Download presentation

1
**INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE **

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 Ethan W. Cooper, Lead Tutor © PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE WRITTEN PERMISSION OF THE AUTHOR.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:

2
**Key Terms: Don’t Forget Notecards**

Hypothesis Test (p. 233) Null Hypothesis (p. 236) Alternative Hypothesis (p. 236) Alpha Level (level of significance) (pp. 238 & 245) Critical Region (p. 238) Type I Error (p. 244) Type II Error (p. 245) Statistically Significant (p. 251) Directional (one-tailed) Hypothesis Test (p. 256) Effect Size (p. 262) Power (p. 265)

3
**Formulas Standard Error of M: 𝜎 𝑀 = 𝜎 𝑛 = 𝜎 2 𝑛 = 𝜎 2 𝑛**

z-Score Formula: 𝑧= 𝑀−𝜇 𝜎 𝑀 Cohen’s d: 𝑚𝑒𝑎𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝜇 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇 𝑛𝑜 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝜎 estimated Cohen’s d: 𝑚𝑒𝑎𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑀 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇 𝑛𝑜 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝜎

4
**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?

5
**Logic of Hypothesis Testing**

Question 1 Answers: For a two-tailed test: 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. For a one-tailed test: 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.

6
**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?)

7
**Alpha Level and the Critical Region**

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.

8
**Possible Outcomes of a Hypothesis Test**

Question 3: Define Type 1 and Type II Error.

9
**Possible Outcomes of a Hypothesis Test**

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 = 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 = false (-) = beta error = (β) A Type II error is likely to occur when a treatment effect is very small.

10
**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 = 0.01.

11
**Two-Tailed Hypothesis Test**

Question 4 Answer: Step 1: State hypotheses H0: Texting has no effect on driving. (µ = 10.5) H1: Texting has an effect on driving. (µ ≠ 10.5) Step 2: Set Criteria for Decision (α = 0.01) z = ± 2.58 Reject H0 Reject H0 z = z = 2.58

12
**Two-Tailed Hypothesis Test**

Question 4 Answer: Step 3: Compute sample statistic 𝜎 𝑀 = 𝜎 𝑛 = = =1.20 𝑧= 𝑀−𝜇 𝜎 𝑀 = 15.9− = =4.50

13
**Two-Tailed Hypothesis Test**

Question 4 Answer Step 4: Make a decision For a Two-tailed Test: zsample (4.50) > zcritical (2.58) Thus, we reject the null and note that texting has a significant effect on driving. If < zsample < 2.58, fail to reject H0 If zsample ≤ or zsample ≥ 2.58, reject H0

14
**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?)

15
**Factors that Influence a Hypothesis Test**

Question 5 Answer: True. A larger sample produces a smaller standard error, which leads to a larger z-score. For 𝑧= 𝑀−𝜇 𝜎 𝑀 , where 𝜎 𝑀 = 𝜎 𝑛 , as sample size (n) increases, standard error ( 𝜎 𝑀 ) decreases, which then increases z. Consequently, as z increases so does the probability of rejecting the null hypothesis.

16
**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?

17
**Factors that Influence a Hypothesis Test**

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 𝑀 = 𝜎 𝑛 = = =2 𝜎 𝑀 = 𝜎 𝑛 = = =4

18
**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)

19
**One-tailed Hypothesis Test**

Question 7 Answer: Step 1: State hypotheses H0: Endurance scores are not significantly higher with the sports drink. (µ ≤ 50) H1: Endurance scores are significantly higher with the sports drink. (µ > 50) Step 2: Set Criteria for Decision (α = 0.05) z = 1.65 Reject H0 z = 1.65

20
**One-tailed Hypothesis Test**

Question 7 Answer: Step 3: Compute sample statistic 𝜎 𝑀 = 𝜎 𝑛 = = =1.41 𝑧= 𝑀−𝜇 𝜎 𝑀 = 53− = =2.13

21
**One-tailed Hypothesis Test**

Question 7 Answer: Step 4: Make a decision For a One-tailed Test: zsample (2.13) > zcritical (1.65) Thus, we reject the null and note that the sports drink does raise endurance scores. If zsample ≤ 1.65, fail to reject H0 If zsample > 1.65, reject H0

22
**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.

23
**Effect Size and Cohen’s d**

Question 8 Answer: estimated Cohen’s d: 𝑚𝑒𝑎𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑀 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 − 𝜇 𝑛𝑜 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝜎 d = 47−40 8 = 7 8 =0.875 This is a large effect. 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.

24
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.

25
**Computing Power Question 9 Answer: Step #3: Calculate the z-score**

Step #1: Calculate standard error for sample 𝜎 𝑀 = 𝜎 𝑛 = = =4 Step #2: Locate Boundary of Critical Region 1.96 * 4 = 7.84 points Thus, the critical boundary corresponds to M = = Step #3: Calculate the z-score 𝑧= 𝑀−𝜇 𝜎 𝑀 = − = − =−0.54 z = 1.96, for α = 0.05 Any sample mean greater than falls in the critical region.

26
**Computing Power Step #4: Interpret Power of the Hypothesis Test**

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) 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.

27
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.

28
**Computing Power Question 10 Answer: Step #3: Calculate the z-score**

Step #1: Calculate standard error for sample 𝜎 𝑀 = 𝜎 𝑛 = = =4 Step #2: Locate Boundary of Critical Region 1.96 * 4 = 7.84 points Thus, the critical boundary corresponds to M = = Step #3: Calculate the z-score 𝑧= 𝑀−𝜇 𝜎 𝑀 = −92 4 = − =−1.04 z = 1.96, for α = 0.05 Any sample mean greater than falls in the critical region.

29
**Computing Power Question 10 Answer:**

Step #4: Interpret Power of the Hypothesis Test 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) 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.

30
**Frequently Asked Questions FAQs**

What is power? 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. Step #1: Calculate the standard error. Step #2: Locate the boundary of the critical region. Step #3: Calculate the z-score. Step #4: Find the probability. 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.

31
**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?

32
**Frequently Asked Questions FAQs**

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

33
**Frequently Asked Questions FAQs**

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

34
**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).

35
**Frequently Asked Questions FAQs**

Interpret Power of the Hypothesis Test 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) 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.

Similar presentations

OK

Chapter 8 Hypothesis Testing I. Significant Differences Hypothesis testing is designed to detect significant differences: differences that did not occur.

Chapter 8 Hypothesis Testing I. Significant Differences Hypothesis testing is designed to detect significant differences: differences that did not occur.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on oxygen cycle for class 9 Ppt on total parenteral nutrition contraindications Ppt on power situation in india Jit ppt on manufacturing Ppt on j&k tourism Ppt on interesting facts about planet mars Ppt on retail sales Download ppt on fundamental rights and duties of constitution Ppt on bookkeeping and accounting Ppt on eia report crude