EPY 8214: Statistical Analysis and Hypothesis Testing – Exam 1 EPY 8214 Exam 1 Take - home Exam Spring 2013 Directions 1. Exam is due in one week. Submit to the assignment dropbox by 11:59 PM on March 23 rd , 2013. Late submissions will not be accepted. 2. Record your answers to all questions directly on the test. 3. Please show all formulas that you use as well as your computations. You may attach additional sheets of paper if necessary. 4. Please feel free to consult your notes, the workbook (text) or any other statistical reference except another person. This is not a collaborative exam . 5. Read the pledge below and sign your paper below Pledge On my honor as a student, I have neither given or received aid on this examination _______________________________________________ Signature 1. For a sample with M = 60 and s = 6, find the value corresponding to each of the following z scores. a. z = +1.50 b. z = - 0.50 c. z = +2.00 d. z = - 1/3 Answer: 2. Describe what happens to the mean, the standard deviation and the shape of the distribution when all of the scores are transformed into z scores. Answer: When all of the scores in a distribution are transformed into z - scores, the following changes occur to the mean, the standard deviation, and the shape of the distribution: 1. Mean (M): • New Mean : The mean of the z - scores will always be 0 . This is because a z - score represents how many standard deviations a score is away from the mean. When transforming scores to z - scores, the mean is shifted to 0. • Reason : The formula for z - scores is z=X−Msz = \ frac{X - M}{s}, where MM is the original mean. When all scores are transformed, the new mean becomes 0 because each original score is adjusted based on how far it is from the original mean. 2. Standard Deviation (s): • New Standard Deviation : The standard deviation of the z - scores will always be 1 . This is because the z - score transformation divides each score by the standard deviation of the original distribution. • Reason : In the formula for a z - score, XX is subtracted by the mean, and then divided by the standard deviation. This division normalizes the spread of the scores, so the new standard deviation becomes 1. 3. Shape of the Distribution: • Shape : The shape of the distribution does not change when scores are transformed into z - scores. • Reason : The transformation is linear (a shift and a scale), so the relative positioning of the scores in relation to each other remains the same. Whether the original distribution is normal, skewed, or uniform, the transformed z - scores will retain the same shape as the original distribution. The only difference is that it will now be centered around 0 and have a standard deviation of 1. Summary: • Mean : The mean becomes 0. • Standard Deviation : The standard deviation becomes 1. • Shape : The shape of the distribution remains unchanged. 3. Which of the following will increase the power of a statistical test and why ? a. Change from .05 to .01 b. Change from a one - tailed test to a two - tailed test c. Change the sample size from n = 100 to n = 25 d. None of the other options will increase power. Answer: To determine which of the following will increase the power of a statistical test, we need to understand what power is and how it is influenced. Power of a statistical test refers to the probability that the test will correctly reject the null hypothesis when it is false. In other words, it is the likelihood of finding a true effect when there is one. Power is influenced by several factors, including: • Alpha level (significance level, usually denoted by α \ alpha) • Sample size (n) • Effect size • Directionality of the test (one - tailed or two - tailed) Now, let's analyze each option: a. Change from .05 to .01 • Explanation : Lowering the alpha level from 0.05 to 0.01 means you're becoming more stringent in your criteria for rejecting the null hypothesis (i.e., you're requiring stronger evidence to reject H0H_0). • Effect on Power : This would decrease power , because the stricter alpha level reduces the chance of rejecting the null hypothesis, even when there is a real effect. • Conclusion : This does not increase power. b. Change from a one - tailed test to a two - tailed test • Explanation : A one - tailed test only considers one direction of the effect (e.g., testing if a mean is greater than a certain value), while a two - tailed test considers both directions (e.g., testing if the mean is either greater or less than a certain value). • Effect on Power : A two - tailed test is typically less powerful than a one - tailed test because the alpha level is divided between the two tails of the distribution, making it harder to reject the null hypothesis. • Conclusion : This does not increase power. It actually decreases power because you're considering both directions. c. Change the sample size from n = 100 to n = 25 • Explanation : Increasing the sample size allows for more precise estimates of the population parameters, reducing standard errors and making it easier to detect a true effect. • Effect on Power : Increasing the sample size (e.g., from n=25n = 25 to n=100n = 100) increases power because the larger sample size leads to a more reliable test and a greater likelihood of detecting a true effect if it exists. • Conclusion : This does increase power. d. None of the other options will increase power. • Based on the explanations above, we see that option (c) (increasing sample size) does increase power. Therefore, this statement is false . Final Answer: The correct answer is (c) Change the sample size from n = 100 to n = 25 . Increasing the sample size increases the power of the statistical test. 4. If a treatment has a very small effect, then a hypothesis test is likely to: a. Result in a Type I error b. Result in a Type II error c. Correctly reject the null hypothesis d. Correctly fail to reject the null hypothesis Answer: When a treatment has a very small effect , it means that the actual difference between groups (or the effect size) is minimal. This can impact the outcome of a hypothesis test. Let's break down each of the options: