Module 10.2 Confidence Intervals and t-distribution

Point estimates

- Single (sample) values used to estimate population parameters -The formula used to compute the point estimate is called the estimator page 202 The value generated with this calculation for a given sample is called the point estimate of the mean. A confidence interval is a range of values in which the population parameter is expected to lie. The construction of confidence intervals is described later in this topic review.

Student's t-distribution

-A bell-shaped probability distribution that is symmetrical about its mean. Used when constructing confidence intervals based on SMALL samples (n < 30) from populations with UNKNOWN variance and a normal, or approximately normal, distribution. -Use the t-distribution when the population variance is unknown and the sample size is large enough that the central limit theorem will assure that the sampling distribution is approximately normal.

Table of critical t-values

-Contains one-tailed critical values for the t-distribution at the 0.05 and 0.025 levels of significance with various degrees of freedom (df). -Note that, unlike the z-table, the t-values are contained within the table, and the probabilities are located at the column headings. Also note that the level of significance of a t-test corresponds to the one-tailed probabilities, p, that head the columns in the t-table. Page 204

Confidence interval

-Result in a range of values within which the actual value of a parameter will lie, given the probability of 1 − α. -Here, alpha, α, is called the level of significance for the confidence interval, and the probability 1 − α is referred to as the degree of confidence. Ex: estimate that the population mean of random variables will range from 15 to 25 with a 95% degree of confidence, or at the 5% level of significance.

The most commonly used standard normal distribution reliability factors are:

​ ​​​ = 1.645 for 90% confidence intervals (the significance level is 10%, 5% in each tail). = 1.960 for 95% confidence intervals (the significance level is 5%, 2.5% in each tail). ​ = 2.575 for 99% confidence intervals (the significance level is 1%, 0.5% in each tail).

Distribution nonnormal vs normal

-If the distribution is nonnormal but the population variance is KNOWN, the z-statistic can be used as long as the sample size is large (n ≥ 30). We can do this because the central limit theorem assures us that the distribution of the sample mean is approximately normal when the sample is large. -If the distribution is nonnormal and the population variance is UNKNOWN, the t-statistic can be used as long as the sample size is large (n ≥ 30). It is also acceptable to use the z-statistic, although use of the t-statistic is more conservative.

The t-distribution is a symmetrical distribution

-Is centered about 0. The shape of the t-distribution is DEPENDENT on the number of degrees of freedom, and degrees of freedom are based on the number of sample observations. -The t-distribution is FLATTER and has THICKER tails than the standard normal distribution. As the number of observations INCREASES (i.e., the degrees of freedom increase), the t-distribution becomes more SPIKED and its tails become THINNER. As the number of degrees of freedom increases without bound, the t-distribution converges to the standard normal distribution (z-distribution). -The thickness of the tails relative to those of the z-distribution is important in hypothesis testing because THICKER tails mean more observations away from the center of the distribution (MORE OUTLIERS). Hence, hypothesis testing using the t-distribution makes it MORE DIFFICULT to reject the null relative to hypothesis testing using the z-distribution.

When select the appropriate sample size

-Larger samples may contain observations from a different population (distribution). If we include observations from a different population (one with a different population parameter), we will not necessarily improve, and may reduce, the precision of our population parameter estimates. The other consideration is cost. The costs of using a larger sample must be weighed against the value of the increase in precision from the increase in sample size. Both of these factors suggest that the largest possible sample size is not always the most appropriate choice.

The tendency for the t-distribution

-Look more like the normal distribution as the degrees of freedom INCREASES. Practically speaking, the GREATER the DEGREE of freedom, the GREATER the % of observations near the center of the distribution and the LOWER the percentage of observations in the tails, which are THINNER as degrees of freedom INCREASES. This means that confidence intervals for a random variable that follows a t-distribution must be wider (narrower) when the degrees of freedom are less (more) for a given significance level. page 204

Student's t-distribution has the following properties:

-Symmetrical. -Defined by a single parameter, the degrees of freedom (df), where the degrees of freedom are equal to the number of sample observations minus 1, n − 1, for sample means. -Has more probability in the tails ("fatter tails") than the normal distribution. -As the degrees of freedom (the sample size) gets LARGER the shape of the t-distribution more closely approaches a standard normal distribution.

Example: Confidence intervals for a population mean and for a single observation Annual returns on energy stocks are approximately normally distributed with a mean of 9% and standard deviation of 6%. Construct a 90% confidence interval for the annual returns of a randomly selected energy stock and a 90% confidence interval for the mean of the annual returns for a sample of 12 energy stocks.

A 90% confidence interval for a single observation is 1.645 standard deviations from the sample mean. 9% ± 1.645(6%) = -0.87% to 18.87% A 90% confidence interval for the population mean is 1.645 standard errors from the sample mean. 9% ± 1.645(6%/square rt 12)= 6.15% - 11.85%

Data mining

Analysts repeatedly use the same database to search for patterns or trading rules until one that "works" is discovered. Ex: empirical research has provided evidence that value stocks appear to outperform growth stocks. Some researchers argue that this anomaly is actually the product of data mining. Because the data set of historical stock returns is quite limited, difficult to know for sure whether the difference between value and growth stock returns is a true economic phenomenon, or simply a chance pattern that was stumbled upon after repeatedly looking for any identifiable pattern in the data.

