Chapter 9 Hypothesis Testing Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania

Methods for Drawing Inference We can draw inference on a population parameter in two ways: Estimation (Chapter 8) Hypothesis Testing (Chapter 9)

Hypothesis Testing In essence, hypothesis testing is the process of making decisions about the value of a population parameter.

Establishing the Hypotheses Null Hypothesis: A hypothesis about the parameter in question that often denotes a theoretical value, an historical value, or a production specification. Denoted as H0 Alternate Hypothesis: A hypothesis that differs from the null hypothesis, such that if we reject the null hypothesis, we will accept the alternate hypothesis. Denoted as H1 (in other sources HA).

Hypotheses Restated

Types of Tests The null hypothesis is always a statement of equality. H0: μ = k, where k is a specified value The alternate hypothesis states that the parameter (μ, p) is less than, greater than, or not equal to a specified value.

Types of Tests Left-Tailed Tests: H1: μ < k H1: p < k Right-Tailed Tests: H1: μ > k H1: p > k Two-Tailed Tests: H1: μ ≠ k H1: p ≠ k

Hypothesis Testing Procedure Select appropriate hypotheses. Draw a random sample. Calculate the test statistic. Assess the compatibility of the test statistic with H0. Make a conclusion in the context of the problem.

Hypothesis Test of μ x is Normal, σ is Known

P-Value P-values are sometimes called the probability of chance. Low P-values are a good indication that your test results are not due to chance.

P-Value for Left-Tailed Test

P-Value for Right-Tailed Test

P-Value for Two-Tailed Test

Types of Errors in Statistical Testing Since we are making decisions with incomplete information (sample data), we can make the wrong conclusion!! Type I Error: Rejecting the null hypothesis when the null hypothesis is true. Type II Error: Accepting the null hypothesis when the null hypothesis is false.

Errors in Statistical Testing Unfortunately, we usually will not know when we have made an error!! We can only talk about the probability of making an error. Decreasing the probability of making a type I error will increase the probability of making a type II error (and vice versa). We can only decrease the probability of both types of errors by increasing the sample size (obtain more information), but this may not be feasible in practice.

Type I and Type II Errors

Level of Significance Good practice is to specify in advance the level of type I error we are willing to risk. The probability of type I error is the level of significance for the test, denoted by α (alpha).

Type II Error The probability of making a type II error is denoted by β (Beta). 1 – β is called the power of the test. 1 – β is the probability of rejecting H0 when H0 is false (a correct decision).

The Probabilities Associated with Testing

Concluding a Statistical Test For our purposes, significant is defined as follows: At our predetermined α level of risk, the evidence against H0 is sufficient to discredit H0. Thus we adopt H1.

Statistical Testing Comments In most statistical applications, α = 0.05 or α = 0.01 is used When we “accept” the null hypothesis, we are not proving the null hypothesis to be true. We are only saying that the sample evidence is not strong enough to justify the rejection of the null Some statisticians prefer to say “fail to reject the null” rather than “accept the null.”

Interpretation of Testing Terms

Testing µ When σ is Known State the null hypothesis, alternate hypothesis, and level of significance. If x is normally distributed, any sample size will suffice. If not, n ≥ 30 is required. Calculate:

Testing µ When σ is Known Use the standard normal table and the type of test (one or two-tailed) to determine the P-value. Make a statistical conclusion: If P-value ≤ α, reject H0. If P-value > α, do not reject H0. Make a context-specific conclusion.

Testing µ When σ is Unknown State the null hypothesis, alternate hypothesis, and level of significance. If x is normally distributed (or mound-shaped), any sample size will suffice. If not, n ≥ 30 is required. Calculate:

Testing µ When σ is Unknown Use the Student’s t table and the type of test (one or two-tailed) to determine (or estimate) the P-value. Make a statistical conclusion: If P-value ≤ α, reject H0. If P-value > α, do not reject H0. Make a context-specific conclusion.

Using Table 6 to Estimate P-Values Suppose we calculate t = 2.22 for a one-tailed test from a sample size of 6. Thus, df = n – 1 = 5. We obtain: 0.025 < P-Value < 0.050

Testing µ Using the Critical Value Method The values of that will result in the rejection of the null hypothesis are called the critical region of the distribution When we use a predetermined significance level α, the Critical Value Method and the P-Value Method are logically equivalent.

