# Rr for a level α 0 05 test is rr z obs ˆ p p 0

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RR for a level α = 0 . 05 test is: RR : z obs = ˆ p p 0 radicalbig p 0 (1 p 0 ) /n > 1 . 645 , where p 0 = 1 / 3 , ˆ p = Y/n , Y is the number of tasters that answered correctly. That is, RR : ˆ p > k, where k = p 0 + radicalbigg p 0 (1 p 0 ) n · z α Calculate β (0 . 5) , the Type II error rate when p a = 0 . 5 . Recall n = 100 . 10 Hypothesis Testing 44 Solution:
10 Hypothesis Testing 45 Q: Calculate the minimize sample size n to control β (0 . 5) at 0.01. 10 Hypothesis Testing 46
10 Hypothesis Testing 47 Type II error rate for Beer example (for p a = 0 . 5 ) 10 Hypothesis Testing 48 10.4 Test/confidence interval relationship Hypothesis Testing and Confidence Intervals Hypothesis testing has a close connection to confidence intervals (CI) in the sense that confidence intervals are often the complement of rejection regions The complement of the RR is sometimes called the acceptance region Consider the problem of two-tailed alternatives: H 0 : θ = θ 0 H a : θ negationslash = θ 0 Test statistic: Z = ˆ θ θ 0 σ ˆ θ Rejection region: RR = { Z : | Z | > z α/ 2 }
10 Hypothesis Testing 49 From the above, the Acceptance Region is: RR = braceleftbig | Z | ≤ z α/ 2 bracerightbig = braceleftBig θ 0 ˆ θ ± z α/ 2 σ ˆ θ bracerightBig . Notice that a 100 (1 α )% CI for θ is: ˆ θ ± z α/ 2 σ ˆ θ . Therefore, Reject H 0 if and only if θ 0 / CI In other words, testing for a level α two-tailed alternative is equivalent to checking if the hypothesized value of θ ( = θ 0 ) lies in the 100(1 α )% CI for θ . A similar relationship exists between one-sided alternative hypotheses and one-sided confidence intervals. 10 Hypothesis Testing 50 Test-confidence interval relationship–Example Refer to Example 10.2.5 (Male and Female Temperature Experience). Males: n m = 36 , σ 2 m = 4 . 0 , ¯ y m = 74 . 6 Females: n f = 40 , σ 2 f = 2 . 5 , ¯ y f = 76 . 5 Let μ m and μ f denote the mean temperature preference of males and females, respectively. Construct a 99% confidence interval for μ m μ f . Is the value μ m μ f = 0 contained in the confidence interval? Based on the interval, should we reject H 0 : μ m μ f = 0 ? 10 Hypothesis Testing 51 10.5 The p -value Observed significance level Often misunderstood P ( these data or more extreme ; H 0 is true ) Reject H 0 at level α p value < α Definition: the smallest level of significance α at which H 0 can be rejected. 10 Hypothesis Testing 52 Refer to Example 10.2.4 . H 0 : β = 10 versus H a : β < 10 ; z obs = 1 . 84 If α = 0 . 1 , z α = 1 . 28 , RR: z obs < 1 . 28 , conclusion: Reject H 0 If α = 0 . 05 , z α = 1 . 65 , RR: z obs < 1 . 65 , conclusion: Reject H 0 ...