Question

Mr. Allen, the candidate for political Party A will run against Mr. Baker of Party B for office. Past races between these parties for this office were always close, and it seems that this one will be no exception— Party A candidates always have gotten between 40% and 60% of the vote and have won about half of the elections. Allen needs to know, defining ? as the proportion of voters favoring him today, whether ?0: ? = 0.5 ?? ?1: ? > 0.5 Is true. A random sample of ? voters is taken with ? voters favoring Allen. The population is large, and it is justifiable to assume that ?~???(?, ?), binomially distributed. The estimate ?̂ = ?/? will be used. Which of the three outcomes, all having the same p-value, should be most encouraging to candidate Allen? a. ? = 15, ? = 20, ?̂ = 0.75 b. ? = 115, ? = 200, ?̂ = 0.575 c. ? = 1,046, ? = 2,000, ?̂ = 0.523 Facts: The p-values are all about 0.021, with values of ? = (?̂−0.5) ( ? √? ) , ?ℎ??? ? = 0.5 being 2.03, 2.05, and 2.03. Standard 95% confidence intervals are (0.560, 0.940), (0.506, 0.644), and (0.501, 0.545), respectively.

Answer 1

Mr. Allen, the candidate for political Party A will run against Mr. Baker of Party B for office. Past races between these parties for this office were always close, and it seems that this one will be no exception— Party A candidates always have gotten between 40% and 60% of the vote and have won about half of the elections. Allen needs to know, defining ? as the proportion of voters favoring him today, whether ?0: ? = 0.5 ?? ?1: ? > 0.5 Is true. A random sample of ? voters is taken with ? voters favoring Allen. The population is large, and it is justifiable to assume that ?~???(?, ?), binomially distributed. The estimate ?̂ = ?/? will be used. Which of the three outcomes, all having the same p-value, should be most encouraging to candidate Allen? a. ? = 15, ? = 20, ?̂ = 0.75 b. ? = 115, ? = 200, ?̂ = 0.575 c. ? = 1,046, ? = 2,000, ?̂ = 0.523 Facts: The p-values are all about 0.021, with values of ? = (?̂−0.5) ( ? √? ) , ?ℎ??? ? = 0.5 being 2.03, 2.05, and 2.03. Standard 95% confidence intervals are (0.560, 0.940), (0.506, 0.644), and (0.501, 0.545), respectively.