Math 227 Elementary Statistics Sullivan, 4th ed.

Copyright of the definitions and examples is reserved to Pearson Education, Inc.. In order to use this PowerPoint presentation, the required textbook for the class is the Fundamentals of Statistics, Informed Decisions Using Data, Michael Sullivan, III, fourth edition.

Hypothesis Tests Regarding a Parameter CHAPTER 10 Hypothesis Tests Regarding a Parameter

Ch 10.1 The Language of Hypothesis Testing Objective A : Set up a Hypothesis Testing Objective B : Type I or Type II Error Objective C : State conclusions for Hypothesis Tests Ch 10.2 Hypothesis Test for a Population Mean Objective A : Classical Approach

Ch 10.1 The Language of Hypothesis Testing Objective A : Set up a Hypothesis Testing Hypothesis testing is a procedure, based on a sample evidence and probability, used to test statements regarding a characteristic of one or more populations. – The null hypothesis is a statement to be tested. – The alternate hypothesis is a statement that we are trying to find evidence to support.

Example 1 : Set up and . In the past, student average income was $6000 per year. (a) An administrator believes the average income has increased. (No change) (A right-tailed test) (b) The percentage of passing a Math course was 50%. A Math professor believes there is a decrease in the passing rate. (A left-tailed test)

Ch 10.1 The Language of Hypothesis Testing Objective A : Set up a Hypothesis Testing Objective B : Type I or Type II Error Objective C : State conclusions for Hypothesis Tests Ch 10.2 Hypothesis Test for a Population Mean Objective A : Classical Approach

Ch 10.1 The Language of Hypothesis Testing Objective B : Type I or Type II Error Type I Error → Rejecting when is true. Type II Error → Not rejecting when is true. We use for the probability of making Type I error. We use for the probability of making Type II error. For this statistics class, we only control the Type I error. is also called the level of significance.

Ch 10.1 The Language of Hypothesis Testing Objective A : Set up a Hypothesis Testing Objective B : Type I or Type II Error Objective C : State conclusions for Hypothesis Tests Ch 10.2 Hypothesis Test for a Population Mean Objective A : Classical Approach

Section 10.1 The Language of Hypothesis Testing Objective C : State Conclusions for Hypothesis Tests If is rejected, there is sufficient evidence to support the statement in . If is NOT rejected, there is NOT sufficient evidence to support the statement in .

Example 1 : In 2007, the mean SAT score on the reasoning test for all students was 710. A teacher believes that, due to the heavy use of multiple choice test questions, the mean SAT reasoning test has decreased. (a) Determine and . (b) Explain what it would mean to make a Type I error. A Type I error is made when the sample evidence leads the teacher to believe is not true when in fact is true.

(c) Explain what it would mean to make a Type II error. A Type II error is made when the sample evidence leads the teacher to believe is true when in fact is less than 710. (d) State the conclusion if the null hypothesis is rejected. There is enough evidence to support the teacher’s claim that the mean SAT reasoning test score has decreased.

Example 2 : The mean score on the SAT Math Reasoning exam is 516 Example 2 : The mean score on the SAT Math Reasoning exam is 516. A test preparation company states that the mean score of students who take its course is higher than 516. (a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the marketing campaign. (b) If sample data indicate that the null hypothesis should not be rejected, state the conclusion of the company. There is not sufficient evidence to support the mean score of the reasoning exam is higher than 516.

(c) Suppose, in fact, that the mean score of students taking the preparatory course is 522. Has a Type I or Type II error been made? Type II error. If we tested this hypothesis at the = 0.01 level, what is the probability of committing a Type I error? Since is the probability of committing Type I error, it is 0.01. (d) If we wanted to decrease the probability of making a Type II error, would we need to increase or decrease the level of significance? If we increase , we decrease and vice versa. Therefore, we need to increase the level of significance in order to decrease the probability of Type II error.

Example 3 : According to the Centers for Disease Control, 15 Example 3 : According to the Centers for Disease Control, 15.2% of adults experience migraine headaches. Stress is a major contributor to the frequency and intensity of headaches. A massage therapist feels that she has a technique that can reduce the frequency and intensity of migraine headaches. (a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the massage therapist's techniques. (b) A sample of 500 American adults who participated in the massage therapist's program results in data that indicate that the null hypothesis should be rejected. Provide a statement that supports the massage therapist's program. There is sufficient evidence to support the massage therapist’s program that can reduce the percentage of migraine headache from 15.2%.

(c) Suppose, in fact, that the percentage of patients in the program who experience migraine headaches is 15.2%. Was a Type I or Type II error committed? Type I error.

