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STAT 301 TA : Lisa Chung [email protected]
DISCUSSION 9
(Mar. 28. 2004)
- Test for mean when σ is known or large sample
Z = barX σ/√−(nμ)^0
(1 − α)100% Acceptance Region
- H 0 : μ = μ 0 vs.HA : μ 6 = μ 0
(μ 0 − z α 2 σ √ n , μ 0 + z α 2 σ √ n
- H 0 : μ = μ 0 vs. HA : μ > μ 0
(−∞, μ 0 + zα
σ √ n
- H 0 : μ = μ 0 vs.HA : μ < μ 0
(μ 0 − zα σ √ n
- Test for mean with unknown σ and small sample T= X¯−μ s/√n ∼^ tn−^1 If it’s reasonable to assume that the population is normal, then for small n, a 100(1-α)% confidence interval for μ is: ( X¯ − t α 2 ,n− 1 s √ n , X¯ + t α 2 ,n− 1 s √ n
with degree of freedom n − 1.
- Proportion X has a binomial distribution: X ∼ Bin(n, p) , where p is unknown. Let x be the observed value of X, and use the number x to make an inference about the unknown value of p. Point estimate is ˆp = x n
The confidence interval for π is
(ˆπ − zα/ 2
ˆπ(1 − πˆ) n
, ˆπ + zα/ 2
ˆπ(1 − πˆ) n
Office: 1335 MSC, 263-5948 1 Office Hour: Wed.1:00-2:00 and Thurs. 11:00-12:
STAT 301 TA : Lisa Chung [email protected]
(1 − α)100%Acceptanceregionf orH 0 : π = π 0
(π 0 − zα/ 2
π 0 (1 − π 0 ) n , π 0 + zα/ 2
π 0 (1 − π 0 ) n
Example 1. Consider the distribution of serum cholesterol levels for all 20- to 74-year-old males living in the United States. The mean of this population is 211 mg/dL, and the standard deviation is 46.0mg/dL. In a study of a subpopulation of such males who smoke and are hypertensive, it is assumed that the distribution of serum cholesterol levels is normally distributed with unknown mean μ , but with the same s.d. σ as the original population. a. Construct the hypothesis for testing whether the serum cholesterol level of smokers is equal to the known population mean. b. Sample mean of ¯x=217 mg/dL is observed from a sample of n=12 hypertensive smokers. Construct 95% C.I. for the true meanof this subpopulation. c. Calculate [-value of this sample. d. Check whether the null hypothesis is rejected at α=0.05. e. Determine 95% acceptance region and complementary rejection region for the null hypothesis.
Example 2. Two physicians are having a disagreement about the effectiveness of chicken soup in relieving common cold symptoms. While both agree that the number of symptomatoc days generally follows a normal distribution, one claims most colds last about a week, soup makes no difference, whereas the other argues that it does. a. Construct the hypothesis for testing. b. After treating a random sample of 16 patients with chicken soup, they get a mean number of symp- tomatic days ¯x=5.47 and standard deviation s=3.6 days. Test the hypothesis. c. One claims The sample size was too small.¨ There was not enough power to detect a statistically significant difference between μ=7 days and say μ=5.5 days, even if there was one present. Calculate the¨ minimum sample size required in order to achieve about 80% power of detecting such a genuine difference, if needed one actually exists.
Example 3. Proportion -posted exam
Office: 1335 MSC, 263-5948 2 Office Hour: Wed.1:00-2:00 and Thurs. 11:00-12:
