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Pennsylvania State University *

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Statistics

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Jun 7, 2024

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LAB 4.2 Statistics 200: Lab Activity for Section 4.2 Measuring Evidence with P-values - Learning objectives:  Recognize that a randomization distribution shows what is likely to happen by random chance if the null hypothesis is true  Use technology to create a randomization distribution  Interpret a p-value as the proportion of samples that would give a statistic as extreme as the observed sample, if the null hypothesis is true  Distinguish between one-tailed and two-tailed tests in finding p-values  Find a p-value from a randomization distribution Activity 1: Create a Randomization Distribution This activity is meant to have you participate in the creation of a randomization distribution to understand that it shows a distribution of sample statistics that were created assuming the null hypothesis is true. Every year in Punxsutawney, Pennsylvania, a famous groundhog, Phil, makes a prediction about the end of winter. If he comes out of his burrow and sees his shadow, he predicts six more weeks of winter. If he does not see his shadow, his prediction is early spring. In the ten years from 2011 to 2020, Phil made the following predictions: Year Prediction February temperature Prediction accuracy 2020 End of winter Above normal Correct 2019 End of winter Below normal Incorrect 2018 More winter Above normal Incorrect 2017 More winter Above normal Incorrect 2016 End of winter Above Normal Correct 2015 More winter Below normal Correct 2014 More winter Below normal Correct 2013 End of winter Above normal Correct 2012 More winter Below normal Correct 2011 End of winter Below normal Incorrect Are his predictions better than a random 50-50 chance? 1. What are the correct null and alternative hypotheses? Hint – what is p if his predictions are random? H 0 :p=0.5, 50% correct H a :p>0.5, more than 50% 2. What is p-hat when considering this example (round your answer to 4 decimal places, 0.xxxx) ? Sample proportion, phat=# of events/number of trials, 6/10=0.60 3. To create a randomization distribution, we must determine what the distribution of p-hat is if his prediction is random. We will use virtual coins, which have a true 50% chance of being heads. Go to justflipacoin.com a. 2/19/20 © - Pennsylvania State University

LAB 4.2 How many times will you need to flip this penny to create one sample statistic for the randomization distribution? 16 4. Now flip the penny that many times. Pretend that getting a heads with the coin is equivalent to Phil making a correct prediction. What was your p-hat? a. 5/16 5. Did your sample make as many correct predictions as Phil? a. no 6. Now we will go big and have StatKey create many many more statistics for our randomization distribution. Verify that the null is set to the correct proportion and edit the data as necessary. Now generate at least 5000 samples. a. Where is the randomization distribution centered? a. 0.6 b. Find the p-value. In StatKey, click on the correct tail (right or left), then click on the box along the x-axis. Enter in our original sample statistic (from part 2), correct to 3 decimal places, 0.xxx. What was the p-value? 0.167 c. Interpret the p-value in context: If Phil is choosing randomly, the chance that he would correctly predict at least 6 out of 10 years is ___1.67%_____. 2/19/20 © - Pennsylvania State University

LAB 4.2 (Continue on to next page) 7. In 2021 Phil correctly predicted that there will be more winter, bringing his record to 7 correct of 11. The randomization distribution for this scenario is below: Using the new data and randomization distribution, what is our sample p-hat and the approximate p-value for testing the same hypothesis we wrote in question 1? (Choose the correct answer from below) a. p-hat = 0.636, p-value = 0.276 b. p-hat = 0.636, p-value = 0.724 c. p-hat = 1, p-value = 0.007 d. p-hat = 1, p-value = 0.039 Activity 2: Where is The Middle? For the settings below, determine a) where the middle of the randomization will be and b) whether the hypothesis test is right-tailed, left-tailed, or two-tailed. Finally consider c) how to find the p-value. 1. To test H 0 : m = 45 vs H a : m > 45 using sample data with x = 53.7: a. Where will the randomization distribution be centered? a. Mu = 45 b. Is this a left-tail test, a right-tail test, or a two-tail test? a. Right tailed c. How can we find the p-value once we have the randomization distribution? a. Example answer: Find the proportion of randomization statistics that are to the left of the sample statistic of 2. (use this as a guide when answering questions 1.c, 2.c and 3.c). 2/19/20 © - Pennsylvania State University

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