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Hypothesis Testing Testing Statistical Significance

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Statistical Decision Making Public managers are often faced with decisions about program effectiveness, personnel productivity, and procedural changes Public managers are often faced with decisions about program effectiveness, personnel productivity, and procedural changes Is Patty Roberts an effective supervisor? Is Patty Roberts an effective supervisor? If we redesign form 54b, will it result in faster processing times? If we redesign form 54b, will it result in faster processing times? Is the Head Start program resulting in better reading scores for its participants? Is the Head Start program resulting in better reading scores for its participants? When we change these questions into statements, we have made hypotheses When we change these questions into statements, we have made hypotheses The head start program has resulted in higher reading scores for its participants The head start program has resulted in higher reading scores for its participants

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Statistical Decision Making However, in research we dont directly test our hypotheses However, in research we dont directly test our hypotheses Instead we test the negative of the hypothesis Instead we test the negative of the hypothesis Our research hypothesis may be that the new speeding fines in Virginia are resulting in fewer highway fatalities Our research hypothesis may be that the new speeding fines in Virginia are resulting in fewer highway fatalities However, what we test is the statement the new speeding fines in Virginia have NOT reduced highway fatalities However, what we test is the statement the new speeding fines in Virginia have NOT reduced highway fatalities This is called the Null Hypothesis This is called the Null Hypothesis Why? Why?

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Statistical Decision Making How do you prove something? Can you prove anything? Can you prove anything? Can you fail to prove something? Can you fail to prove something? All we can do is triangulate on the truth by eliminating what, most likely, is not the truth. All we can do is triangulate on the truth by eliminating what, most likely, is not the truth.

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Huh? Its all based on the concept of disconfirming evidence Its all based on the concept of disconfirming evidence Hypothesis testing relies on disconfirming evidence Hypothesis testing relies on disconfirming evidence An investigator does not directly assert that his/her data support the hypothesis An investigator does not directly assert that his/her data support the hypothesis Instead, investigator states that evidence shows that the null hypothesis is probably false Instead, investigator states that evidence shows that the null hypothesis is probably false Sherlock Holmes got it Sherlock Holmes got it Eliminate the impossible, whatever remains, however improbable, is the truth. Eliminate the impossible, whatever remains, however improbable, is the truth.

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Examples of H 1 and H 0 H1: Some job training programs are more successful than other programs in placing trainees in permanent employment. H1: Some job training programs are more successful than other programs in placing trainees in permanent employment. H0: All job training programs are equally likely to place trainees in permanent employment. H0: All job training programs are equally likely to place trainees in permanent employment. H1: Male planners earn higher salaries than female planners. H1: Male planners earn higher salaries than female planners. H0: Gender is not related to planners salaries. H0: Gender is not related to planners salaries. H1: Dr. Schroeder is smarter than the average Virginia Tech Research Professor H1: Dr. Schroeder is smarter than the average Virginia Tech Research Professor H0: Dr. Schroeder is no smarter than the average Virginia Tech Research Professor (his intelligence is a random error) H0: Dr. Schroeder is no smarter than the average Virginia Tech Research Professor (his intelligence is a random error)

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Disconfirming example We cannot prove that Dr. Schroeder is smarter than the average Virginia Tech Research Professor We cannot prove that Dr. Schroeder is smarter than the average Virginia Tech Research Professor What we can do is reject the null hypothesis – we can prove that Dr. Schroeders intelligence is not within the random error surrounding the scores of a random sample of other research professors – If H 0 is not true, than there is more evidence that H 1 might be true! What we can do is reject the null hypothesis – we can prove that Dr. Schroeders intelligence is not within the random error surrounding the scores of a random sample of other research professors – If H 0 is not true, than there is more evidence that H 1 might be true!

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Importance of Stating the Hypothesis Correctly The ability to state the null (H0) and research hypotheses (H1) correctly is essential The ability to state the null (H0) and research hypotheses (H1) correctly is essential The statistical techniques used in significance testing will have little meaning if not stated correctly The statistical techniques used in significance testing will have little meaning if not stated correctly Lets practice making some research and null hypotheses: Lets practice making some research and null hypotheses:

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Hypotheses Example 1 Six months after the local newspaper ran a weeklong series of articles on the Northlake, Virginia, Community Pride Center, the director wants to see whether the positive media coverage improved turnout at the centers after school recreation programs, compared to turnout before the media coverage took place Six months after the local newspaper ran a weeklong series of articles on the Northlake, Virginia, Community Pride Center, the director wants to see whether the positive media coverage improved turnout at the centers after school recreation programs, compared to turnout before the media coverage took place H1 H1 Media coverage increased turnout at the Community Pride Center Media coverage increased turnout at the Community Pride Center H0 H0 Media coverage did not increase turnout at the Community Pride Center Media coverage did not increase turnout at the Community Pride Center

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Hypotheses Example 2 The head of the Alton, New York, Public Works Department has installed security cameras in the public yard in hopes of lowering the large number of illegal after hours dumping incidents. After 90 days, officials want to assess the impact this measure has had on the number of illegal dumping incidents. The head of the Alton, New York, Public Works Department has installed security cameras in the public yard in hopes of lowering the large number of illegal after hours dumping incidents. After 90 days, officials want to assess the impact this measure has had on the number of illegal dumping incidents. H1 H1 The Installation of security cameras has led to a decrease in the number of illegal dumping incidents The Installation of security cameras has led to a decrease in the number of illegal dumping incidents H0 H0 The installation of security cameras has not led to a decrease in the number of illegal dumping incidents The installation of security cameras has not led to a decrease in the number of illegal dumping incidents

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Hypotheses Example 3 The principal of the Oaklawn Charter School claims that the Oaklawn method of mathematics instruction produces higher scores on standardized math skills tests compared to those of students in the district who are taught the old math. The principal of the Oaklawn Charter School claims that the Oaklawn method of mathematics instruction produces higher scores on standardized math skills tests compared to those of students in the district who are taught the old math. H1 H1 Math scores at Oaklawn are higher than those at other schools in the district. Math scores at Oaklawn are higher than those at other schools in the district. H0 H0 Math scores at Oaklawn are not higher than those at other schools in the district Math scores at Oaklawn are not higher than those at other schools in the district

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Testing Hypotheses Now that we have the idea about how to state research and null hypotheses, we can start looking at the statistical techniques used to test them Now that we have the idea about how to state research and null hypotheses, we can start looking at the statistical techniques used to test them Three situations you will be in when needing to test your hypotheses Three situations you will be in when needing to test your hypotheses 1.Population Parameter vs. Population Parameter 2.Sample Statistic vs. Population Parameter 3.Sample Statistic vs. Sample Statistic

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The Hypothesis Testing System 1.State the research and null hypotheses H 1 – Research Hypothesis H 1 – Research Hypothesis H 0 – Null Hypothesis H 0 – Null Hypothesis 2.Select an alpha level - 2.Select an alpha level - % willing to incorrectly reject the null hypothesis % willing to incorrectly reject the null hypothesis 3.Select and compute a test statistic Chi-Squared Chi-Squared T-test T-test 4.Accept or reject the null hypothesis 5.Make a decision

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Type I vs. Type II errors These are the mirror of each other – one goes up, the other goes down These are the mirror of each other – one goes up, the other goes down If you increase your sample size, both go down, but the relationship between them remains the same If you increase your sample size, both go down, but the relationship between them remains the same Type I error: rejecting a true Null Hypothesis Type I error: rejecting a true Null Hypothesis Finding that Dr. Schroeder IS more intelligent (when in fact he is not)! Finding that Dr. Schroeder IS more intelligent (when in fact he is not)! Type II error: not rejecting a false Null Hypothesis Type II error: not rejecting a false Null Hypothesis Finding that Dr. Schroeders intelligence is no different than the average (when in fact he is MUCH smarter)! Finding that Dr. Schroeders intelligence is no different than the average (when in fact he is MUCH smarter)! Why do we call it not rejecting or failing to reject the null hypothesis? Why dont we just accept the null hypothesis or find the null hypothesis to be true? Why do we call it not rejecting or failing to reject the null hypothesis? Why dont we just accept the null hypothesis or find the null hypothesis to be true? Which type of error is it generally worse to make? Which type of error is it generally worse to make? Type I (finding false evidence that your hypothesis may be true, as opposed to failing to find more evidence – can always try again: more subjects, different alpha level, etc.) Type I (finding false evidence that your hypothesis may be true, as opposed to failing to find more evidence – can always try again: more subjects, different alpha level, etc.)

