Reading 11: Hypothesis Testing

Hypothesis testing process

S1: State a null hypothesis (what is to be rejected, must have equal sign) and alternative hypothesis (what is to be supported)

S2: Select test statistics

S3: Specify the significance level.

S4: State the decision rule

S5: Calculate sample statistics

S6: Make decision about the hypothesis and conclusion based on test result.

1-tailed vs. 2 tailed tests

1-tailed test: results from a 1-tailed alternative hypothesis (e.g., $H^{a}: \mu > \mu _{0}$ )

2-tailed test: results from a 2-sided alternative hypothesis.
(e.g., $H^{a}: \mu \neq \mu _{0}$ )

Test statistics, type I & II errors, and Significance level

Test statistic: is the value that a decision about the hypothesis will be based on.
For a test about the value of distribution mean:
$Test Statistics = \frac{\bar{x} - \bar{x_{hypothesized}}}{Sd}$

Type I and II errors

Type I: rejecting the true Null hypothesis.

Type II: Not rejecting a false null hypothesis

Significance level: probability that Type I Error happens. A significance level must be specified to select critical values for the test.

Decision rule, power of a test, and confidence interval

Decision rule: a critical value at a stated level of significance.

Power of A Test: Probability of accurately rejecting a false null hypothesis.
Power of A Test = 1 - P(Type II error)

Relation with Confidence interval：a hypothesis is rejected when the sample statistic lies outside the confidence interval (100x(1- $\alpha$))% around the hypothesized value for the chosen level of significance

Statistical result vs. Economically meaningful result

the benefit of Statistical significant result sometimes is not enough to outweigh the costs

P-value
The smallest significance level for which the Null hypothesis would be rejected.

Identify appropriate test
USE TABLES

Concerning 1 Mean
(Normally distributed population)

Known Population Variance

Unknown Population Variance

Test whether (Population) Means are different
(for samples from 2 independent, normally distributed populations)

Unequal Variances

Calculating the Mean Value of Difference
(2 DEPENDENT, equal-size, samples from normal populations)

Concerning 1 Variance
(Normally distributed population)

Comparing Variance
(Normally distributed populations)

Non-parametric test

S1: State hypothesis

S3: Identify significance level

S4: Find the rejection points

S5: Calculate statistic: Use T-statistic: $T=\frac{\bar{x}-\mu _{0}}{s\div \sqrt{n}}$

S6: Make conlcusion: if Test statistic is outside the range of critical values, reject Null Hypothesis

S2 & 4: Use Z-statistic: $Z=\frac{\bar{x}-\mu _{0}}{\sigma\div \sqrt{n}}$

Equal Varriance

S1: State hypothesis:
$ H_{0}: \mu_{1}-\mu_{2}=0$
$ H_{1}: \mu_{1}-\mu_{2}\neq 0$

S2: select t-statistic and Degree of Freedom
degree of freedom: $df=n_{1}+n_{2}-2$

S4: Find the rejection points

S5: t-test: $ t=\frac{(\bar{X_{1}}-\bar{X_{2}})-(\mu _{1}-\mu _{2})}{\sqrt{\frac{s_{p}^{2}}{n_{1}}+\frac{s_{p}^{2}}{n_{2}} }}$

S6: Make conlcusion: Reject of T-stat > critical value or T-stat < negative critical value

S2: T-statistic and Degree of Freedom : $df=\frac{\left ( \frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}} \right )^{2}}{\frac{\left ( \frac{s_{1}^{2}}{n_{1}} \right )^{2}}{n_{1}}+\frac{\left ( \frac{s_{2}^{2}}{n_{2}} \right )^{2}}{n_{2}}}$

Can use Z-statistic if n is large (n > 30) but prefer T-statistic.

S2: Select Test statistic

S3: Find significance level

S5: $ t= \frac{\left ( \bar{X_{1}}-\bar{X_{2}} \right )-\left ( \mu _{1}-\mu _{2} \right )}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}} }}$

Unknown Population Variance

Step 0: Find Sample Mean Difference, Sample Variance, and Standard Error

Sample Mean Difference: $\bar{d}=\frac{\sum_{i=1}^{n}(d_{i})}{n}$

Sample Variance: $ s_{d}^{2}=\frac{\sum_{i=1}^{n}(d_{i}-\bar{d})^{2}}{n-1}$

S5: $t=\frac{\bar{d}-\mu _{d0}}{s_{\bar{d}}}$

Standard Error: $ s_{\bar{d}}=\frac{s_{d}}{\sqrt{n}}$

S2: Degree of freedom = (n-1)

S1: State hypothesis with $\mu_{d}$ and $\mu_{d0}$

S5: Use Chi-square statistic: $\chi ^{2}=\frac{(n-1 )\times S^{2}}{\sigma _{0}^{2}}$

Rejection Points

"Not equal to" $H_{a}$: reject null hypothesis if test statistic > $\chi _{\alpha /2}^{2}$ or <$\chi _{1-\alpha /2}^{2}$ with df = n-1

"Greater than" $H_{a}$: reject null hypothesis if test statistic > $\chi _{\alpha}^{2}$ with df= n-1

"Less than" $H_{a}$: reject null hypothesis if test statistic < $\chi _{1-\alpha}^{2}$ with df= n-1

Degree of freedom: $(n_{1}-1)$ for $S_{1}^{2}$ and $(n_{2}-1)$ for $S_{2}^{2}$ (usually equal)

S4: Rejection points

"Not equal to" $H_{a}$: reject null hypothesis if test statistic > $F_{\alpha /2}$ or <$F_{1-\alpha /2}$ with specified number of numerator and denominator df

"Greater than" $H_{a}$: reject null hypothesis if test statistic > $F_{\alpha}$

"Less than" $H_{a}$: reject null hypothesis if test statistic < $F_{1-\alpha}$

S5: F-distribution: $F=\frac{s_{1}^{2}}{s_{2}^{2}}$ (whichever larger sample variance is on numerator)

Used when assumptions of parametric test cannot be supported or when data is not suitable for parametric test.

Transaction cost

Taxes

Risk

$\bar{x}$: Sample mean

$\mu _{0}$: Hypothesized population mean

$s$: Sample Standard Deviation

$n$: Sample Size

$\bar{x}$: Sample mean

$\mu _{0}$: Hypothesized sample mean

$\sigma$: Population Standard Deviation

$n$: Sample Size

$ (\mu _{1}-\mu _{2}) $: this is the hypothesized difference in Means which is usually equal 0

$ \mu _{d0}$: is usually = 0

with Sample Variance:
$S^{2}=\frac{\sum_{i=1}^{n}\left (X_{i}-\overline{X} \right )}{n-1}$

With $s_{p}^{2}=\frac{\left ( n_{1}-1 \right )\times s_{1}^{2}+\left ( n_{2}-1 \right )\times s_{2}^{2}}{n_{1}+n_{2}-2}$

Comparing Correlation

Test of whether population correlation coefficient equal 0

T-stat = $ \frac{r \sqrt{n-2}}{\sqrt{1-r^{2}}} $

r = sample correlation coefficient

(n-2) degree of freedom

Run test

Rank Correlation Test

From df and $\alpha$ find t-value

Rejection point: from $\alpha$ find z-value so that P(Zvalue) = $1-\frac{\alpha}{2} $ for 2-tail test or $1-\alpha$ for 1-tail test

$\frac{\alpha}{2}$ for 2 tail test

$\alpha$ for 1 tail test