Ch 10.1 The Language of Hypothesis Testing Objective A : Set up a Hypothesis Testing Objective B : Type I or Type II Error Objective C : State conclusions for Hypothesis Tests Ch 10.2 Hypothesis Test for a Population Mean Objective A : Classical Approach

Ch 10.2 Hypothesis Tests for a Population Proportion Objective A : Classical Approach – Test for a Proportion A hypothesis test involving a population proportion can be considered as a binomial experiment. As we learned from Ch 8.2, the best point estimate of , the population proportion, is a sample proportion, . There are two methods for testing hypothesis. Method 1 : The Classical Approach

Method 2 : The – Value Approach We are going to focus on the Classical Approach for today’s lecture.

Testing Hypotheses Regarding a Population Proportion, Testing Hypotheses Regarding a Population Proportion, . Use the following steps to perform a proportion – Test provided that

Example 1: Use the Classical Approach to test the following hypotheses. versus Step 1 : (left-tailed test) Step 2 : Step 3 : Sample : * (test statistic)

Use Table V to determine critical value for : Recall : (left-tailed test) From Table V * (Closer to 0.1) Step 4 : Since – statistic, , is not within the critical region, , fail to reject . Step 5 : There is not enough evidence to support .

Example 2: The percentage of physicians who are women is 27. 9% Example 2: The percentage of physicians who are women is 27.9%. In a survey of physicians employed by a large university health system, 45 of 120 randomly selected physicians were women. Use the Classical Approach to determine whether there is sufficient evidence at the 0.05 level of significance to conclude that the proportion of women physicians at the university health system exceeds 27.9%? Step 1 : (right-tailed test) Step 2 :

Step 3 : Sample : * (test statistic) Use Table V to determine critical value for : (right-tailed test)

From Table V * Step 4 : Since – statistic, ,falls within the critical region, , reject . Step 5 : There is sufficient evidence to support the claim that the proportion of female physicians exceeds 27.9%.

Ch 10.3 Part I Classical Approach

Ch 10.3 Hypothesis Test about a Population Mean with unknown Objective A : Classical Approach

Ch 10.3 Hypothesis Test about a Population Mean with unknown

The – test procedure requires either that the sample was drawn from a normally distributed population or the sample size was greater than 30. Minor departures from normality will not adversely affect the results of the test. However, if the data include outliers, the – test procedure should not be used. A normality plot is used to test whether the sample was drawn from a normally distributed population. A boxplot is used to detect outliers.

Ch 10.3 Hypothesis Test about a Population Mean with unknown Objective A : Classical Approach

Ch 10.3 Hypothesis Test about a Population Mean with unknown Objective A : Classical Approach Example 1: Determine the critical value(s) for (a) a right-tailed test of a population mean at the level of significance based on a sample size of . From Table VI (b) a left-tailed test of a population mean at the level of significance based on a sample size of . From Table VI

(c) a two-tailed test of a population mean at the level of significance based on a sample size of . From Table VI

Example 2: Determine the – statistic and the tail(s) of testing. (a) versus A random sample of size was obtained from a population that was normally distributed produced a mean of 16.5 with a standard deviation of 0.4 . (b) versus A random sample of size was obtained from a population that was normally distributed produced a mean of 73.5 with a standard deviation of 17.1 .

Example 3: Use the classical approach to determine whether to reject or fail to reject . (a) versus for a right-tailed test Reject . * (b) versus for a left-tailed test Fail to reject . *

(c) versus for a two-tailed test reject . *

Example 4: To test versus , a random sample of size is obtained from a population that is known to be normally distributed. (a) If , , compute the test statistic and round the answer to three decimal places. * (b) If the researcher decides to test this hypothesis at the level of significance, determine the critical value(s). From Table VI Recall : (left-tailed test)

(c) Draw a – distribution that depicts the critical region and the – statistic. * (d) What conclusion can be drawn? Fail to reject . There is not enough evidence to support .

Example 5: An Fast Oil Change Company advertised a 15-minute wait for an oil change. A sample of 25 oil changes showed a mean time of 16.8 minutes and a standard deviation of 4.3 minutes. At the 2% level of significance, is there evidence that the mean time for an oil change is different from 15 minutes? Assume the population is approximately normal. Use the Classical Approach. Population : Sample: (two-tailed test) *

Recall : -- > two-tailed test -- > From Table VI * Fail to reject . There is not enough evidence to support the mean waiting time for an oil change is different from 15 minutes.

