14 Lesson 2: Point and Interval Estimates of the Population Mean, Student's t-distribution, Sample Size and Biases

Why are point estimates and confidence intervals fundamental concepts in statistics?

Because they help make inferences about the population parameters based on sample data.

Define point estimate

Is a statistic calculated from sample data that serves as an approximation of an unknown parameter.

Give an example of a point estimate

the sample mean ¯x is a point estimate of the population mean μ. The formula used to calculate a point estimate is known as an estimator.

Define confidence interval

A confidence interval is a range of values constructed using sample data that is likely to contain the true (unknown) population parameter with a certain level of confidence.

How can we express a confidence interval?

A confidence interval is expressed as Point estimate ± Margin of Error.

How is the margin of error determined?

The margin of error is determined by the reliability factor (related to the desired confidence level) and the standard error of the point estimate → (reliability factor × standard error)

What are the factors that affect the reliability factor?

The reliability factor depends on the assumed distribution of the point estimate and the chosen confidence level (often denoted as (1 - α), where α is the level of significance).

State the desirable properties of an estimator

The estimator needs to be:

* Unbiased
* Efficient
* Consistent

Define an unbiased estimator

An estimator is unbiased if, on average, it produces estimates that are equal to the true population parameter.

For example, the sample mean ¯x is an unbiased estimator of the population mean μ, as E(¯x) = μ.

Define an efficient estimator

An efficient unbiased estimator is the one that has the *lowest* variance among all unbiased estimators of the same parameter.

Define a consistent estimator

A consistent estimator is one for which the probability of producing estimates close to the true parameter value increases as the sample size increases. As sample size grows, the estimator approaches the true parameter.

Where does consistency apply the most?

Consistency is particularly important in large datasets (Big Data).

State the properties of the t-Distribution

* It is symmetrical.
* It is characterized by a single parameter, degrees of freedom (df), where df = sample size - 1 (n - 1).
* It has a lower peak and fatter tails compared to the normal distribution.
* As degrees of freedom increase, the t-distribution approaches the shape of the standard normal distribution (z-distribution)

How are the degrees of freedom related to the sample size?

Higher degrees of freedom lead to a t-distribution that closely resembles the normal distribution.

When can we use the t-Distribution?

* To construct confidence intervals for a *normally* distributed population whose variance is *unknown* when the sample size is small (n ﹤ 30).
* *Non-normally* distributed population whose variance is *unknown* if the sample size is *large* (n ≥ 30).

How do we calculate a confidence interval for the population mean when the population follows a *normal* distribution and its variance is *known?*

We use the z-Distribution

The following reliability factors are used frequently when constructing confidence intervals based on the standard normal distribution:



* For a 90% confidence interval we use z0.05 = 1.65
* For a 95% confidence interval we use z0.025 = 1.96
* For a 99% confidence interval we use z0.005 = 2.58

How can we interpret the results when we use z-Distribution?

How do we calculate a confidence interval for the population mean when the population follows a *normal* distribution and its variance is *not* *known?*

We use the t-Distribution.

Remember that we need to use the table for the T-Distribution and to calculate the reliability factor we need: degrees of freedom and level of significance/2 (for both tails)

What can we use when the population distribution is non normal?

* If the population variance is *known* and the sample size is large (n ≥ 30) we use the z-statistic. This is because the central limit theorem tells us that the distribution of the sample mean is approximately normal when sample size is large.
* If the population variance is *not known* and sample size is large, we can use the z-statistic or the t-statistic. However, in this scenario the use of the t-statistic is encouraged because it results in a more conservative measure.

Summarize when to apply the z and t-Distributions

a: z-statistic also accepted

What is the process of bootstrapping?

Bootstrap is a technique that simulates the process of repeatedly drawing samples from the original observed data sample, with replacement, to estimate the sampling distribution of a statistic.

What does the resampling with replacement implies on bootstrapping?

Some items may appear multiple times, and others may not appear at all in the resamples.

Does bootstrap rely on analytical formulas?

No. That is why is known for its accuracy.

Where can we apply bootstrapping?

Bootstrap finds extensive applications in historical simulations, asset allocation, and assessing investment strategy performance against benchmarks.

What is the jackknife resampling method?

Unlike bootstrap, jackknife does not involve sampling with replacement; instead, it leaves out chosen observations (one at a time) from subsequent resamples.

Why does jackknife uses resampling without replacement

To reduce the bias of an estimator, particularly when dealing with small sample sizes.

When is jackknife useful?

Jackknife typically produces similar results for every run, making it a useful method when consistency in outcomes is preferred.

What facto affect confidence intervals?

* The choice of test statistic: A t-statistic gives a wider confidence interval.
* The degree of confidence: A higher desired level of confidence increases the size of the confidence interval.

What is the relationship between standard errors, sample size and confidence intervals?

The *larger* the standard error, the *wider* the confidence interval. The standard error, in turn, is a function of sample size. More specifically, a *larger* sample size results in a *smaller* standard error and *reduces* the width of the confidence interval. Therefore, large sample sizes are desirable as they increase the precision with which we can estimate a population parameter.

What are some disadvantages of increasing sample size?

* Increasing the size of the sample may result in drawing observations from a different population.
* Increasing the sample size may involve additional expenses that outweigh the benefit of increased accuracy of estimates.

What is the data snooping bias?

Occurs when researchers extensively search through a dataset for statistically significant relationships without a predetermined hypothesis, leading to the discovery of apparent correlations that are, in fact, coincidental.

How can we avoid data snooping bias?

Statistically significant relationships should be tested on "out-of-sample" data to see if they hold.

What is the sample-selection bias?

Sample-selection bias occurs when certain assets are excluded from a study due to unavailability of data, leading to a biased sample.

What is the look-ahead bias?

Look-ahead bias arises when a study uses information that was not available on the test date, potentially leading to inaccurate conclusions.

How can we mitigate look-ahead bias?

To mitigate this bias, researchers can use point-in-time (PIT) data whenever possible, which reflects the information available at a specific point in time.

What is the time-period bias?

Time-period bias occurs when a study is based on a specific time period, potentially making the results time-period specific and not applicable to other periods.