Overview 9.1 Introduction to Hypothesis Testing 9.2 OMIT 9.3 Z Test for the Population Mean: Critical-Value Method 9.4 t Test for the Population Mean 9.5 Z Test for the Population Proportion p 9.6 OMIT

9.1 Introduction to Hypothesis Testing Objectives: By the end of this section, I will be able to… Construct the null hypothesis and the alternative hypothesis from the statement of the problem. Explain the role of chance variation in establishing reasonable doubt. State the two types of errors made in hypothesis tests: the Type I error, made with probability a, and the Type II error, made with probability b.

Researchers Investigate Many Types of Questions Accountant - evidence exists for corporate tax fraud. Department of Homeland Security - test whether a new surveillance method will uncover terrorist activity. Sociologist - examine whether the mayor’s economic policy is increasing poverty in the city. Questions such as these can be tackled using statistical hypothesis testing.

Constructing the Hypotheses Basic idea of hypothesis testing: Make a decision about the value of a population parameter (such as μ or p). True value of parameter unknown. May be different hypotheses about the true value of this parameter.

Hypothesis testing is a procedure where claims about the value of a population parameter may be investigated using the sample evidence. Craft two competing statements (hypotheses) about the value of the population parameter (either μ, p, or σ ). The two hypotheses cover all possible values of the parameter, and they cannot both be correct.

The Hypotheses Status quo hypothesis - what has been assumed about the value of the parameter Called the null hypothesis, denoted as H0. Alternative hypothesis, or research hypothesis, denoted as Ha Represents an alternative claim about the value of the parameter.

Hypothesis Testing Is Like Criminal Trial. In U.S. defendant is innocent until proven guilty, Jury must evaluate the truth of two competing hypotheses: H0: defendant is not guilty versus Ha: defendant is guilty

Table 9.1 The three possible forms for the hypotheses for a test for μ.

Convert a Word Problem Recall that each hypothesis is a claim that the statement is true. Find evidence that is consistent with one of the two claims H0 or Ha. The first task in hypothesis testing is to form hypotheses.

Key English Words, with Mathematical Symbols and Synonyms Table 9.2

Strategy for Constructing the Hypotheses About μ Step 1 : Search problem for keywords using Table 9.2. Step 2: Determine form of hypotheses using Table 9.1. Step 3: Find value of m0 (the number that answers the question: “greater than what?” or “less than what?”) and write hypotheses.

Practicing the Techniques Provide the null and alternative hypotheses. 1) Test whether μ is equal to 70. 2) Test whether μ is at most 50. 3) Test whether μ is greater than 150.

Practicing the Techniques Solution to: 1) Test whether μ is equal to 70. The keyword is equal. This means we will write a hypothesis that contains the = symbol. The form of the hypotheses is a two-tailed test: H0: μ = μ0 versus Ha: μ ≠ μ0 Find the value for μ0 and write your Hypotheses. μ0 = 70. H0: μ = 70 versus Ha: μ ≠ 70

Practicing the Techniques Solution to: 2) Test whether μ is at most 50. The keyword is at most. This means we will write a hypothesis that contains the ≤ symbol. The form of the hypotheses is a Right-tailed test: H0: μ ≤ μ0 versus Ha: μ > μ0 Find the value for μ0 and write your Hypotheses. μ0 = 50. H0: μ ≤ 50 versus Ha: μ > 50

Practicing the Techniques Solution to: 3) Test whether μ is greater than150. The keyword is greater than. This means we will write a hypothesis that contains the > symbol. The form of the hypotheses is a Right-tailed test: H0: μ ≤ μ0 versus Ha: μ > μ0 Find the value for μ0 and write your Hypotheses. μ0 = 150. H0: μ ≤ 150 versus Ha: μ > 150

Significant Difference or Chance Variation Statistical Significance: Result that is unlikely to have occurred due to chance.

Type I and Type II Errors Let’s return to the example of a criminal trial.

Type I and Type II Errors Two Ways of Making the Correct Decision To not reject H0 when H0 is true. Example: To find the defendant not guilty when in reality he did not commit the crime. To reject H0 when H0 is false. Example: To find the defendant guilty when in reality he did commit the crime.

Two Types of Errors Type I error: To reject H0 when H0 is true. Example: To find the defendant guilty when in reality he did not commit the crime. Type II error: To not reject H0 when H0 is false. Example: To find the defendant not guilty when in reality he did commit the

Summary Statistical hypothesis testing is a way of formalizing the decision-making process so that a decision can be rendered about the unknown value of the parameter. The status quo hypothesis that represents what has been tentatively assumed about the value of the parameter is called the null hypothesis and is denoted as H0.

Summary The alternative hypothesis, or research hypothesis, denoted as Ha, represents an alternative conjecture about the value of the parameter. In a hypothesis test, we compare the sample mean x with the value m0 of the population mean used in the H0 hypothesis. If the difference is large, then H0 is rejected. If the difference is not large, then H0 is not rejected. —

Summary When performing a hypothesis test, there are two ways of making a correct decision: to not reject H0 when H0 is true and to reject H0 when H0 is false. There are two types of error: a Type I error is to reject H0 when H0 is true, and a Type II error is to not reject H0 when H0 is false. The probability of a Type I error is denoted as a (alpha). The probability of a Type II error is denoted as b (beta).

Level of Significance, a a is boundary between results that are statistically significant (where we reject H0) and results that are not statistically significant (where we do not reject H0). a is called the level of significance of the hypothesis test.

9.3 Z Test for the Population Mean: Critical-Value Method Objectives: By the end of this section, I will be able to… 1) Explain the meaning of the critical region and the critical value Zcrit. 2) Perform and interpret the Z test for the population mean using the critical-value method. 3) Use confidence intervals for μ to perform two- tailed hypothesis tests about μ.

Critical Regions and the Critical Value Zcrit Critical region: values of the test statistic Zdata for which we reject the null hypothesis. Noncritical region: values of the test statistic Zdata for which we do not reject the null hypothesis. Zcrit : value of Z that separates critical region from the noncritical region

Example 9.13 Finding the critical value Zcrit Do you have a debit card? How often do you use it? ATM network operator Star System of San Diego reported in 2006 that active users of debit cards used them an average of 11 times per month. Suppose that we are interested in testing whether debit card usage has increased since 2006. For level of significance a = 0.01, find the critical value Zcrit for this hypothesis test.

Example 9.13 Finding the critical value Zcrit Continued Solution Key word “increased” leads to hypotheses H0: m ≤ 11 versus Ha: m > 11 Because level of significance is 0.01, critical region is values of Zdata in the uppermost 1% of the Z distribution. Zcrit separates uppermost 1% of the values from the remaining 99%.

Example 9.13 continued Figure 9.23

Example 9.13 continued Look up the area 0.99 on the inside of Z table and take closest area which is 0.9901. Working backward, find that Zcrit = 2.33.

Example 9.13 continued Values of Zdata greater than Zcrit = 2.33: reject the null hypothesis. Values of Zdata less than Zcrit = 2.33: do not reject the null hypothesis.

Critical Region for a left-tailed test H0: μ ≥ μ0 versus Ha: μ < μ0 Critical region: values of Zdata that are less than Zcrit because alternative hypothesis states that the population mean m is less than m0.

Critical Region for a two-tailed test H0: μ = μ0 versus Ha: μ ≠ μ0 Critical region: values of Zdata that are either extremely large (in the right tail) or extremely large negatively (in the left tail).

Example 9.14 - Finding the Critical Region for a Left-tailed Test According to a report in the New England Journal of Medicine, a regimen of estrogen plus progestin therapy can help reduce cholesterol levels in postmenopausal women with high cholesterol. The study examined whether the population mean cholesterol level could be lowered by this regimen to below 305 milligrams per deciliter (mg/dl). Find the critical region for this hypothesis test, using level of significance a = 0.05.

Example 9.14 - Finding the Critical Region for a Left-tailed Test Continued Solution Hypotheses: H0: μ ≥ 305 versus Ha: μ < 305 μ = population mean cholesterol level in milligrams per deciliter Since this is a left-tailed test, and a = 0.05, we obtain from Table 9.6 page 484 that the critical value Zcrit is -1.645. Since alternative hypothesis states that population mean is less than m0, reject the null hypothesis Zdata < -1.645.

Z Test for the Population Mean m: Critical-Value Method Random sample of size n taken from a population and either of the following conditions is satisfied: Case 1: the population is normal Case 2: the sample size is large (n ≥ 30).

Z Test for the Population Mean m: Critical-Value Method Continued Step 1: State hypotheses and rejection rule. Use one of the forms from Table 9.1. Define μ.

Z Test for the Population Mean m: Critical-Value Method Continued Step 2: Find Zcrit State rejection rule using Tables 9.6 (page 484) and 9.7 (page 485).

Z Test for the Population Mean m: Critical-Value Method Continued Step 3: Use technology or calculate the value of Zdata:

Z Test for the Population Mean m: Critical-Value Method Continued Step 4: State conclusion and interpretation If Zdata falls in critical region, then reject H0. Otherwise, do not reject H0. Interpret your conclusion so that a nonspecialist can understand.

Example 9.16 - Critical-value method for the Z test for m Let’s use the critical-value method to test, using level of significance a = 0.01, whether people use debit cards on average more than 11 times per month (Example 9.13). Suppose a random sample of 36 people used debit cards last month an average of x = 11.5 times. Assume that the population standard deviation equals 3. _

Example 9.16 continued Step One: State hypotheses H0: m ≤ 11 versus Ha: m > 11 Step Two: Find Zcrit and state rejection rule.

Example 9.16 continued right-tailed test a = 0.01, Table 9.6 indicates Zcrit = 2.33 Table 9.7, rejection rule: Reject H0 if Zdata > 2.33 FIGURE 9.27

Example 9.16 continued Step Three: Find Zdata

Example 9.16 continued Step 4: State the conclusion and the interpretation. Zdata = 1.0, which is not greater than 2.33, do not reject H0 Final conclusion: There is insufficient evidence that the population mean monthly debit card use is greater than 11 times per month.

