Hypothesis Testing

The Summary from class is outlined below.  It outlines our textbook, " Lind. (2005). Statistical techniques in business & economics (11th ed). New York: McGraw-Hill, Chapter 9 and 10."

Hypothesis Testing is a statistical methodology that seeks to prove or disprove if a certain belief about a series of data is true. It is a procedure, based on sample evidence and probability theory, used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected.

A Hypothesis is a statement about the value of a population parameter developed for the purpose of testing.  The testing party begins by setting up a null hypothesis, which is typically the original belief that is potentially incorrect given the new data; this represents the statement being tested. The alternative hypothesis is the new possibility that will potentially disprove the null hypothesis. If there is sufficient evidence after testing to accept the alternative hypothesis, then it will be considered true; if it is disproved, then the null hypothesis will be accepted.

One Sample Hypothesis Testing

Steps to Hypothesis Testing:

Step 1:  State the Null and Alternative Hypothesis to Test: 

Null Hypothesis, Ho, is a statement about the value of a population parameter.

Alternative Hypothesis, H1, is a statement that is accepted if the sample data provide evidence that the null hypothesis is false.

Step 2:  Select the Level of Significance

The probability of rejecting the null hypothesis when it is actually true.

You could have two types of errors.

• Type I Error: Rejecting the null hypothesis when it is actually true.
• Type II Error: Accepting the null hypothesis when it is actually false.

Step 3:  Identify the Test Statistic

Test statistic: A value, determined from sample information, used to determine whether or not to reject the null hypothesis.

Critical value: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

Step 4:  Formulate the Decision Rule

Decide on type of test.

1. One-Tailed Tests of Significance:  A test is one-tailed when the alternate hypothesis, H1 , states a direction (i.e, > or < a certain value).
2. Two-Tailed Tests of Significance: A test is two-tailed when no direction is specified in the alternate hypothesis H1 (i.e., = or not = to).

Testing for the Population Mean: Large Sample, Population Standard Deviation Known

Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown.

In both cases as long as the sample size n  30, z can be approximated

Testing for a Population Mean: Small Sample, Population Standard Deviation Unknown

Step 5:  Take a Sample,

Step 6:  Analyze the Sample,

A p-Value in Hypothesis Test is the probability, assuming that the null hypothesis is true, of finding a value of the test statistic at least as extreme as the computed value for the test.

1. If the p-Value is smaller than the significance level, H0 is rejected.
2. If the p-Value is larger than the significance level, H0 is not rejected.

Computation of the p-Value

1. One-Tailed Test: p-Value = P{z absolute value of the computed test statistic value}
2. Two-Tailed Test: p-Value = 2P{z absolute value of the computed test statistic value}

Step 7:  Make a Decision (accept the null or reject the null)

Tests Concerning Proportion

A Proportion is the fraction or percentage that indicates the part of the population or sample having a particular trait of interest.

p = (Number of Successes in the Sample/ Number Sampled)

Test Statistic for Testing a Single Population Proportion

The sample proportion is p and B is the population proportion.

Two Sample Tests of Hypothesis

The purpose is to Test or Compare Two Sets of Data.  We wish to know whether the distribution of the differences in sample means has a mean of 0.

    2.1 One Variable, One or Two Populations/Treatments
        2.1.1 Test of Averages/Means
        2.1.2 Variances are Unknown but assumed Equal, Two Treatment Means, Completely Randomized Design, Small Sample Size (<30).
            2.1.2.1 t Test - Two Samples
            2.1.2.2 t Test - One Sample
        2.1.3 Variances are Known, Sampling From Independent Normal Distributions. If both samples contain at least 30 observations we use the z distribution as the test statistic.
            2.1.3.1 Z Test - Two Samples
            2.1.3.2 Z Test - One Sample
        2.1.4 Asked For Other
            2.1.4.1 Use Confidence Interval
            2.1.4.2 Use P-Value

    2.2 Test of Variability
        2.2.1 One Normal Population, Variance Constant Known
            2.2.1.1 c^2 Test - One Sample Test
        2.2.2 Two normal Populations, Independent Random Samples, Estimated Means and Variances Unknown for Both.
            2.2.2.1 F Test - More than one Sample

    2.3 Significant Difference Between Means
        2.3.1 md Test
        2.3.2 Ask For Other
            2.3.2.1 Use Confidence Interval

    2.4 One Variable, More Than Two Populations/Treatments
        2.4.1 Single Factor Analysis of Variance (ANOVA) Analysis of Fixed Effects Model to Test more than 2 means where means are equal.
        2.4.2 Equality of Variance
            2.4.2.1 Bartiett's Test, to test Variances, Requires Normality.
        2.4.3 Confidence Interval
            2.4.3.1 to Test - On differences in two means, Normal Distribution with Variances Unknown.
        2.4.4 Sample Size
            2.4.4.1 to Test - Difference in Two Means Unknown Variances
            2.4.4.2 Using OC Curves
            2.4.4.3 Given SDV
        2.4.5 Kruskal-Wallis Test - Normality assumption is unjustified, Good for test of differences in means - A nonparametric Alternative
        2.4.6 Randomized Complete Block Design ANOVA

    2.5 Comparing Pairs of Treatment Means after the ANOVA F test where Ho is rejected. Test each Ho: mI = mj for all I=j.
        2.5.1 Least Significant Difference (LSD) Method very effective for detecting true differences in means if it is applied only after F test in ANOVA where level of significant is at 5%.
        2.5.2 Duncan's Multiple Range Test
        2.5.3 Newman-Keuls Test - More conservative than Duncan's Test in that the Type I error is smaller and the method is less powerful.
        2.5.4 Tukey's Test - Results are somewhat more difficult to interpret than LSD or Newman-Keals Test. Tukey has a type I error rate of a which is conservative (smaller rate) than either Newman-Keuls or Duncan Test. (Therefor is less Powerful.)

    2.6 Estimation of Model Parameters
        2.6.1 Single Factor Model - Yij = m + tI + Îij
        2.6.2 Overall Mean - E(m) = Average Y..
        2.6.3 Treatment Effects -- tI = Average Yi. - Average Y.. (I=1,2,…..a)
        2.6.4 Confidence Interval of Mean for some 1-a CI º [ Average Yi. ± ((ta/2,N-a)*(MSE/n)^1/2)] = LL £ mI £ UL
        2.6.5 Confidence Interval of Mean Difference for some 1-a CI º [(Average of Yi. - Average of Yj.) ± ((ta/2,N-a)*(2*MSE/n)^1/2)] = LL £ (mI- mj) £ UL