chapter 9 bstat

p-Value Approach to Two-Tailed Hypothesis Testing

Compute the p-value using the following three steps: ● 1.Compute the value of the test statistic z. 2.If z is in the upper tail (z > 0), compute the probability that z is greater than or equal to the value of the test statistic. If z is in the lower tail (z < 0), compute the probability that z is less than or equal to the value of the test statistic. 3.Double the tail area obtained in step 2 to obtain the p-value. The rejection rule: Reject H0 if the p-value < a .

Tests About a Population Mean: s Unknown

Test Statistic: t=(x ̅-μ_0)/(s∕√n) This test statistic has a t distribution with n - 1 degrees of freedom. §Rejection Rule: p -Value Approach Reject H0 if p -value < a § §Rejection Rule: Critical Value Approach § H0: m > m0 Reject H0 if t < -ta H0: m < m0 Reject H0 if t > ta H0: m = m0 Reject H0 if t < - ta/2 or t > ta/2

Developing Null and Alternative Hypotheses examples

qAlternative Hypothesis as a Research Hypothesis Example: A new teaching method is developed that is believed to be better than the current method. üAlternative Hypothesis: The new teaching method is better. üNull Hypothesis: The new method is no better than the old method. qAlternative Hypothesis as a Research Hypothesis Example: A new sales force bonus plan is developed in an attempt to increase sales. üAlternative Hypothesis: The new bonus plan increases sales. üNull Hypothesis: The new bonus plan does not increase sales. qAlternative Hypothesis as a Research Hypothesis Example: A new drug is developed with the goal of lowering blood pressure more than the existing drug. üAlternative Hypothesis: The new drug lowers blood pressure more than the existing drug. üNull Hypothesis: The new drug does not lower blood pressure more than the existing drug. §Null Hypothesis as an Assumption to be Challenged Example: The label on a soft drink bottle states that it contains at least 67.6 fluid ounces. ü Null Hypothesis: The label is correct. m > 67.6 ounces. ü Alternative Hypothesis: The label is incorrect. m < 67.6 ounces.

One-Tailed Test About a Population Mean: s Unknown

qp -Value and Critical Value Approaches q 1.Determine the hypotheses. H0: m < 7 Ha: m > 7 2.Specify the level of significance. α= .05 3.Compute the value of the test statistic. t=(x ̅-μ_0)/(s∕√n)=(7.25 -7)/(1.052/√60) = 1.84 qp -Value Approach 4.Compute the p -value. For t = 1.84, p-value=1-T.DIST(1.84,60-1,TRUE)=0.0353 5.Determine whether to reject H0. Because p-value < a = .05, we reject H0. We reject the null hypothesis. Heathrow should be classified as a superior service airport. qCritical Value Approach 4.Determine the critical value and rejection rule. For a = .05 and d.f. = 60 - 1 = 59, t.05 =T.INV(1-0.05,59)=1.67 Reject H0 if t > 1.67 5.Determine whether to reject H0. Because 1.84 >1.67, we reject H0. Heathrow should be classified as a superior service airport.

Type I Error

§ §Because hypothesis tests are based on sample data, we must allow for the possibility of errors. §A Type I error is rejecting H0 when it is true. §The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance. §Applications of hypothesis testing that only control the Type I error are often called significance tests.

Critical Value Approach to Two-Tailed Hypothesis Testing

§ §The critical values will occur in both the lower and upper tails of the standard normal curve. §Find za/2 (the z-value with an area of a/2 in the upper tail of the distribution). §The rejection rule is: Reject H0 if z < -za/2 or z > za/2.

Type II Error

§A Type II error happens when one fails to reject H0 when it is false. §It is difficult to control the probability of making a Type II error. §Statisticians avoid the risk of making a Type II error by using "do not reject H0" and not "accept H0".

Chapter 9Hypothesis Testing

§Developing Null and Alternative Hypotheses §Type I and Type II Errors §Population Mean: s Known §Population Mean: s Unknown §Population Proportion §Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. §The null hypothesis, denoted by H0 , is a tentative assumption about a population parameter. §The alternative hypothesis, denoted by Ha, is the opposite of what is stated in the null hypothesis. §The hypothesis testing procedure uses data from a sample to test the two competing statements indicated by H0 and Ha.