Using Confidence Intervals for μ to Perform Two-Tailed Hypothesis Tests About μ FIGURE 9.31 Reject H0 for values of μ0 that lie outside confidence interval.

9.4 t Test for the Population Mean Objectives: By the end of this section, I will be able to… Perform the t test for the mean using the critical-value method. Calculate the standard error, and use it to explain the meaning of the test statistic.

t Test for using Critical-Value method Random sample of size n taken from a population and either of the following conditions is satisfied: Case 1: the population is normal Case 2: the sample size is large (n ≥ 30).

t Test for μ using the Critical-Value Method Because there is different t curve for every different sample size, need to find the following: The form of the hypothesis test (one-tailed or two-tailed) Degrees of freedom (df = n - 1) The level of significance a

t Test for μ using the Critical-Value Method Step 1: State the hypotheses. Use one of the forms from Table 9.11. Define μ.

t Test for μ using the Critical-Value Method Step 2: Find tcrit. State the rejection rule. Use Table 9.11.

t Test for μ using the Critical-Value Method Step 3: Find tdata. Either use technology to find the value of the test statistic tdata or calculate the value of tdata as follows: tdata =

t Test for μ using the Critical-Value Method Step 4: State the conclusion and the interpretation. If tdata falls within the critical region, then reject H0. Otherwise, do not reject H0. Interpret your conclusion so that a non- specialist can understand.

Table 9.11 Critical regions and rejection rules for various forms of the t test for µ

Standard Error of the Sample Mean x Thus, the test statistic tdata is written tdata =

Interpretation of the Meaning of the Test Statistic tdata tdata is positive: tdata = number of standard errors that the sample mean x lies above the hypothesized mean m0. tdata is negative: tdata = number of standard errors that the sample mean x lies below the hypothesized mean m0. — —

Case Study: The Golden Ratio Test if there is evidence for the use of the golden ratio in artistic traditions of Shoshone, a Native American tribe. If, A > B > 0, A/B is called the golden ratio if Golden ratio approximately 1.6180339887 Figure 9.46

Case Study Continued Figure 9.49 Detail of nineteenth century Shoshone beaded dress, including five beaded rectangles.

Table 9.12 Ratio of length to width of Shoshone beaded rectangles Case Study Continued Perform a hypothesis test to determine whether the population mean ratio of Shoshone beaded rectangles equals the golden ratio of 1.6180339887. Level of significance = 0.05 Table 9.12 Ratio of length to width of Shoshone beaded rectangles

Case Study Continued Population standard deviation for rectangles unknown Use t test rather than a Z test Sample size n =18 is not large Figure 9.50 indicates evidence for normality

Hypothesis Test Step One State hypotheses and rejection rule. versus Ha: μ ≠ 1.6180339887 Rejection rule: reject H0 if p-value < 0.05.

Hypothesis Test Continued Step Two Find tdata. tdata = =

9.5 Z Test for the Population Proportion p Objectives: By the end of this section, I will be able to… Explain the essential idea about hypothesis testing for the proportion. Perform and interpret the Z test for the proportion using the critical-value method. Use confidence intervals for p to perform two-tailed hypothesis tests about p.

The Essential Idea About Hypothesis Testing for the Proportion If sample proportion p is unusual or extreme in the sampling distribution of p assuming that H0 is correct, reject H0. Otherwise, there is insufficient evidence against H0, and do not reject H0.

Z Test for the Population Proportion p: Critical-Value Method Critical-value method for the Z test for the proportion equivalent to the p-value method for the Z test for the mean. Normality conditions for Z test critical-value method same as the conditions for Z test p-value method: np0 ≥ 5 and n(1 - p0) ≥ 5. Compare one Z-value (Zdata) with another Z- value (Zcrit).

Z Test for the Population Proportion p: Critical-Value Method Rejection rules for Z test for a proportion Table 9.15

Z Test for the Population Proportion p: Critical-Value Method Step 1: State the hypotheses. Use one of the forms from Table 9.15. Clearly describe the meaning of p.

Z Test for the Population Proportion p: Critical-Value Method Step 2: Find Zcrit. State the rejection rule. Use Tables 9.14 and 9.15.

Z Test for the Population Proportion p: Critical-Value Method Step 3: Find Zdata. Either use technology to find the value of Zdata or calculate Zdata as follows:

Z Test for the Population Proportion p: Critical-Value Method Step 4: State the conclusion and the interpretation. If Zdata falls in the critical region, then reject H0. Otherwise, do not reject H0. Interpret the conclusion so that a non- specialist can understand.

Using Confidence Intervals for p to Perform Two-Tailed Hypothesis Tests About p Just as for μ, use a 100(1 - a)% confidence interval for p to perform two-tailed hypothesis tests for p.