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Reading 11: Hypothesis Testing (Hypothesis testing process (S1: State a…
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
Transaction cost
Taxes
Risk
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
S2 & 4: Use Z-statistic: \(Z=\frac{\bar{x}-\mu _{0}}{\sigma\div \sqrt{n}}\)
\(\bar{x}\): Sample mean
\(\mu _{0}\): Hypothesized sample mean
\(\sigma\): Population Standard Deviation
\(n\): Sample Size
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
Unknown Population Variance
S1: State hypothesis
S3: Identify significance level
S4: Find the rejection points
From df and \(\alpha\) find t-value
\(\frac{\alpha}{2}\) for 2 tail test
\(\alpha\) for 1 tail test
S5: Calculate statistic: Use T-statistic: \(T=\frac{\bar{x}-\mu _{0}}{s\div \sqrt{n}}\)
\(\bar{x}\): Sample mean
\(\mu _{0}\): Hypothesized population mean
\(s\): Sample Standard Deviation
\(n\): Sample Size
S6: Make conlcusion: if Test statistic is outside the range of critical values, reject Null Hypothesis
Can use Z-statistic if n is large (n > 30) but prefer T-statistic.
S2: Select Test statistic
Test whether (Population) Means are different
(for samples from 2 independent, normally distributed populations)
Unequal Variances
S2
: T-statistic and Degree of Freedom : \(df=\frac{\left ( \frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}} \right )^{2}}{\frac{\left ( \frac{s_{1}^{2}}{n_{1}} \right )^{2}}{n_{1}}+\frac{\left ( \frac{s_{2}^{2}}{n_{2}} \right )^{2}}{n_{2}}}\)
S5:
\( t= \frac{\left ( \bar{X_{1}}-\bar{X_{2}} \right )-\left ( \mu _{1}-\mu _{2} \right )}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}} }}\)
\( (\mu _{1}-\mu _{2}) \): this is the hypothesized difference in Means which is usually equal 0
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:
\( t=\frac{(\bar{X_{1}}-\bar{X_{2}})-(\mu _{1}-\mu _{2})}{\sqrt{\frac{s_{p}^{2}}{n_{1}}+\frac{s_{p}^{2}}{n_{2}} }}\)
\( (\mu _{1}-\mu _{2}) \): this is the hypothesized difference in Means which is usually equal 0
With \(s_{p}^{2}=\frac{\left ( n_{1}-1 \right )\times s_{1}^{2}+\left ( n_{2}-1 \right )\times s_{2}^{2}}{n_{1}+n_{2}-2}\)
S6: Make conlcusion: Reject of T-stat > critical value or T-stat < negative critical value
S3: Find significance level
Calculating the Mean Value of Difference
(2 DEPENDENT, equal-size, samples from normal populations)
Unknown Population Variance
S5: \(t=\frac{\bar{d}-\mu _{d0}}{s_{\bar{d}}}\)
\( \mu _{d0}\): is usually = 0
S2: Degree of freedom = (n-1)
S1: State hypothesis with \(\mu_{d}\) and \(\mu_{d0}\)
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
: \( s_{d}^{2}=\frac{\sum_{i=1}^{n}(d_{i}-\bar{d})^{2}}{n-1}\)
Standard Error
: \( s_{\bar{d}}=\frac{s_{d}}{\sqrt{n}}\)
Concerning 1 Variance
(Normally distributed population)
S5: Use Chi-square statistic: \(\chi ^{2}=\frac{(n-1 )\times S^{2}}{\sigma _{0}^{2}}\)
with Sample Variance:
\(S^{2}=\frac{\sum_{i=1}^{n}\left (X_{i}-\overline{X} \right )}{n-1}\)
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
Comparing Variance
(Normally distributed populations)
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)
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
Non-parametric test
Used when assumptions of parametric test cannot be supported or when data is not suitable for parametric test.
Run test
Rank Correlation Test