Confidence Interval

By adding or subtracting an appropriate value from the point estimate. In general, confidence intervals take on the following form: point estimate ± (reliability factor × standard error) -----point estimate = value of a sample statistic of the population parameter -----reliability factor = number that depends on the sampling distribution of the point estimate and the probability that the point estimate falls in the confidence interval, (1 − α) -----standard error = standard error of the point estimate

Sample selection bias

Data is systematically excluded from the analysis, because of the lack of availability. This practice renders the observed sample to be nonrandom, and any conclusions drawn from this sample can't be applied to the population because the observed sample and the portion of the population that was not observed are different.

Criteria for selecting the appropriate test statistic

If the sample isn't random, the central limit theorem doesn't apply, our estimates won't have the desirable properties, and we can't form unbiased confidence intervals. Creating a random sample is not as easy as one might believe. There are a number of potential mistakes in sampling methods that can bias the results. These biases are particularly problematic in financial research, where available historical data are plentiful, but the creation of new sample data by experimentation is restricted.

Survivorship bias

Most common form of sample selection bias. Ex: in the study of mutual fund performance. Most mutual fund databases, like Morningstar®'s, only include funds currently in existence—the "survivors." They do not include funds that have ceased to exist due to closure or merger. Not a problem if characteristics of the surviving funds and the missing funds the SAME; then the sample of survivor funds would still be a random sample drawn from the population of mutual funds. As one would expect, however, and as evidence has shown, the funds that are dropped from the sample have lower returns relative to the surviving funds. Thus, the surviving sample is biased toward BETTER funds (i.e., it is not random). The analysis of a mutual fund sample with survivorship bias will yield results that overestimate the average mutual fund return because the database only includes the better-performing funds. The solution to survivorship bias is to use a sample of funds that all started at the same time and not drop funds that have been dropped from the sample.

Consider a practice exam that was administered to 36 Level I candidates. The mean score on this practice exam was 80. Assuming a population standard deviation equal to 15, construct and interpret a 99% confidence interval for the mean score on the practice exam for 36 candidates. Note that in this example the population standard deviation is known, so we don't have to estimate it.

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If the distribution of the population is normal with unknown variance, we can use the t-distribution to construct a confidence interval:

Page 207 Unlike the standard normal distribution, the reliability factors for the t-distribution DEPEND on the sample size, so we can't rely on a commonly used set of reliability factors. Instead, reliability factors for the t-distribution have to be looked up in a table of Student's t-distribution, like the one at the back of this book. Owing to the relatively fatter tails of the t-distribution, confidence intervals constructed using t-reliability factors (tα/2) will be more conservative (wider) than those constructed using z-reliability factors (zα/2).

Confidence intervals can be interpreted from a probabilistic perspective or a practical perspective.

Probabilistic interpretation: After repeatedly taking samples of CFA candidates, administering the practice exam, and constructing confidence intervals for each sample's mean, 99% of the resulting confidence intervals will, in the long run, include the population mean. -Practical interpretation. We are 99% confident that the population mean score is between 73.55 and 86.45 for candidates from this population.

Larger sample

REDUCES the sampling error and the standard deviation of the sample statistic around its true (population) value. Confidence intervals are NARROWER when samples are LARGER and the standard errors of the point estimates of population parameters are LESS.

Data-mining bias

Results where the statistical significance of the pattern is overestimated because the results were found through data mining. When reading research findings that suggest a profitable trading strategy, make sure you heed the following warning signs of data mining: -Evidence that many different variables were tested, most of which are unreported, until significant ones were found. -The lack of any economic theory that is consistent with the empirical results. The best way to avoid data mining is to test a potentially profitable trading rule on a data set different from the one you used to develop the rule (i.e., use out-of-sample data).

When compared with normal distribution

T-distribution is FLATTER with more area under the tails (i.e., it has fatter tails). As the degrees of freedom for the t-distribution INCREASES however, its shape approaches that of the normal distribution. The degrees of freedom for tests based on sample means are n − 1 because, given the mean, only n − 1 observations can be unique.

Time-period bias

The data is gathered is either too short or too long. If the time period is too short, research results may reflect phenomena specific to that time period, or perhaps even data mining. If the time period is too long, the fundamental economic relationships that underlie the results may have changed. Ex: small stocks outperformed large stocks during 90-85, may be a result of time-period bias, this case used too short time period. Not clear whether this will continue in the future or just an isolated occurence Ex: relationship between inflation and unemployment 1940-2000, time-period bias because period is too long. Data should be divided into 2 subsamples that span the period b4 and after the change.

This means that if we are sampling from a nonnormal distribution (which is sometimes the case in finance),

We cannot create a confidence interval if the sample size is less than 30. So, all else equal, make sure you have a sample of at least 30, and the larger, the better.

z-statistic vs t-statistic

We now know that the z-statistic should be used to construct confidence intervals when the population distribution is normal and the variance is known, and the t-statistic should be used when the distribution is normal but the variance is unknown.

Look a-head bias

When a study tests a relationship using sample data that was NOT available on the test date. Ex: example, consider the test of a trading rule that is based on the price-to-book ratio at the end of the fiscal year. Stock prices are available for all companies at the same point in time, while end-of-year book values may not be available until 30 to 60 days after the fiscal year ends. In order to account for this bias, a study that uses price-to-book value ratios to test trading strategies might estimate the book value as reported at fiscal year end and the market value two months later.

The population has a normal distribution with a known variance, a confidence interval for the population mean can be calculated as:

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Let's return to the McCreary, Inc. example. Recall that we took a sample of the past 30 monthly stock returns for McCreary, Inc. and determined that the mean return was 2% and the sample standard deviation was 20%. Since the population variance is unknown, the standard error of the sample was estimated to be:

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