Homework 5.docx - Homework 5 The homework is due on Friday Please submit your work in class 1 You are given a coin with unknown success probability pp

Homework 5.docx - Homework 5 The homework is due on Friday...

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Homework 5 The homework is due on Friday, 4/27/2018. Please submit your work in class. 1. You are given a coin with unknown success probability p p . You repeatedly toss the coin for 100 times. When head shows up, you record value 1; otherwise you record value 0. The sample mean of all your coin-tossing results is 0.55. You would like to test whether the coin is fair at significance level 5\%5% . Answer the following questions manually using a z z -score table first, then verify your answers using R example script on Blackboard. (1) What is the null hypothesis of the test? (2) What is the critical value of the test? (3) What test statistic should you use in the test? What distribution does it follow? What is the actual value of the test statistic? (4) What is the P -value of the test statistic? Can we reject the null hypothesis? (5) If actually p=0.45 p =0.45 , what is the power of the test? 2. Let X X represent BMI (body mass index) of a person, and \text{E}\left(X\right)E( X ) be the average BMI of the whole population. Using data provided in the Body Data.csv file, answer the following questions: (1) What is the sample size? (2) Please find the sample mean and sample standard deviation. (3) Please find the 95\%95% confidence interval estimate of \text{E}\left(X\right)E( X ) . Next you would like to test the claim that \text{E}\left(X\right)=27E( X )=27 at significance level 5\%5% . Answer the following questions manually using a z z - score table first, then verify your answers using R example script on Blackboard. (4) What is the critical value of the test? (5) What test statistic should you use in the test? What distribution does it follow? What is the actual value of the test statistic? (6) What is the P -value of the test statistic? Can we reject the null hypothesis? (7) If actually \text{E}\left(X\right)=25E( X )=25 , what is the power of the test? 3. Let X X represent height of a female, and \text{E}\left(X\right)E( X ) be the average height of the whole female population. Using the height data of the first 20 females in the Body Data.csv file, answer the following questions: (1) What is the sample size? Is it a large sample? (2) Please find the sample mean and sample standard deviation. (3) Please find the 95\%95% confidence interval estimate of \text{E}\left(X\right)E( X ) . (4) (bonus question) Please find the 95\%95% confidence interval estimate
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of \text{E}\left(X\right)E( X ) using bootstrap. Next you would like to test the claim that \text{E}\left(X\right)=160E( X )=160 at significance level 5\%5% . Answer the following questions using R example script on Blackboard. (5) What is the critical value of the test? (6) What test statistic should you use in the test? What distribution does it follow? (7) What is the actual value of the test statistic? Can we reject the null hypothesis? 4. Let X X represent height of a female, and \sigma^{2} σ 2 be the variance of height of the whole female population. Using the height data of the first 20 females in the Body Data.csv file, answer the following questions: (1) Please find the sample variance.
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