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Selecting an alpha level ( ) alpha, what is it? - the probability that you will make a Type I error alpha, what is it? - the probability that you will make a Type I error.05 means 5% chance of committing error.05 means 5% chance of committing error.01 means 1%, etc..01 means 1%, etc. Alpha, whats it for? – used with test statistic to determine threshold a score must be above in order to be accepted as non-random Alpha, whats it for? – used with test statistic to determine threshold a score must be above in order to be accepted as non-random Chosen prior to begging analysis, why? Chosen prior to begging analysis, why? Because, depends on practical consequences of committing Type I or II error, NOT on what the data collected shows – need to think this through first! Because, depends on practical consequences of committing Type I or II error, NOT on what the data collected shows – need to think this through first!

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How SURE do you need to be? Social scientists routinely use.05 for alpha Social scientists routinely use.05 for alpha In managerial situations, however, that may be two big In managerial situations, however, that may be two big

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How SURE do you need to be? A rape crisis center may decide that the probability that one staff member cannot handle all the possible rape calls in a single day is.05 A rape crisis center may decide that the probability that one staff member cannot handle all the possible rape calls in a single day is.05 This means, however, that 1 day in 20, or once every three weeks, the rape crisis center will fail to meet a crisis This means, however, that 1 day in 20, or once every three weeks, the rape crisis center will fail to meet a crisis In this situation, you might instead pick.001 (which comes out to about one failure every three years) In this situation, you might instead pick.001 (which comes out to about one failure every three years)

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How SURE do you need to be? A police department. On the other hand, may be able to accept a.05 probability that one of its cars may be out of service A police department. On the other hand, may be able to accept a.05 probability that one of its cars may be out of service But the fire department may require a probability of.0001 that a fire hose will fail to operate (1 in 10,000 chance) But the fire department may require a probability of.0001 that a fire hose will fail to operate (1 in 10,000 chance)

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Selecting a Test Statistic Most commonly used in social sciences: chi- square and t-test Most commonly used in social sciences: chi- square and t-test Which one to use? Depends on level of data being investigated. Which one to use? Depends on level of data being investigated. chi-square: for nominal level data predicting nominal level data (usually in contingency tables) chi-square: for nominal level data predicting nominal level data (usually in contingency tables) e.g. type of training program [a nominal category] vs. working status [another nominal category] – see Table 12.3 e.g. type of training program [a nominal category] vs. working status [another nominal category] – see Table 12.3 t-test: for nominal level data predicting interval level data t-test: for nominal level data predicting interval level data e.g. gender [a nominal level category] determining salary [an interval level category] e.g. gender [a nominal level category] determining salary [an interval level category]

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Testing Hypotheses with Population Parameters Parameter vs. Parameter If you have access to the population parameters, then hypothesis testing is pretty easy If you have access to the population parameters, then hypothesis testing is pretty easy Its like deciding whom should start at center if Shaquille ONeal plays for your team Its like deciding whom should start at center if Shaquille ONeal plays for your team

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Testing Hypotheses with Population Parameters Parameter vs. Parameter Suppose Jerry Green, the governor of a large eastern state, wants to know whether a former governors executive reorganization has had any impact on the states expenditures Suppose Jerry Green, the governor of a large eastern state, wants to know whether a former governors executive reorganization has had any impact on the states expenditures After some thought, he postulates the following After some thought, he postulates the following H1: State expenditures decreased after the executive reorganization, compared with the state budgets long-run growth rate H1: State expenditures decreased after the executive reorganization, compared with the state budgets long-run growth rate H0: State expenditures did not decrease after the executive reorganization, compared with the state budgets long-run growth rate H0: State expenditures did not decrease after the executive reorganization, compared with the state budgets long-run growth rate

