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100BHWS1 - STAT 100B HWI Solution Problem 1 Suppose in a...

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STAT 100B HWI Solution Problem 1: Suppose in a population of voters, the proportion of those who support a candidate is p . Suppose we get a random sample of 1000 people, and within this sample, the number of people who support this candidate is 530. (1) Calculate the 95% confidence interval for p . Calculate the margin of error. A: ˆ p = 530 / 1000 = . 53. The standard error of ˆ p is p ˆ p (1 - ˆ p ) /n = p . 53 × (1 - . 53) / 1000 = . 0158. The 95% confidence interval is [ . 53 - 2 × . 0158 , . 53 + 2 × . 0158] = [ . 498 , . 562]. The margin of error is . 0158 × 2 = . 0316. (2) Interpret the confidence level of 95% (so that your roommate can understand you). A: Suppose we repeatedly sample 1000 people, and calculate the interval, the interval will change from time to time. In the long run, 95% of times the interval covers the true value of p . Problem 2: Continue from Problem 1. Suppose we want to test the hypotheses: H 0 : p = . 5 versus H 1 : p > . 5. (1) Calculate the p-value. A: The Z -score is (ˆ p - p 0 ) / p p 0 (1 - p 0 ) /n = ( . 53 - . 5) / p . 5 × . 5 / 1000 = 1 . 90. The p-value is .0287. (2) Suppose the significance level is 2 . 5%. Should we reject H 0 ? A: We accept H 0 because . 0287 > 2 . 5%. Problem 3: Suppose we want to test whether a die is fair or not. In particular, we suspect that
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