Confidence Intervals for Means and Proportions in Statistics, Study notes of Data Analysis & Statistical Methods

Download Confidence Intervals for Means and Proportions in Statistics and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! STATISTICS 571 TA: Perla Reyes DISCUSSION 6 Review 1. Confidence Interval for µ. (a) Suppose X ∼ N(µ, σ2), and X1, X2, ..., Xn is a random sample from this distribution. i. If σ is known, the (1− α) C.I. for µ is x̄− Zα/2 σ√ n ≤ µ ≤ x̄ + Zα/2 σ√ n , where Zα/2 is such that P (Z ≥ Zα/2) = α/2. ii. If σ is unknown, then the (1− α) C.I. for µ is x̄− Tn−1,α/2 s√ n ≤ µ ≤ x̄ + Tn−1,α/2 s√ n , where Tn−1,α/2 is such that P (Tn−1 ≥ Tn−1,α/2) = α/2. (b) Suppose the distribution of X is unknown, but the sample size n is large. Then if E(X)=µ, Var(X)=σ2. i. If σ is known, the (1− α) C.I. for µ is x̄− Zα/2 σ√ n ≤ µ ≤ x̄ + Zα/2 σ√ n , where Zα/2 is such that P (Z ≥ Zα/2) = α/2. ii. If σ is unknown, then the (1− α) C.I. for µ is x̄− Zα/2 s√ n ≤ µ ≤ x̄ + Zα/2 s√ n , where Zα/2 is such that P (Z ≥ Zα/2) = α/2. 2. Confidence Interval for p. If X is distributed as B(n, p), and np̂ > 5 and n(1− p̂) > 5, then the (1− α) C.I. for p is p̂− Zα/2 √ p̂(1− p̂) n ≤ p ≤ p̂ + Zα/2 √ p̂(1− p̂) n Note that in hypothesis testing, we use the hypothesized value p0 to calculate p-value, but we use p̂ in computing C.I. for p. 3. Relation between C.I. and two-sided hypothesis testing (not for binomial distribu- tion): (a) If the (1−α) C.I. for µ contains the hypothesized value µ0, then we do not reject the null hypothesis H0 at level α. (b) If the (1− α) C.I. for µ does not contains the hypothesized value µ0, then we reject the null hypothesis H0 at level α. email: reyes@stat.wisc.edu 1 Office: 248 MSC M2:30-3:30 R3:30-4:30