# Chapter 1: Introduction to Statistics - ECU Blogs COURSE: JUST 3900 TIPS FOR APLIA Chapter 8: Hypothesis Testing Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick Key Terms: Dont 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) Formulas Standard Error of M: z-Score Formula: Cohens d:

estimated Cohens d: 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? 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. Alpha Level and the Critical Region Question 2: If the alpha level is decreased from = 0.01 to = 0.001, then the boundaries for the critical region move farther away from the center of the distribution. (True or False?) 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. Possible Outcomes of a Hypothesis Test Question 3: Define Type 1 and Type II Error.

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 doesnt. Type I error = false (+) = Alpha () = 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 = () A Type II error is likely to occur when a treatment effect is very small. Two-Tailed Hypothesis Test Question 4: After years of teaching drivers education, an instructor knows that students hit an average of = 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 = 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 = 0.01. Two-Tailed Hypothesis Test

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

z = 2.58 Reject H0 Reject H0 z = - 2.58 z = 2.58 Two-Tailed Hypothesis Test Question 4 Answer:

Step 3: Compute sample statistic Two-Tailed Hypothesis Test Question 4 Answer Step 4: Make a decision For a Two-tailed Test: If -2.58 < zsample < 2.58, fail to reject H0 If zsample -2.58 or zsample 2.58, reject H0 zsample (4.50) > zcritical (2.58) Thus, we reject the null and note that texting has a significant effect on driving. 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?) 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 , where , as sample size (n) increases,

standard error () decreases, which then increases z. Consequently, as z increases so does the probability of rejecting the null hypothesis. 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 = 2 or = 10?

Factors that Influence a Hypothesis Test Question 6 answer: = 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. 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 = 50 with a standard deviation of = 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, = 0.05) One-tailed Hypothesis Test

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

Reject H0 z = 1.65 One-tailed Hypothesis Test Question 7 Answer: Step 3: Compute sample statistic One-tailed Hypothesis Test

Question 7 Answer: Step 4: Make a decision For a One-tailed Test: If zsample 1.65, fail to reject H0 If zsample > 1.65, reject H0 zsample (2.13) > zcritical (1.65)

Thus, we reject the null and note that the sports drink does raise endurance scores. Effect Size and Cohens d Question 8: A researcher selects a sample from a population with = 40 and = 8. A treatment is administered to the sample and, after treatment, the sample mean is found to be M = 47. Compute Cohens d to measure the size of the treatment effect.

Effect Size and Cohens d Question 8 Answer: estimated Cohens d: d= 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. 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 = 100 and a standard deviation of = 20. The researcher expects a 10-point treatment effect and plans to use a two-tailed hypothesis test with = 0.05. Compute the power of the test if the researcher uses a sample of n = 25 individuals. Computing Power

Question 9 Answer: Step #1: Calculate standard error for sample Step #2: Locate Boundary of Critical Region z = 1.96, for = 0.05 1.96 * 4 = 7.84 points Thus, the critical boundary corresponds to M = 100 + 7.84 = 107.84.

Any sample mean greater than 107.84 falls in the critical region. Step #3: Calculate the z-score Computing Power Step #4: Interpret Power of the Hypothesis Test Find probability associated with a z-score > - 0.54 Look this probability up as the proportion in the body of the normal distribution (column B in your textbook)

p(z > -0.54) = 0.7054 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. 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 = 80 and a standard deviation of = 20. The researcher expects a 12-point treatment effect and plans to use a two-tailed hypothesis test with = 0.05. Compute the power of the

test if the researcher uses a sample of n = 25 individuals. Computing Power Question 10 Answer: Step #1: Calculate standard error for sample Step #2: Locate Boundary of Critical Region

z = 1.96, for = 0.05 1.96 * 4 = 7.84 points Thus, the critical boundary corresponds to M = 80 + 7.84 = 87.84. Any sample mean greater than 87.84 falls in the critical region. Step #3: Calculate the z-score Computing Power

Question 10 Answer: Step #4: Interpret Power of the Hypothesis Test Find probability associated with a z-score > - 1.04 Look this probability up as the proportion in the body of the normal distribution (column B in your textbook) p(z > -1.04) = 0.8508 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. 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. is the probability of a type II error (false negative). Therefore, power is 1 . 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, lets go through each step. Frequently Asked Questions FAQs The previous slide was based upon a study from your book with = 80, = 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? Frequently Asked Questions FAQs Step #1: Calculate standard error for sample In this step, we work from the populations standard deviation () and the sample size (n) 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 ( =.05) 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 ( = 88).

Frequently Asked Questions FAQs Interpret Power of the Hypothesis Test Find probability associated with a z-score > - 2.04 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.