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COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Chapter 8: Hypothesis Testing Peer Tutor Slides Instructor: Mr. Ethan W. Cooper, Lead Tutor © PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE WRITTEN PERMISSION OF THE AUTHOR. FOR PERMISSION OR QUESTIONS, PLEASE MR. COOPER AT THE FOLLWING:

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Key Terms: Don’t Forget Notecards Hypothesis Test (p. 233) Hypothesis Test (p. 233) Null Hypothesis (p. 236) Null Hypothesis (p. 236) Alternative Hypothesis (p. 236) Alternative Hypothesis (p. 236) Alpha Level (level of significance) (pp. 238 & 245) Alpha Level (level of significance) (pp. 238 & 245) Critical Region (p. 238) Critical Region (p. 238) Type I Error (p. 244) Type I Error (p. 244) Type II Error (p. 245) Type II Error (p. 245) Statistically Significant (p. 251) Statistically Significant (p. 251) Directional (one-tailed) Hypothesis Test (p. 256) Directional (one-tailed) Hypothesis Test (p. 256) Effect Size (p. 262) Effect Size (p. 262) Power (p. 265) Power (p. 265)

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Formulas

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Logic of Hypothesis Testing Question 1: The city school district is considering increasing class size in the elementary schools. However, some members of the school board are concerned that larger classes may have a negative effect on student learning. In words, what would the null hypothesis say about the effect of class size on student learning? Question 1: The city school district is considering increasing class size in the elementary schools. However, some members of the school board are concerned that larger classes may have a negative effect on student learning. In words, what would the null hypothesis say about the effect of class size on student learning?

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Logic of Hypothesis Testing Question 1 Answers: Question 1 Answers: For a two-tailed test: For a two-tailed test: The null hypothesis would say that class size has no effect on student learning. The null hypothesis would say that class size has no effect on student learning. The alternative hypothesis would say that class size does have an effect on student learning. The alternative hypothesis would say that class size does have an effect on student learning. For a one-tailed test: For a one-tailed test: The null hypothesis would say that class size does not have a negative effect on student learning. The null hypothesis would say that class size does not have a negative effect on student learning. The alternative hypothesis would say that class size has a negative effect on student learning. The alternative hypothesis would say that class size has a negative effect on student learning.

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Alpha Level and the Critical Region Question 2: If the alpha level is decreased from α = 0.01 to α = 0.001, then the boundaries for the critical region move farther away from the center of the distribution. (True or False?) Question 2: If the alpha level is decreased from α = 0.01 to α = 0.001, then the boundaries for the critical region move farther away from the center of the distribution. (True or False?)

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Alpha Level and the Critical Region Question 2 Answer: Question 2 Answer: True. A smaller alpha level means that the boundaries for the critical region move further away from the center of the distribution. True. A smaller alpha level means that the boundaries for the critical region move further away from the center of the distribution.

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Possible Outcomes of a Hypothesis Test Question 3: Define Type 1 and Type II Error. Question 3: Define Type 1 and Type II Error.

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Possible Outcomes of a Hypothesis Test Question 3 Answer: Question 3 Answer: Type I error is rejecting a true null hypothesis – that is, saying that treatment has an effect when, in fact, it doesn’t. Type I error is rejecting a true null hypothesis – that is, saying that treatment has an effect when, in fact, it doesn’t. Type I error = false (+) = Alpha (α) = level of significance Type I error = false (+) = Alpha (α) = level of significance Type II error is the failure to reject a null hypothesis. In terms of a research study, a Type II error occurs when a study fails to detect a treatment that really exists. Type II error is the failure to reject a null hypothesis. In terms of a research study, a Type II error occurs when a study fails to detect a treatment that really exists. Type II error = false (-) = beta error = (β) Type II error = false (-) = beta error = (β) A Type II error is likely to occur when a treatment effect is very small.

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Two-Tailed Hypothesis Test Question 4: After years of teaching driver’s education, an instructor knows that students hit an average of µ = 10.5 orange cones while driving the obstacle course in their final exam. The distribution of run-over cones is approximately normal with a standard deviation of σ = 4.8. To test a theory about text messaging and driving, the instructor recruits a sample of n = 16 student drivers to attempt the obstacle course while sending a text message. The individuals in this sample hit an average of M = 15.9 cones. Do the data indicate that texting has a significant effect on driving? Test with α = Question 4: After years of teaching driver’s education, an instructor knows that students hit an average of µ = 10.5 orange cones while driving the obstacle course in their final exam. The distribution of run-over cones is approximately normal with a standard deviation of σ = 4.8. To test a theory about text messaging and driving, the instructor recruits a sample of n = 16 student drivers to attempt the obstacle course while sending a text message. The individuals in this sample hit an average of M = 15.9 cones. Do the data indicate that texting has a significant effect on driving? Test with α = 0.01.

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Two-Tailed Hypothesis Test Question 4 Answer: Question 4 Answer: Step 1: State hypotheses Step 1: State hypotheses H 0 : Texting has no effect on driving. (µ = 10.5) H 0 : Texting has no effect on driving. (µ = 10.5) H 1 : Texting has an effect on driving. (µ ≠ 10.5) H 1 : Texting has an effect on driving. (µ ≠ 10.5) Step 2: Set Criteria for Decision (α = 0.01) Step 2: Set Criteria for Decision (α = 0.01) z = ± 2.58 z = 2.58 z = Reject H 0

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Two-Tailed Hypothesis Test

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Question 4 Answer Question 4 Answer Step 4: Make a decision Step 4: Make a decision For a Two-tailed Test: For a Two-tailed Test: z sample (4.50) > z critical (2.58) z sample (4.50) > z critical (2.58) Thus, we reject the null and note that texting has a significant effect on driving. Thus, we reject the null and note that texting has a significant effect on driving. If < z sample < 2.58, fail to reject H 0 If z sample ≤ or z sample ≥ 2.58, reject H 0

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Factors that Influence a Hypothesis Test Question 5: If other factors are held constant, increasing the size of a sample increases the likelihood of rejecting the null hypothesis. (True or False?) Question 5: If other factors are held constant, increasing the size of a sample increases the likelihood of rejecting the null hypothesis. (True or False?)

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Factors that Influence a Hypothesis Test Question 5 Answer: Question 5 Answer: True. A larger sample produces a smaller standard error, which leads to a larger z-score. True. A larger sample produces a smaller standard error, which leads to a larger z-score. Consequently, as z increases so does the probability of rejecting the null hypothesis.

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Factors that Influence a Hypothesis Test Question 6: If other factors remain constant, are you more likely to reject the null hypothesis with a standard deviation of σ = 2 or σ = 10? Question 6: If other factors remain constant, are you more likely to reject the null hypothesis with a standard deviation of σ = 2 or σ = 10?

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Factors that Influence a Hypothesis Test Question 6 answer: Question 6 answer: σ = 2. A smaller standard deviation produces a smaller standard error, which leads to a larger z-score. Thus, increasing the probability of rejecting the null hypothesis. σ = 2. A smaller standard deviation produces a smaller standard error, which leads to a larger z-score. Thus, increasing the probability of rejecting the null hypothesis.

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One-tailed Hypothesis Test Question 7: A researcher is testing the hypothesis that consuming a sports drink during exercise improves endurance. A sample of n = 50 male college students is obtained and each student is given a series of three endurance tasks and asked to consume 4 ounces of the drink during each break between tasks. The overall endurance score for this sample is M = 53. For the general population of male college students, without any sports drink, the scores average μ = 50 with a standard deviation of σ = 10. Can the researcher conclude that endurance scores with the sports drink are significantly higher than score without the drink? (Use a one-tailed test, α = 0.05) Question 7: A researcher is testing the hypothesis that consuming a sports drink during exercise improves endurance. A sample of n = 50 male college students is obtained and each student is given a series of three endurance tasks and asked to consume 4 ounces of the drink during each break between tasks. The overall endurance score for this sample is M = 53. For the general population of male college students, without any sports drink, the scores average μ = 50 with a standard deviation of σ = 10. Can the researcher conclude that endurance scores with the sports drink are significantly higher than score without the drink? (Use a one-tailed test, α = 0.05)

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One-tailed Hypothesis Test Question 7 Answer: Question 7 Answer: Step 1: State hypotheses Step 1: State hypotheses H 0 : Endurance scores are not significantly higher with the sports drink. (µ ≤ 50) H 0 : Endurance scores are not significantly higher with the sports drink. (µ ≤ 50) H 1 : Endurance scores are significantly higher with the sports drink. (µ > 50) H 1 : Endurance scores are significantly higher with the sports drink. (µ > 50) Step 2: Set Criteria for Decision (α = 0.05) Step 2: Set Criteria for Decision (α = 0.05) z = 1.65 Reject H 0

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One-tailed Hypothesis Test

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Question 7 Answer: Question 7 Answer: Step 4: Make a decision Step 4: Make a decision For a One-tailed Test: For a One-tailed Test: z sample (2.13) > z critical (1.65) z sample (2.13) > z critical (1.65) Thus, we reject the null and note that the sports drink does raise endurance scores. Thus, we reject the null and note that the sports drink does raise endurance scores. If z sample ≤ 1.65, fail to reject H 0 If z sample > 1.65, reject H 0

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Effect Size and Cohen’s d Question 8: A researcher selects a sample from a population with µ = 40 and σ = 8. A treatment is administered to the sample and, after treatment, the sample mean is found to be M = 47. Compute Cohen’s d to measure the size of the treatment effect. Question 8: A researcher selects a sample from a population with µ = 40 and σ = 8. A treatment is administered to the sample and, after treatment, the sample mean is found to be M = 47. Compute Cohen’s d to measure the size of the treatment effect.

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Effect Size and Cohen’s d Remember: These are thresholds. Any effect less than d = 0.2 is a trivial effect and should be treated as having no effect. Any effect between d = 0.2 and d = 0.5 is a small effect. And between d = 0.5 and d = 0.8 is a medium effect.