Null and Alternative Hypotheses

§H0: m < 12 The emergency service is meeting the response goal; no follow-up action is necessary. §Ha: m > 12 The emergency service is not meeting the response goal; appropriate follow-up action is necessary. § where: m = mean response time for the population of medical emergency requests

Developing Null and Alternative Hypotheses

§It is not always obvious how the null and alternative hypotheses should be formulated. §Care must be taken to structure the hypotheses appropriately so that the test conclusion provides the information the researcher wants. §The context of the situation is very important in determining how the hypotheses should be stated. §In some cases it is easier to identify the alternative hypothesis first. In other cases the null is easier. §Correct hypothesis formulation will take practice. qAlternative Hypothesis as a Research Hypothesis § §Many applications of hypothesis testing involve an attempt to gather evidence in support of a research hypothesis. §In such cases, it is often best to begin with the alternative hypothesis and make it the conclusion that the researcher hopes to support. §The conclusion that the research hypothesis is true is made if the sample data provide sufficient evidence to show that the null hypothesis can be rejected. §We might begin with a belief or assumption that a statement about the value of a population parameter is true. §We then using a hypothesis test to challenge the assumption and determine if there is statistical evidence to conclude that the assumption is incorrect. §In these situations, it is helpful to develop the null hypothesis first.

Steps of Hypothesis Testing

§Step 1. Develop the null and alternative hypotheses. §Step 2. Specify the level of significance a. §Step 3. Collect the sample data and compute the value of the test statistic. p-Value Approach §Step 4. Use the value of the test statistic to compute the p-value. §Step 5. Reject H0 if p-value < a. Critical Value Approach §Step 4. Use the level of significance a to determine the critical value and the rejection rule. §Step 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0.

Tests About a Population Proportion

§Test Statistic: z=(p ̅-p_0)/σ_p ̅ § where: σ_p ̅ =√((p_0 (1-p_0 ))/n) assuming np > 5 and n(1 - p) > 5 §Rejection Rule: p -Value Approach Reject H0 if p -value < a §Rejection Rule: Critical Value Approach H0: p < p0 Reject H0 if z > za H0: p > p0 Reject H0 if z < -za H0: p = p0 Reject H0 if z < -za/2 or z > za/2 qp -Value and Critical Value Approaches 1. Determine the hypotheses : H_0:p ≤.2 and〖 H〗_a:p ">.2" 2. Specify the level of significance. a = .05 3. Compute the value of the test statistic. σ_p ̅ =√((p_0 (1-p_0 ))/n)=√((.2 (1-.2))/400) = .02 z=(p ̅-p_0)/σ_p ̅ =((.25)-.20)/.02 = 2.50 4. Compute the p -value. For z = 2.5, p-value = 1 - NORM.S.DIST(2.5,TRUE) = .0062 5. Determine whether to reject H0. Because p-value = .0062≤a = .05, we reject H0. qCritical Value Approach 4.Determine the critical values and rejection rule. z.05 = NORM.S.INV(1-0.05)=1.64 Reject H0 z > 1.64 5.Determine whether to reject H0. Because z = 2.5>1.64 we reject H0.

Summary of Forms for Null and Alternative Hypotheses about a Population Mean

§The equality part of the hypotheses always appears in the null hypothesis. §In general, a hypothesis test about the value of a population mean m must take one of the following three forms (where m0 is the hypothesized value of the population mean). H_0: μ ≥ μ_0 H_0: μ ≤ μ_0 H_a: μ < μ_0 H_a: μ > μ_0 One-tailed (lower-tail) One-tailed (upper-tail) H_0: μ = μ_0 H_a: μ ≠ μ_0 Two-tailed test

A Summary of Forms for Null and Alternative Hypotheses About a Population Proportion

§The equality part of the hypotheses always appears in the null hypothesis. §In general, a hypothesis test about the value of a population proportion p must take one of the following three forms (where p0 is the hypothesized value of the population proportion). One-tailed (lower tail) - H_0:p ≥ p_0 H_a:p < p_0 One-tailed (upper tail) - H_0:p ≤ p_0 〖 H〗_a:p > p_0 Two-tailed - H_0:p = p_0H_a:p ≠ p_0

p-Value Approach to One-Tailed Hypothesis Testing

§The p-value is the probability computed using the test statistic, that measures the support (or lack of support) provided by the sample for the null hypothesis. §If the p-value is less than or equal to the level of significance a, the value of the test statistic is in the rejection region. §Reject H0 if the p-value < a .

Critical Value Approach to One-Tailed Hypothesis Testing

§z has a standard normal probability distribution. §za is the z-value with an area of a in the upper tail of the distribution. §The value that establishes the boundary of the rejection region is called the critical value for the test. §The rejection rule is: Lower tail: Reject H0 if z < -za Upper tail: Reject H0 if z > za