distribution is assumed normal. If the population variance is known, use the z-statistic. Otherwise, use the t-statistic. If the population isn’t normal or unknown, check if the sample size is at least 30. If it’s less than 30, the problem can’t be solved. If it’s at least 30, use the central limit theorem to determine whether to use the z or t statistic.

## Calculating Confidence Intervals with the t-statistic

EXAMPLE

Using the earlier example (Veron Foo), let’s change the problem a little. We now do not know the standard deviation of the population, which is the more likely scenario. We also now make the assumption that the underlying distribution is normal. We also have a smaller sample size of 5 random observations. Find the 95% confidence interval for the mean monthly return of the stock.

1. Determine if you should use the z-statistic or t-statistic. In this case, use the t-statistic since the population variance is unknown.
2. Use the t-distribution with 4 degrees of freedom (n-1).
3. Find the t-value for a 95% confidence interval: 2.776.
4. Calculate the standard error: 2.34.
5. Find the confidence interval: -5.08% to 7.96%.

We are 95% confident that the mean monthly return falls between -5.08% and 7.96%.

## Summary

To calculate confidence intervals, determine these 3 key components:

Remember this framework to calculate confidence intervals effectively. In the next lesson, we’ll discuss biases in sampling.

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