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Computing Power Question 9: A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of µ = 100 and a standard deviation of σ = 20. The researcher expects a 10-point treatment effect and plans to use a two-tailed hypothesis test with α = Compute the power of the test if the researcher uses a sample of n = 25 individuals. Question 9: A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of µ = 100 and a standard deviation of σ = 20. The researcher expects a 10-point treatment effect and plans to use a two-tailed hypothesis test with α = Compute the power of the test if the researcher uses a sample of n = 25 individuals.

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Computing Power z = 1.96, for α = 0.05 Any sample mean greater than falls in the critical region.

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Computing Power Step #4: Interpret Power of the Hypothesis Test Step #4: Interpret Power of the Hypothesis Test Find probability associated with a z-score > Find probability associated with a z-score > Look this probability up as the proportion in the body of the normal distribution (column B in your textbook) Look this probability up as the proportion in the body of the normal distribution (column B in your textbook) p(z > -0.54) = p(z > -0.54) = Thus, with a sample of 25 people and a 10-point treatment effect, 70.54% of the time the hypothesis test will conclude that there is a significant effect. Thus, with a sample of 25 people and a 10-point treatment effect, 70.54% of the time the hypothesis test will conclude that there is a significant effect.

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Computing Power Question 10: A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of µ = 80 and a standard deviation of σ = 20. The researcher expects a 12-point treatment effect and plans to use a two-tailed hypothesis test with α = Compute the power of the test if the researcher uses a sample of n = 25 individuals. Question 10: A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of µ = 80 and a standard deviation of σ = 20. The researcher expects a 12-point treatment effect and plans to use a two-tailed hypothesis test with α = Compute the power of the test if the researcher uses a sample of n = 25 individuals.

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Computing Power z = 1.96, for α = 0.05 Any sample mean greater than falls in the critical region.

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Computing Power Question 10 Answer: Question 10 Answer: Step #4: Interpret Power of the Hypothesis Test Step #4: Interpret Power of the Hypothesis Test Find probability associated with a z-score > Find probability associated with a z-score > Look this probability up as the proportion in the body of the normal distribution (column B in your textbook) Look this probability up as the proportion in the body of the normal distribution (column B in your textbook) p(z > -1.04) = p(z > -1.04) = Thus, with a sample of 25 people and a 12-point treatment effect, 85.08% of the time the hypothesis test will conclude that there is a significant effect. Thus, with a sample of 25 people and a 12-point treatment effect, 85.08% of the time the hypothesis test will conclude that there is a significant effect.

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Frequently Asked Questions FAQs What is power? What is power? Power is the probability that a hypothesis test will reject the null hypothesis, if there is a treatment effect. Power is the probability that a hypothesis test will reject the null hypothesis, if there is a treatment effect. There are 4 steps involved in finding power. There are 4 steps involved in finding power. Step #1: Calculate the standard error. Step #1: Calculate the standard error. Step #2: Locate the boundary of the critical region. Step #2: Locate the boundary of the critical region. Step #3: Calculate the z-score. Step #3: Calculate the z-score. Step #4: Find the probability. Step #4: Find the probability. Using the example from the lecture notes, let’s go through each step. Using the example from the lecture notes, let’s go through each step. β is the probability of a type II error (false negative). Therefore, power is 1 – β.

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Frequently Asked Questions FAQs The previous slide was based upon a study from your book with μ = 80, σ = 10, and a sample (n=25) that is drawn with an 8-point treatment effect (M=88). What is the power of the related statistical test for detecting the difference between the population and sample mean? The previous slide was based upon a study from your book with μ = 80, σ = 10, and a sample (n=25) that is drawn with an 8-point treatment effect (M=88). What is the power of the related statistical test for detecting the difference between the population and sample mean?

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Frequently Asked Questions FAQs Step #1: Calculate standard error for sample Step #1: Calculate standard error for sample In this step, we work from the population’s standard deviation (σ) and the sample size (n) In this step, we work from the population’s standard deviation (σ) and the sample size (n)

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Frequently Asked Questions FAQs Step #2: Locate Boundary of Critical Region Step #2: Locate Boundary of Critical Region In this step, we find the exact boundary of the critical region In this step, we find the exact boundary of the critical region Pick a critical z-score based upon alpha (α =.05) Pick a critical z-score based upon alpha (α =.05)

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Frequently Asked Questions FAQs Step #3: Calculate the z-score for the difference between the treated sample mean (M=83.92) for the critical region boundary and the population mean with an 8-point treatment effect (μ = 88). Step #3: Calculate the z-score for the difference between the treated sample mean (M=83.92) for the critical region boundary and the population mean with an 8-point treatment effect (μ = 88).

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Frequently Asked Questions FAQs Interpret Power of the Hypothesis Test Interpret Power of the Hypothesis Test Find probability associated with a z-score > Find probability associated with a z-score > Look this probability up as the proportion in the body of the normal distribution (column B in your textbook) Look this probability up as the proportion in the body of the normal distribution (column B in your textbook) p =.9793 p =.9793 Thus, with a sample of 25 people and an 8-point treatment effect, 97.93% of the time the hypothesis test will conclude that there is a significant effect. Thus, with a sample of 25 people and an 8-point treatment effect, 97.93% of the time the hypothesis test will conclude that there is a significant effect.

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