# Level 1 CFA® Exam:

Hypothesis Testing – Introduction

We start with the definition of hypothesis, only to proceed to hypothesis testing and the steps that hypothesis testing involves.

A hypothesis is a statement about the values of parameters of one or more populations.

Hypothesis testing involves the following steps:

- Formulate two hypotheses – the null hypothesis and the alternative hypothesis.
- Identify the appropriate test statistic and its probability distribution.
- Specify the significance level.
- Formulate the decision rule.
- Gather the data and calculate the test statistic.
- Make the statistical decision.
- Finally, make the investment or economic decision based on the statistical decision and relevant data.

Now, let’s look at each of the steps of hypothesis testing one by one.

The first step of hypothesis testing is stating the null and alternative hypothesis.

We always have to state two hypotheses. The null hypothesis, also called the null for short, is designated by the symbol \(H_0\), and the alternative hypothesis is usually designated as \(H_1\) (or sometimes as \(H_A\)). The null and the alternative hypothesis form a pair of hypotheses that complement each other and exhaust all possible values of the parameter or parameters.

The null hypothesis is one that is tested. We assume it is true unless there is convincing evidence that it is, in fact, false. The alternative hypothesis is accepted when the null hypothesis is rejected.

There are two types of hypothesis tests:

- a two-sided hypothesis test (aka. two-tailed hypothesis test), and
- a one-sided hypothesis test (aka. one-tailed hypothesis test).

Depending on what we want to test, we choose either the former or the latter. If we want to test whether a parameter is equal to a given value, we use two-sided hypothesis test. If, however, we want to test whether a parameter is lower or greater than a given value, we use the one-sided test.

- A two-sided hypothesis test: \(H_0:\Theta=\Theta_0 \ H_1:\Theta\neq\Theta_0\)

The null is that a parameter is equal to a certain value. If we reject the null hypothesis, we accept the alternative one, namely that the parameter is not equal to this value. For example, if we want to test whether a diameter of a steel rod is on average equal to 13 millimeters, we will use two-tailed test.

- A right-tailed hypothesis test: \(H_0:\Theta\text{ ≤ }\Theta_0 \ H_1:\Theta\text{ > }\Theta_0\)

The null is that a parameter is lower or equal to a certain value. If we reject the null hypothesis, we accept the alternative one, namely that the parameter is greater the given value. So, if we suspect that on average a stock fund’s return was greater than 7%, we should do a right-tailed test. If we reject the null, which states that the average return was lower or equal to 7%, we will statistically prove that the average return was greater than 7%.

- A left-tailed hypothesis test: \(H_0:\Theta\text{ ≥ }\Theta_0 \ H_1:\Theta\text{ < }\Theta_0\)

The null is that a parameter is greater or equal to a certain value. If we reject the null, we accept the alternative hypothesis which says that the parameter is lower than the given value. For example if we think that on average the marginal tax rate paid by listed companies is lower than 18%, we will do a left-tailed test. If we reject the null, we will be able to accept the alternative hypothesis, namely that average marginal tax rate is lower than 18%.

The second step in hypothesis testing involves identifying the appropriate test statistic and appropriate distribution. We’ll discuss application of different tests in the next lessons.

In the level 1 CFA exam all test statistics regarding population mean are calculated according to the following formula:

\(TS = \frac{S_{stat}-P_{H_{0}}}{E}\)

- \(TS\) - test statistic
- \(S_{stat}\) - sample statistic
- \(P_{H_{0}}\) - value of the population parameter under null hypothesis
- \(H_{0}\) - null hypothesis
- \(E\) - standard error of the sample statistic

The formula that CFA candidates use to compute the standard error looks as follows:

\(\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} \ or \ s_{\overline{x}} = \frac{s}{\sqrt{n}}\)

- \(\overline{x}\) - sample mean
- \(\sigma\) - standard deviation for population
- \(s\) - sample standard deviation
- \(n\) - sample size

Apart from identifying the appropriate test statistic, the second step in hypothesis testing also involves identifying the appropriate probability distribution for the chosen test statistic.