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Testing Hypotheses with Population Parameters Parameter vs. Parameter A management review shows that the states expenditure grew at a rate of 10.7% per year before the reorganization and 10.4% after the reorganization A management review shows that the states expenditure grew at a rate of 10.7% per year before the reorganization and 10.4% after the reorganization What do these figures say about the null hypothesis? What do these figures say about the null hypothesis? Because 10.4% is less than 10.7%, we REJECT the NULL hypothesis Because 10.4% is less than 10.7%, we REJECT the NULL hypothesis We conclude that the growth rate in state expenditures declined after the reorganization We conclude that the growth rate in state expenditures declined after the reorganization

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Testing Hypotheses with Population Parameters Parameter vs. Parameter Is a 0.3% decrease in the growth rate of expenditures significant? Is a 0.3% decrease in the growth rate of expenditures significant? Of course! These are population parameters which means the probability that the difference between the two conditions is real is 100%! Of course! These are population parameters which means the probability that the difference between the two conditions is real is 100%! Is this statistically significant difference trivial? Is this statistically significant difference trivial? Probably Probably We could have made the hypothesis more specific – …expenditures decreased by more than 5%... We could have made the hypothesis more specific – …expenditures decreased by more than 5%...

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Testing Hypotheses with Population Parameters Parameter vs. Parameter Dont really have to use the hypothesis testing system in this scenario Dont really have to use the hypothesis testing system in this scenario No need for an alpha or statistical test (we are not dealing with statistics, we are dealing only with parameters) No need for an alpha or statistical test (we are not dealing with statistics, we are dealing only with parameters) Only when we add unknowns into the mix via sampling do we need to resort to statistical tests Only when we add unknowns into the mix via sampling do we need to resort to statistical tests You DO, however, need to take care to state logical hypotheses You DO, however, need to take care to state logical hypotheses

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Testing Hypotheses with Samples Statistic vs. Parameter Referred to as One-Sample tests Referred to as One-Sample tests Comparing one sample to a known population Comparing one sample to a known population Whats the likelihood that the mean I just obtained from my sample is representative of the population as a whole? Whats the likelihood that the mean I just obtained from my sample is representative of the population as a whole? I already know the population of M&M handfuls for the class, so, how well does the mean of a specific sample of four handfuls represent the population? I already know the population of M&M handfuls for the class, so, how well does the mean of a specific sample of four handfuls represent the population?

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square The One Sample Chi-Square The One Sample Chi-Square The Chi-square test used with one sample is described as a "goodness of fit" test. It can help you decide whether a distribution of frequencies for a variable in a sample is representative of, or "fits", a specified population distribution. For example, you can use this test to decide whether your data are approximately normal or not. The Chi-square test used with one sample is described as a "goodness of fit" test. It can help you decide whether a distribution of frequencies for a variable in a sample is representative of, or "fits", a specified population distribution. For example, you can use this test to decide whether your data are approximately normal or not.

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square Suppose the relative frequencies of marital status for the population of adult American females under 40 years of age are as follows: Suppose the relative frequencies of marital status for the population of adult American females under 40 years of age are as follows: General Population Sample (N=200) Marital Status Relative Frequency Observed Frequencies Married Single Separated Divorced Widowed0.034

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square Then suppose an investigator wanted to know whether a particular sample of 200 adult females under age 40 was drawn from a population that is representative of the general population Then suppose an investigator wanted to know whether a particular sample of 200 adult females under age 40 was drawn from a population that is representative of the general population By applying the procedures of Chi Square and the steps of hypothesis testing, we can decide whether the sample distribution is close enough to the population distribution to be considered representative of it. By applying the procedures of Chi Square and the steps of hypothesis testing, we can decide whether the sample distribution is close enough to the population distribution to be considered representative of it. General Population Sample (N=200) Marital Status Relative Frequency Observed Frequencies Married Single Separated Divorced Widowed0.034

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square State the Research and Null Hypotheses State the Research and Null Hypotheses H1: The sample does not represent the population distribution H1: The sample does not represent the population distribution H0: The sample does represent the population distribution H0: The sample does represent the population distribution Why is H1 stated negatively? Why is H1 stated negatively? What does Chi-Square show us? What does Chi-Square show us? General Population Sample (N=200) Marital Status Relative Frequency Observed Frequencies Married Single Separated Divorced Widowed0.034