Critical Regions for H0: µ = k

Critical Regions for H0: µ = k

Critical Regions for H0: µ = k

Testing µ When σ is Known (Critical Region Method) State the null hypothesis, alternate hypothesis, and level of significance. If x is normally distributed, any sample size will suffice. If not, n ≥ 30 is required. Calculate:

Testing µ When σ is Known (Critical Region Method) Show the critical region and critical value(s) on a graph (determined by the alternate hypothesis and α). Conclude in favor of the alternate hypothesis if z is in the critical region. State a conclusion within the context of the problem.

Left-Tailed Tests

Right-Tailed Tests

Two-Tailed Tests

Testing a Proportion p Test assumptions: r is a binomial variable n is the number of independent trials p is the probability of success on each trial np > 5 and n(1-p) > 5

Types of Proportion Tests

The Distribution of the Sample Proportion

Converting the Sample Proportion to z

Testing p State the null hypothesis, alternate hypothesis, and level of significance. Check np > 5 and nq > 5 (recall q = 1 – p). Compute: p = the specified value in H0

Testing p Use the standard normal table and the type of test (one or two-tailed) to determine the P-value. Make a statistical conclusion: If P-value ≤ α, reject H0. If P-value > α, do not reject H0. Make a context-specific conclusion.

Using the Critical Value Method for p As when testing for means, we can use the critical value method when testing for p. Use the critical value graphs exactly as when testing µ.

Critical Thinking: Issues Related to Hypothesis Testing Central question – Is the value of sample test statistics too far away from the value of the population parameter proposed in H0 to occur by chance alone? – P-value tells the probability for that to occur by chance alone.

Critical Thinking: Issues Related to Hypothesis Testing If the P-value is so close to α, then we might attempt to clarify the results by - Increasing the sample size - controlling the experiment to reduce the standard deviation. How reliable is the study and the measurements in the sample? – consider the source of the data and the reliability of the organization doing the study.

Tests Involving Paired Differences Data pairs occur naturally in many settings: Before and after measurements on the same observation after a treatment. Be sure to have a definite and uniform method for creating pairs of data points.

Advantages to Using Paired Data Reduces the danger of extraneous or uncontrollable variables Theoretically reduces measurement variability. Increases the accuracy of statistical conclusions.

Testing for Differences We take the difference between each pair of data points. Denoted by d We then test the average difference against the Student’s t distribution.

Hypotheses for Differences H0: µd = 0

Sample Test Statistic for Differences with

Finding the P-Value Just as in the test for µ when σ is unknown, use Table 6 to estimate the P-Value of the test.

Testing d State the null hypothesis, alternate hypothesis, and level of significance. If you can assume d is normal (mound-shaped), any sample size will do. If not, make sure n ≥ 30. Calculate: df = n-1

Testing d Use the Student’s t table and the type of test (one or two-tailed) to determine (or estimate) the P-value. Make a statistical conclusion: If P-value ≤ α, reject H0. If P-value > α, do not reject H0. Make a context-specific conclusion.

Testing the Differences Between Independent Samples Many practical applications involve testing the difference between population means or population proportions.

Testing µ1 - µ2 when σ1, σ2 are Known

Hypotheses for Testing µ1 - µ2 when σ1, σ2 are Known

Testing µ1 - µ2 when σ1, σ2 are Known

Testing µ1 - µ2 when σ1, σ2 are Known

Testing µ1 - µ2 when σ1, σ2 are Known

Testing µ1 - µ2 when σ1, σ2 are Unknown Just as in the one-sample test for the mean, we will resort to the Student’s t distribution and proceed in a similar fashion. Remark: in practice, the population standard deviation will be unknown in most cases .

Testing µ1 - µ2 when σ1, σ2 are Unknown

Testing µ1 - µ2 when σ1, σ2 are Unknown

Testing µ1 - µ2 when σ1, σ2 are Unknown

Deciding Which Test to Use for H0: µ1 - µ2

Testing for a Difference in Proportions, p1 – p2 Suppose we have two independent binomial experiments. We would like to test if the two population proportions are equal.

Testing for a Difference in Proportions

Testing for a Difference in Proportions

Testing for a Difference in Proportions The test statistic is as follows:

The Test Procedure

The Test Procedure

The Test Procedure

The Test Procedure

Critical Regions For Tests of Differences Recall, our emphasis is on the P-Value method. Most scientific studies use this technique. Also, for a fixed α-level test, the methods are equivalent and lead to identical results.