Binomial Distribution and Hypothesis Testing in Statistics: A Practice Problem, Exams of Statistics

Download Binomial Distribution and Hypothesis Testing in Statistics: A Practice Problem and more Exams Statistics in PDF only on Docsity! STATISTICS 301 TA: Perla E. Reyes DISCUSSION 10 Pag. 1 Review 1. Proportion Y a binary random variable -experiment trial- with 2 posible outcomes, Y = { 1 Success with probability: π 0 Failure with probability: 1− π where π is unknown. Let X be the number of successes obtained from repeat the experiment n independent times, X has a binomial distribution: X ∼ Bin(n, π), X can be used to make inferences about the unknown value of π. • Point estimate π̂ = X n • Sampling Distribution Recall Assuming nπ ≥ 15 and n(1− π) ≥ 15 X ∼ N(nπ, √ nπ(1− π)) approximately Therefore, dividing everything by n... π̂ = X n ∼ N ( π, √ π(1− π) n ) approximately • Standard error of π̂ s.e.(π̂) = √ π(1− π) n • The (1− α)100% confidence interval for π is( π̂ − zα/2 √ π̂(1− π̂) n , π̂ + zα/2 √ π̂(1− π̂) n ) • (1− α)100% Acceptance region for H0 : π = π0 vs. HA : π 6= π0( π0 − zα/2 √ π0(1− π0) n , π0 + zα/2 √ π0(1− π0) n ) • Test Statistic Z = π̂ − π0√ π0(1−π0) n ∼ N(0, 1) • p-value for H0 : π = π0 vs. HA : π 6= π0 p− value = 2P Z ≥ |π̂ − π0|√ π0(1−π0) n  reyes@stat.wisc.edu www.stat.wisc.edu/∼reyes/ B248MSC, TTh 11:30-12:30 STATISTICS 301 TA: Perla E. Reyes DISCUSSION 10 Pag. 2 Practice Problem 1. A standard insecticide is used to control a particular insect pest in soy beans. The probability with which this insecticide eliminates the insect from an individual plant infested with the insect is 0.7. A new insecticide and it is desired to determine if the new insecticide’s perform differes from the standard’s. An experiment is to be conducted in which 500 randomly selected plants (known to be infested with the insect) are sprayed with the new insecticide. After a fixed period of time, the plant for which the new insecticide eliminate the insect will be counted. (a) State symbolically the null and alternative hypotheses. (b) Determine the acceptance region for an α = 0.05 significance level. (c) When the experiment was actually carried out, it was found that new insecticide elimi- nates insect in 395 plants. Find the 99% C.I. for π̂. (d) Compute the p− value for this sample. (e) State the conclusion in words. reyes@stat.wisc.edu www.stat.wisc.edu/∼reyes/ B248MSC, TTh 11:30-12:30