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square Select an alpha level Select an alpha level How willing are we to make a mistake and say that the sample IS representative of the population when it actually isnt? How willing are we to make a mistake and say that the sample IS representative of the population when it actually isnt? Really depends on why you are getting the sample in the first place, but lets assume 5% for now Really depends on why you are getting the sample in the first place, but lets assume 5% for now Stated another way wed say there is a probability of.05 that we will mistakenly accept the research hypothesis Stated another way wed say there is a probability of.05 that we will mistakenly accept the research hypothesis General Population Sample (N=200) Marital Status Relative Frequency Observed Frequencies Married Single Separated Divorced Widowed0.034

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square Select and Compute a Test Statistic Select and Compute a Test Statistic In this case we are dealing with what level of data? In this case we are dealing with what level of data? Nominal Nominal Chi Square is selected Chi Square is selected General Population Sample (N=200) Marital Status Relative Frequency Observed Frequencies Married Single Separated Divorced Widowed0.034

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square We calculate Expected frequencies for each of the cells in our sample distribution We calculate Expected frequencies for each of the cells in our sample distribution If in our general population, 55% of such women are married then we would expect 55% of 200 = 110 in our sample to be married If in our general population, 55% of such women are married then we would expect 55% of 200 = 110 in our sample to be married Single women would be 21% of 200 = 42 Single women would be 21% of 200 = 42 Separated 9% of 200 = 18 Separated 9% of 200 = 18 Divorced = 12% of 200 = 24 Divorced = 12% of 200 = 24 Widowed 3% of 200 = 6 Widowed 3% of 200 = 6 We then get the difference between each Expected and each Observed, square this, and then divide this result by the Expected. We then get the difference between each Expected and each Observed, square this, and then divide this result by the Expected. General Population Sample (N=200) Marital Status Relative Frequency Observed Frequencies Married Single Separated Divorced Widowed0.034

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square General Population Sample (N=200) Marital Status Relative Frequency Observed Frequencies Married Single Separated Divorced Widowed0.034 = =7.90

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample Chi Square General Population Sample (N=200) Marital Status Relative Frequency Observed Frequencies Married Single Separated Divorced Widowed0.034 Reject or Accept the Null Hypothesis Reject or Accept the Null Hypothesis You then refer to your X 2 table under the 0.05 heading with df = C-1 = 4 You then refer to your X 2 table under the 0.05 heading with df = C-1 = 4 You find a critical value of 9.49 You find a critical value of 9.49 Is our calculated value of 7.9 significant? Is our calculated value of 7.9 significant? No, so do we accept or reject the null hypothesis? No, so do we accept or reject the null hypothesis? Accept H0 – what does that mean? Accept H0 – what does that mean?

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test The One-Sample t Test The One-Sample t Test A professor wants to know if her introductory statistics class has a good grasp of basic math A professor wants to know if her introductory statistics class has a good grasp of basic math Six students are chosen at random from the class and given a math proficiency test Six students are chosen at random from the class and given a math proficiency test The professor wants the class to be able to score at least 70 on the test The professor wants the class to be able to score at least 70 on the test The six students get scores of 62, 92, 75, 68, 83, and 95 The six students get scores of 62, 92, 75, 68, 83, and 95 Can the professor be at least 90 percent certain that the mean score for the class on the test would be at least 70? Can the professor be at least 90 percent certain that the mean score for the class on the test would be at least 70? Scores mean: sd: 13.17

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test State your research and null hypotheses State your research and null hypotheses H1: μ 70 H1: μ 70 H0: μ < 70 H0: μ < 70 Scores mean: sd: 13.17

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test Select and Compute Statistic Select and Compute Statistic Dealing with Interval/Ratio level data Dealing with Interval/Ratio level data Select t-test Select t-test Scores mean: sd: 13.17

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test To test the hypothesis, the computed t-value of 1.71 will be compared to the critical value in the t-table. To test the hypothesis, the computed t-value of 1.71 will be compared to the critical value in the t-table. Scores mean: sd: 13.17

