1.6 Hypothesis Testing Flashcards
Hypothesis testing
used to determine whether a sample statistic is likely from a population with the hypothesized value of the population parameter
aims to provide an insight to this question by examining how a sample statistic describes a population parameter
Hypothesis testing
used to determine whether a sample statistic is likely from a population with the hypothesized value of the population parameter
aims to provide an insight to this question by examining how a sample statistic describes a population parameter
hypothesis
a statement about one or more populations tested using sample statistics
The steps in the hypothesis testing process
- State the hypotheses.
- Identify the appropriate test statistic.
- Specify the level of significance.
- State the decision rule.
- Collect data and calculate the test statistic.
- Make a decision.
The two hypotheses always stated:
Null hypothesis: H0
Alternative hypothesis: Ha
Null hypothesis: H0
This is assumed true until the test proves otherwise.
Alternative hypothesis: Ha
This is only accepted if there is sufficient evidence to reject the null hypothesis
a two-sided hypothesis test
two-tailed
ex:
H0: μ = 10%
Ha: μ ≠ 10%
This is a two-sided hypothesis test because the null hypothesis will be rejected if the sample mean return is significantly different from 10%.
–> It could be a lot greater than or less than 10%, so it is a two-tailed test
a one-sided hypothesis test
one-tailed
ex:
H0: μ ≤ 10%
Ha: μ > 10%
This is a one-sided hypothesis test because the null hypothesis will be rejected only if the sample mean return is significantly greater than 10%
It does not matter if the sample mean is a lot smaller than 10%
The analyst is only interested in whether the population mean is greater than 10% (instead of being different from 10%).
–> Therefore, it is a one-tailed test
the two important rules in forming the hypotheses:
- The null and alternative hypotheses should be mutually exclusive (i.e., do not overlap) and collectively exhaustive (i.e., cover all possibilities)
- The null hypothesis includes the point of equality (i.e., H0 always contains an equal sign).
The choice of null and alternative hypotheses should be based on what?
should be based on the hoped-for condition.
For example:
if an analyst is attempting to show that the mean annual return of a stock index has exceeded 10%, the null hypothesis (H0) should be that the mean return is less than or equal to 10%.
The alternative hypothesis (Ha) should only be accepted if statistical tests provide sufficient evidence that the mean return is not less than or equal to 10%
The test statistic
the quantity calculated from the sample used to evaluate the hypothesis
can be calculated as follows:
z = (X¯ − μ0) / (σ/√nz)
X¯: Sample mean
μ0: Hypothesized mean
σ: Population standard deviation
n: Sample size
The null hypothesis can be rejected or not rejected after the test statistic has been calculated.
The decision is based on what?
based on a comparison that assumes a specific significance level, which establishes how much evidence is required to reject the null hypothesis
the four possible outcomes when we see whether a null hypothesis is to be rejected or not
Decision: Do not reject H0
–> H0 is True: Correct Decision
–> H0 is False: Type II Error (False negative)
Decision: reject H0
–> H0 is True: Type I Error (False positive)
–> H0 is False: Correct decision
A Type I error
occurs if a true null hypothesis is mistakenly rejected
A Type II error
occurs if a false null hypothesis is mistakenly accepted
The probability of a Type I error
the level of significance of the test, which is denoted as α
the confidence level
The complement of the level of significance of the test
1− α
The complement of the probability of a Type I error
The probability of a Type II error
hard to quantify but is symbolized as β
the power of a test
The complement of the probability of a Type II error
his is the probability of correctly rejecting the false null hypothesis
The power equals 1 − β
explain the power of Test matrix
Decision:
Do not reject H0:
–> Ho is True: Confidence level (1 − α)
–> Ho is False:
reject H0:
–> Ho is True: Level of significance α
–> Ho is False: the power of test 1 − β
It is common to set the level of significance to be at which levels
10%, 5%, and 1%.
when must the decision rule be stated?
when comparing the test statistic’s calculated value to a given value based on the significance level of the test
The critical value of the test statistic
the rejection point of the null hypothesis
The critical value of the test statistic is based on what?
based on the level of significance and the probability distribution associated with the test statistic
when is the null hypothesis rejected?
when the test statistic is calculated to be more extreme than the critical value(s)
when is a result known to be statistically significant?
when the test statistic is calculated to be more extreme than the critical value(s)
when the null hypothesis is rejected
For a two-tailed test, which are the two ways to reject the null hypothesis?
- The sample/estimator is significantly smaller than the hypothesized value of the population parameter.
- The sample/estimator is significantly larger than the hypothesized value of the population parameter
In order to capture these two rejection regions, two critical values are needed
how do we find the two critical values needed to reject a two-tailed test?
the z-distribution (i.e., standard normal distribution) is used
–> the values would correspond to ±zα/2
The confidence interval
Other than using critical values, which are based on the stated level of significance, the decision of a hypothesis test can be determined based on the confidence interval
The confidence interval explains what hypothesized values are acceptable so that the null hypothesis will not be rejected
there are two ways to make a decision (whether to reject the null hypothesis) in a two-sided hypothesis test. Explain the following:
Metric: Critical Values
Explain the procedure and the decision
Procedure: Compare the test statistic with the critical values
Decision: Reject the null hypothesis if the test statistic is less than the lower critical value or greater than the upper critical value
there are two ways to make a decision (whether to reject the null hypothesis) in a two-sided hypothesis test. Explain the following:
Metric: confidence interval
Explain the procedure and the decision
Procedure: Compare the hypothesized value of the population parameter with the confidence interval
Decision: Reject the null hypothesis if the hypothesized value falls outside of the confidence interval
A statistical decision
can be made based on the conclusion of the hypothesis test
For example, an investor may perform a test on the null hypothesis (based on market consensus) that the mean return of a security is no greater than 5%.
–> If the test statistic exceeds the critical value, the investor makes the statistical decision to reject the null hypothesis. The mean return is expected to be greater than 5%.
An economic decision
made based on the statistical decision
For example, if the investor rejects the null hypothesis that the mean return is no greater than 5% (in favor of the alternative hypothesis that the mean return is greater than 5%), the investor may consider investing in this security as it is currently undervalued.
Different economic factors such as the risk tolerance and the financial positions should be considered
Statistically Significant but Not Economically Significant decision?
Statistical significance does not necessarily imply economic significance.
For example, if a security is only slightly undervalued (test statistic only slightly exceeds the critical value), transaction costs and other fees could prevent successful implementation.
Which of the following statements is most accurate? A Type I error:
A
occurs when a false null hypothesis is not rejected.
B
is less likely to be committed if the level of significance is lowered.
C
is less likely to be committed if the likelihood of committing a Type II error is reduced.
B
is less likely to be committed if the level of significance is lowered.