We use:

- t-distribution for t-test,
- standard normal distribution or z-distribution for z-test,
- chi-square distribution for chi-square test, and
- F-distribution for F-test.

After we decide which test statistic and distribution is appropriate in a given situation, we should specify the significance level. Before we define the significance level, let’s first talk about the consequences of our decision with respect to whether we reject the null hypothesis or not.

A decision rule consists in determining conditions under which we can reject the null hypothesis. It pretty much comes down to deciding whether to reject or not to reject the null hypothesis. Now, remember that when you decide to reject or not to reject the null hypothesis the following can happen.

(...)

Now let’s move to the fourth step of hypothesis testing, namely to stating the decision rule.

When testing the null hypothesis, we must find the appropriate critical values, also called rejection points. They form the border between the acceptance and rejection areas. Depending on the test, whether it is a two-sided test or a right-sided test or a left-sided test, we reject the null hypothesis if the calculated test statistic is smaller or greater than the critical values.

We can state that the result is statistically significant at the level of significance \(\alpha\), when we reject the null hypothesis. In other words, we do not reject the null hypothesis when the result is statistically insignificant. Now, let's take a closer look at critical values and the decision-making process for one-sided and two-sided hypothesis tests.

If we are dealing with two-sided hypothesis test, we've got two critical values:

- one negative, under the left tail, and
- one positive, under the left tail.

Let’s assume the level of significance of 5% and normal distribution. Since we've got two-tailed test and the distribution is normal, under each of the tails the probability is 2.5%. In consequence, we've got two rejection points:

(...)

The fifth step in hypothesis testing involves gathering the data.

We often forget that the conclusions we draw depend not only on the appropriateness of the statistical model but also on the quality of the data we use when carrying out the test. The gathered data must be verified. We need to check for possible errors, for example when drawing a sample.

We've already talked about statistical decision making (rejecting or not the null). However, the statistical decision does not determine the actual decision. Before making the actual decision, be it an investment decision, many other nonstatistical factors should be considered.

For example, an investor should take into consideration factors such as risk tolerance, financial position, investment horizon, and so on. What is more, the statistical decision may not take into consideration many facts present in the real economy like taxes, political risk, transaction costs, etc.

Before reaching the final decision, we should analyze all valid factors pertinent to the actual consequences of the actions.

In practice, we often encounter the so-called p-value or the marginal significant level. The p-value is the smallest level of significance at which the null hypothesis can be rejected.

Remember, if the p-value is higher than \(\alpha\), we have no reason to reject the null hypothesis and the test is not statistically significant. When the p-value is lower than \(\alpha\), the test is statistically significant and the null hypothesis is rejected.

Let's assume that p-value equals 0.03. How can we interpret this p-value?

If p-value equals 0.03, it means that at a level of significance of \(\alpha=0.03\) or higher we must reject the null hypothesis.

As you can see the p-value provides more information than the level of significance. It is so, because it’s unique – it’s the smallest level of significance at which the null hypothesis can be rejected.

- A hypothesis is a statement about the values of parameters of one or more populations.
- The alternative hypothesis is accepted when the null hypothesis is rejected.
- There are two types of hypothesis tests: a two-sided hypothesis test and a one-sided hypothesis test.
- A decision rule consists in determining conditions under which we can reject the null hypothesis.
- If we reject the null and the null is true we commit a type I error.
- If we don’t reject the null and the null is false, we commit a type II error.
- The only way to decrease the probability of both type I and type II errors is to increase the sample size.
- The probability of type I error, which is designated as \(\alpha\), is called the level of significance of the test.
- The probability of type II error is designated as \(\beta\).
- The power of a test, on the other hand, is the probability of not committing a type II error.
- We can state that the result is statistically significant at the level of significance \(\alpha\), when we reject the null hypothesis.
- The p-value is the smallest level of significance at which the null hypothesis can be rejected.