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test Accept or Reject the Null Hypothesis Accept or Reject the Null Hypothesis A 90 percent confidence level is equivalent to an alpha level of.10 A 90 percent confidence level is equivalent to an alpha level of.10 The number of degrees of freedom for the problem is 6 – 1 = 5 The number of degrees of freedom for the problem is 6 – 1 = 5 The value in the t-table for t 10,5 is The value in the t-table for t 10,5 is Because the computed t-value of 1.71 is larger than the critical value in the table, the null hypothesis can be rejected, and the professor can be 90 percent certain that the class mean on the math test would be at least 70 Because the computed t-value of 1.71 is larger than the critical value in the table, the null hypothesis can be rejected, and the professor can be 90 percent certain that the class mean on the math test would be at least 70 Scores mean: sd: 13.17

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t vs. z Note that the formula for the one-sample t-test for a population mean is the same as the z- test, except that the t-test substitutes the sample standard deviation s for the population standard deviation σ and takes critical values from the t-distribution instead of the z- distribution. The t-distribution is particularly useful for tests with small samples ( n < 30) Note that the formula for the one-sample t-test for a population mean is the same as the z- test, except that the t-test substitutes the sample standard deviation s for the population standard deviation σ and takes critical values from the t-distribution instead of the z- distribution. The t-distribution is particularly useful for tests with small samples ( n < 30) Could use either distribution to test your hypothesis about n=30 Could use either distribution to test your hypothesis about n=30

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test Two-Tail Example Two-Tail Example Used when you dont care if something is more or less than – just different than Used when you dont care if something is more or less than – just different than A Little League baseball coach wants to know if his team is representative of other teams in scoring runs A Little League baseball coach wants to know if his team is representative of other teams in scoring runs Nationally, the average number of runs scored by a Little League team in a game is 5.7 Nationally, the average number of runs scored by a Little League team in a game is 5.7 He chooses five games at random in which his team scored 5 9, 4, 11, and 8 runs. Is it likely that his team's scores could have come from the national distribution? He chooses five games at random in which his team scored 5 9, 4, 11, and 8 runs. Is it likely that his team's scores could have come from the national distribution? Scores mean: 7.4 sd: 2.88

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test State Research and Null Hypotheses State Research and Null Hypotheses H1: μ 5.7 H1: μ 5.7 H0: μ = 5.7 H0: μ = 5.7 Select alpha Select alpha Alpha:.05 Alpha:.05 Scores mean: 7.4 sd: 2.88

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test Calculate t Calculate t Scores mean: 7.4 sd: 2.88

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test Now, look up the critical value from the t-table Now, look up the critical value from the t-table The degrees of freedom is 5 – 1 = 4. The overall alpha level is.05 The degrees of freedom is 5 – 1 = 4. The overall alpha level is.05 But because this is a two- tailed test, the alpha level must be divided by two, which yields.025 But because this is a two- tailed test, the alpha level must be divided by two, which yields.025 This means.025 on either end vs.05 on one end This means.025 on either end vs.05 on one end The tabled value for t.025,4 is The tabled value for t.025,4 is Scores mean: 7.4 sd: 2.88

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test In a two-tailed hypothesis, you have to consider BOTH ends, not just one In a two-tailed hypothesis, you have to consider BOTH ends, not just one

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test t must be EITHER more than the positive critical value or less than the negative critical value (±2.776) t must be EITHER more than the positive critical value or less than the negative critical value (±2.776) The computed t of 1.32 is not smaller than or more than The computed t of 1.32 is not smaller than or more than You cannot reject the null hypothesis that the mean of this team is equal to the population mean You cannot reject the null hypothesis that the mean of this team is equal to the population mean The coach can conclude that his team fits in with the national distribution on runs scored. The coach can conclude that his team fits in with the national distribution on runs scored. Scores mean: 7.4 sd: 2.88

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Confidence interval for population mean using t This is exactly the same as when dealing with z! Remember ± 1.96 x s.e.? The 1.96 was the z- score Now we are just using t x s.e.