Ch 10.2 Part II and Ch 10.3 Part II

Ch 10.2 Hypothesis Test for a Population Mean Objective B : – Value Approach Ch 10.3 Hypothesis Test about a Population Mean with unknown Objective B : – Value Approach

Ch 10.2 Hypothesis Test for a Population Mean Objective B : – Value Approach – Test for a Proportion A hypothesis test involving a population proportion can be considered as a binomial experiment. As we learned from Ch 8.2, the best point estimate of , the population proportion, is a sample proportion, , provided the sample is obtained by 1) simple random sampling ; 2) to guarantee that a normal distribution can be used to test hypotheses for ; 3) the sampled values are independent of each other ( ).

Testing Hypotheses Regarding a Population Proportion, Testing Hypotheses Regarding a Population Proportion, . Use the following steps to perform a – Test for a Proportion.

Example 1: The percentage of physicians who are women is 27. 9% Example 1: The percentage of physicians who are women is 27.9%. In a survey of physicians employed by a large university health system, 45 of 120 randomly selected physicians were women. Use the - Value Approach to determine whether there is sufficient evidence at the 0.05 level of significance to conclude that the proportion of women physicians at the university health system exceeds 27.9%? Step 1 : (right-tailed test) Step 2 :

Step 3 : Sample : * (test statistic)

– value approach -- > Let alone. -- > Convert to a tailed area. From Table V – value = 1 – 0.9904 = 0.0096 Step 4 : Is – value < ? Is 0.0096 < 0.05 ? Yes, reject . Step 5 : There is enough evidence to support the claim that the proportion of female physicians exceeds 27.9%.

Example 2: In 1997, 4% of mothers smoked more than 21 cigarettes during their pregnancy. An obstetrician believes that the percentage of mothers who smoke 21 cigarettes or more is less than 4% today. She randomly selects 120 pregnant mothers and finds that 3 of them smoked 21 or more cigarettes during pregnancy. Does the sample data support the obstetrician’s belief? Use the - Value Approach with level of significant. (left-tailed test)

* (test statistic) – value approach -- > Let alone. -- > Convert to a tailed area. From Table V – value = 0.2005

Is – value < ? Is 0.2005 < 0.01 ? No, do not reject . There is not enough evidence to support the claim that the percentages of mothers who smoke 21 cigarettes or more is less than 4%.

Example 3: In 2000, 58% of females aged 15 years of age and older lived alone, according to the U.S. Census Bureau. A sociologist tests whether this percentage is different today by conducting a random sample of 500 females aged 15 years of age and older and finds that 285 are living alone. Use the -Value Approach to determine whether there is sufficient evidence at the level of significance to conclude the proportion has changed since 2000? (Two-tailed test)

* (test statistic) – value approach -- > Let alone. -- > Convert to a two-tailed area. From Table V – value = Double (0.3264) = 0.6528

Is – value < ? Is 0.6528 < 0.1 ? No, do not reject . There is not enough evidence to support the claim.

Ch 10.2 Hypothesis Test for a Population Mean Objective B : – Value Approach Ch 10.3 Hypothesis Test about a Population Mean with unknown Objective B : – Value Approach

Ch 10.3 Hypothesis Test about a Population Mean with unknown Objective B : – Value Approach

The – test procedure requires either that the sample was drawn from a normally distributed population or the sample size was greater than 30. Minor departures from normality will not adversely affect the results of the test. However, if the data include outliers, the – test procedure should not be used. A normality plot is used to test whether the sample was drawn from a normally distributed population. A boxplot is used to detect outliers.

Example 1: Find the – value interval for each test value. (a) right-tailed From Table VI (b) right-tailed From Table VI

(c) left-tailed From Table VI – value is between 0.01 and 0.02.

(d) two-tailed From Table VI is between 0.05 and 0.10.

Example 2: Use the – value approach to determine whether to reject or fail to reject . (a) – value versus for a right-tailed test Since – value is not smaller than , fail to reject . (b) – value versus for a left-tailed test Since – value < , reject .

Example 3: A survey of 15 large U. S Example 3: A survey of 15 large U.S. cities finds that the average commute time one way is 25.4 minutes. A chamber of commerce executive feels that the commute in his city is less and wants to publicize this. He randomly selects 25 commuters and finds the average is 22.1 minutes with a standard deviation of 5.3 minutes. At , is there enough evidence to support the claim? Use the – value approach. Step 1 : Setup : (left-tailed test) Step 2 : Step 3 : *

– value Approach From Table VI – value is between 0.001 and 0.0025. Step 4 : Is – value < ? Yes , reject . Step 5 : There is sufficient evidence to support the executive’s claim that the commute time in his city is less than 25.4 minutes.

As we have experienced from example 1 to 3, we were unable to find exact – values using the table because the – table provides – values only for certain area. To find exact – values, we use statistical software. Example 4: Use StatCrunch to calculate the following question.