EPY 8214: Statistical Analysis and Hypothesis Testing � Exam 1

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EPY 8214: Statistical Analysis and Hypothesis Testing–Exam 1EPY 8214Exam 1 Take-home Exam Spring 2013Directions1.Exam is due in one week. Submit to the assignment dropbox by 11:59 PM onMarch 23rd,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 attachadditional sheets of paper if necessary.4.Please feel free to consult your notes, the workbook (text) or any other statistical referenceexcept another person. This is not a collaborative exam.5.Read the pledge below and sign your paper belowPledgeOn my honor as a student, I have neither given or received aid onthis examination_______________________________________________Signature

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1.For a sample with M = 60 and s = 6, find the value corresponding to each of thefollowing z scores.a.z = +1.50b.z =-0.50c.z = +2.00d.z =-1/3Answer:

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2.Describe what happens to the mean, the standard deviation and the shape of thedistribution 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 changesoccur to the mean, the standard deviation, and the shape of the distribution:

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1. Mean (M):•New Mean: The mean of the z-scores will always be0. This is because a z-scorerepresents how many standard deviations a score is away from the mean. Whentransforming 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 theoriginal mean. When all scores are transformed, the new mean becomes 0 because eachoriginal 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 be1. Thisis because the z-score transformation divides each score by the standard deviation of theoriginal distribution.•Reason: In the formula for a z-score, XX is subtracted by the mean, and then divided bythe standard deviation. This division normalizes the spread of the scores, so the newstandard 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 ofthe scores in relation to each other remains the same. Whether the original distribution isnormal, skewed, or uniform, the transformed z-scores will retain the same shape as theoriginal distribution. The only difference is that it will now be centered around 0 andhave 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 testand why?a.Change from .05 to .01b.Change from aone-tailed test to a two-tailed test

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c.Change the sample size from n = 100 to n = 25d.None of the other options will increase power.Answer:To determine which of the following will increase thepowerof a statistical test, we need tounderstand what power is and how it is influenced.Power of a statistical test refers to the probability that the test will correctly reject the nullhypothesis when it is false. In other words, it is the likelihood of finding a true effect whenthere 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 from0.05 to 0.01means you're becoming morestringent in your criteria for rejecting the null hypothesis (i.e., you're requiring strongerevidence to reject H0H_0).•Effect on Power:This would decrease power, because the stricter alpha level reducesthe chance of rejecting the null hypothesis, even when there is a real effect.•Conclusion: Thisdoes notincrease power.b. Change from a one-tailed test to a two-tailed test•Explanation: Aone-tailed testonly considers one direction of the effect (e.g., testing ifa mean is greater than a certain value), while atwo-tailed testconsiders both directions(e.g., testing if the mean is either greater or less than a certain value).•Effect on Power: Atwo-tailed testis typicallyless powerfulthan a one-tailed testbecause the alpha level is divided between the two tails of the distribution, making itharder to reject the null hypothesis.

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•Conclusion: Thisdoes notincrease power. It actuallydecreases powerbecause you'reconsidering both directions.c. Change the sample size from n = 100 to n = 25•Explanation: Increasing the sample size allows for more precise estimates of thepopulation parameters, reducing standard errors and making it easier to detect a trueeffect.•Effect on Power:Increasing the sample size(e.g., from n=25n = 25 to n=100n = 100)increases powerbecause the larger sample size leads to a more reliable test and a greaterlikelihood of detecting a true effect if it exists.•Conclusion: Thisdoesincrease power.d. None of the other options will increase power.•Based on the explanations above, we see thatoption (c)(increasing sample size)doesincrease power. Therefore, this statement isfalse.Final Answer:The correct answer is(c) Change the samplesize from n = 100 to n = 25. Increasing thesample 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 errorb.Result in a Type II errorc.Correctly reject the null hypothesisd.Correctly fail to reject the null hypothesisAnswer:When a treatment has avery small effect, it means that the actual difference between groups(or the effect size) is minimal. This can impact the outcome of ahypothesis test. Let's break down eachof the options:

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