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Confidence interval for population mean using t Using the previous example, what is a 95 percent confidence interval for runs scored per team per game? First, determine the t-value. A 95 percent confidence level is equivalent to an alpha level of.05 Half of.05 is.025 (Why Half?) The t-value corresponding to an area of.025 at either end of the t-distribution for 4 degrees of freedom ( t.025,4) is

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Testing Hypotheses with Samples Statistic vs. Parameter – 1 Sample t-test In-Class Exercise In-Class Exercise

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Testing Hypotheses with Samples Statistic vs. Statistic Used to test hypotheses that two groups have statistically different means Used to test hypotheses that two groups have statistically different means (two-tailed [non-directional]) (two-tailed [non-directional]) H 1 : Men make a different salary than women H 1 : Men make a different salary than women H 0 : Men and women make the same H 0 : Men and women make the same Or, tests hypotheses that one groups mean is higher than the other groups mean Or, tests hypotheses that one groups mean is higher than the other groups mean (one-tailed [directional]) (one-tailed [directional]) H 1 : Men make more than women H 1 : Men make more than women H 0 : Men make the same or less than women H 0 : Men make the same or less than women

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Testing Hypotheses with Samples Statistic vs. Statistic – Chi Square Youve already done this previously when you were calculating chi square for contingency tables! Youve already done this previously when you were calculating chi square for contingency tables! Now, you are just adding on the proper way to hypothesize Now, you are just adding on the proper way to hypothesize Because chi square looks for the existence of a relationship based on the difference between observed and expected, your null hypothesis is always that there is no difference Because chi square looks for the existence of a relationship based on the difference between observed and expected, your null hypothesis is always that there is no difference

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Previous Example Calculations for Expected Frequencies Table Cell CompetenceHierarchyObservedExpected (O-E) 2 /E LowLow113.50x152= LowMedium31.40x152= LowHigh8.10x152= MediumLow60.50x159= MediumMedium91.40x159= MediumHigh8.10x159= HighLow27.50x89= HighMedium38.40x89= HighHigh24.10x89= Total CHI-SQUARE!

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Previous Example H1: Hierarchy is Related to Competence H1: Hierarchy is Related to Competence H0: Hierarchy is not related to Competence H0: Hierarchy is not related to Competence If Chi Square is Higher than the critical value, you reject the null hypothesis and accept the research hypothesis If Chi Square is Higher than the critical value, you reject the null hypothesis and accept the research hypothesis

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means Tests whether the means of two groups are statistically different from each other Lets look at some graphs so we may visually understand what it is we are looking at

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Statistical Analysis

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Control group mean

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Statistical Analysis Control group mean Treatment group mean

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Statistical Analysis Control group mean Treatment group mean Is there a difference?

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What Does Difference Mean?

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Medium variability

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What Does Difference Mean? Medium variability High variability

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What Does Difference Mean? Medium variability High variability Low variability

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What Does Difference Mean? Medium variability High variability Low variability The mean difference is the same for all three cases.

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What Does Difference Mean? Medium variability High variability Low variability Which one shows the greatest difference?

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What Does Difference Mean? A statistical difference is a function of the difference between means relative to the variability. A statistical difference is a function of the difference between means relative to the variability. A small difference between means with large variability could be due to chance. A small difference between means with large variability could be due to chance. Low variability Which one shows the greatest difference?

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What Do We Estimate? Low variability

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What Do We Estimate? Low variability Signal Noise

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What Do We Estimate? Low variability Signal Noise Difference between group means =

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What Do We Estimate? Low variability Signal Noise Difference between group means Variability of groups = Difference between group means Variability of groups

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What Do We Estimate? Low variability Signal Noise Difference between group means Variability of groups = = X T - X C SE(X T - X C ) __ __ Difference between group means Variability of groups _ Difference between group means Variability of groups __ Difference between group means Variability of groups _ __ Difference between group means Variability of groups __ __ Difference between group means Variability of groups

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What Do We Estimate? Low variability Signal Noise Difference between group means Variability of groups = X T - X C SE(X T - X C ) = = t-value __ __ Difference between group means Variability of groups __ _ __ __ __

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means The Ware County librarian wants to increase circulation from the Ware County bookmobiles The Ware County librarian wants to increase circulation from the Ware County bookmobiles The librarian thins that poster ads in areas where the book mobiles stop will attract more browsers and increase circulation The librarian thins that poster ads in areas where the book mobiles stop will attract more browsers and increase circulation To test this idea, the librarian sets up an experiment To test this idea, the librarian sets up an experiment

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means Ten bookmobile routes are selected at random Ten bookmobile routes are selected at random On those routes, poster ads are posted with bookmobile information On those routes, poster ads are posted with bookmobile information Ten other bookmobile routes are selected at random Ten other bookmobile routes are selected at random On those routes, no advertising is done On those routes, no advertising is done

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means Step 1 - Hypotheses Step 1 - Hypotheses The null hypothesis is that the mean circulation of the experimental group is not higher than the mean circulation of the control group The null hypothesis is that the mean circulation of the experimental group is not higher than the mean circulation of the control group The research hypothesis is that the mean circulation of the experimental group is higher than the mean circulation of the control group The research hypothesis is that the mean circulation of the experimental group is higher than the mean circulation of the control group Step 2 – Alpha Step 2 – Alpha.05.05

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means The following data is obtained: The following data is obtained: Librarians Data Groups Books Experimental Group Control Group Mean Standard Deviation

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means STEP 3 STEP 3 Calculate the s.e. for each group Calculate the s.e. for each group 125/sqrt(10) 125/sqrt(10) =39.5 = /sqrt(10) 115/sqrt(10) =36.4 =36.4 Librarians Data Groups Books Experimental Group Control Group Mean Standard Deviation

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means STEP 4 STEP 4 Create a pooled standard error Create a pooled standard error Librarians Data Groups Books Experiment al Group Control Group Mean Standard Deviation

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means STEP 5 STEP 5 Subtract the mean of the second group from the first Subtract the mean of the second group from the first Then divide by the pooled error Then divide by the pooled error Librarians Data Groups Books Experiment al Group Control Group Mean Standard Deviation

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means STEP 6 STEP 6 Degrees of Freedom equals n 1 + n 2 – 2 Degrees of Freedom equals n 1 + n 2 – – 2 = – 2 = 18 Librarians Data Groups Books Experiment al Group Control Group Mean Standard Deviation

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means STEP 7 STEP 7 Look it up and accept or reject the null hypothesis Look it up and accept or reject the null hypothesis Critical value for 18df at the.05 level of significance is Critical value for 18df at the.05 level of significance is We did not meet that value and, therefore, fail to reject the null hypothesis We did not meet that value and, therefore, fail to reject the null hypothesis We cant say that the advertising increased book circulation We cant say that the advertising increased book circulation Librarians Data Groups Books Experiment al Group Control Group Mean Standard Deviation

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means What if we wanted to just see if advertising had any effect? What if we wanted to just see if advertising had any effect? What values would we be comparing then? What values would we be comparing then? t would have to be outside the range of ±2.10 t would have to be outside the range of ±2.10 How did we get that? How did we get that? Librarians Data Groups Books Experiment al Group Control Group Mean Standard Deviation

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for two sample means In-Class Exercise In-Class Exercise

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for proportions Because we can figure out standard errors for proportions (like we did last 2 weeks), we can use a t-test to also compare two groups proportions Because we can figure out standard errors for proportions (like we did last 2 weeks), we can use a t-test to also compare two groups proportions The formulas are the same, the only difference is the calculation of the standard deviation from the proportions The formulas are the same, the only difference is the calculation of the standard deviation from the proportions

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for proportions If youre trying to see if there is a statistical difference between two groups on whether or not they support an amendment to the state constitution, it would look like this If youre trying to see if there is a statistical difference between two groups on whether or not they support an amendment to the state constitution, it would look like this Once you have the standard deviation, you do everything the same as when comparing means Once you have the standard deviation, you do everything the same as when comparing means Group A Group B n=50n=50 For=60%For=40% p=.60p=.40 s=sqrt(p(1-p)) =.49 s=sqrt(p(1-p))

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Testing Hypotheses with Samples Statistic vs. Statistic – t-test for proportions In-Class Worksheet In-Class Worksheet

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Homework Write 2 scenarios and analyses (make them PA relevant), one for a comparison of sample means, another for a comparison of sample proportions Write 2 scenarios and analyses (make them PA relevant), one for a comparison of sample means, another for a comparison of sample proportions Make up the problem descriptions and data Make up the problem descriptions and data ed to me by Halloween Midnight ed to me by Halloween Midnight No class Halloween night No class Halloween night

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