text: Question
  
  An opinion poll based on a sample of 50 subjects estimated p, the proportion of the...
  
  An opinion poll based on a sample of 50 subjects estimated p, the proportion of the population in favor of the proposition, as 0.72.
  (i) Estimate the true proportion, θ, with a 95% conﬁdence interval. State any assumptions you may have to make in answering this question.
  
  (ii) If the true population proportion is suspected to be θ =0.8, and the estimate from an opinion poll is to be determined to within ±0.05 with 95% conﬁdence, how
  many people, n, should be sampled?
  
  (iii) If the proportion is to be estimated to within the same margin of ±0.05, but with 90% conﬁdence, what is the value of n required? Comment on the eﬀect that
  reducing the conﬁdence level has on the sample size, n required to achieve the desired precision.
  
  Given: Sample size = n = 50, sample proportion = p = 0.72
  
  (i) The random variable has a binomial distribution with probability of success as P. As the sample size is sufficiently large, the normal approximation to binomial distribution can be used.
  
  Thus, the required 95% confidence interval is { p - 1.96*sqrt(p*(1-p)/n) , p +1.96*sqrt(p*(1-p)/n) }
  
  i.e. {0.72 - 1.96*sqrt(0.72*0.28/50), 0.72 + 1.96*sqrt(0.72*0.28/50)}
  
  i.e.(0.72-0.1245 , 0.72+0.1245), i.e. (0.5955,0.8445)
  
  (ii) According to the given condition,
  
  1.96*sqrt(p*(1-p)/n) <= 0.05
  
  i.e. 1.96*sqrt(0.8*0.2/n) <= 0.05
  
  i.e. n >= 0.16*(1.96/0.05)^2
  
  i.e. n >= 245.8624
  
  Thus the sample should be of size at least 246.
  
  (iii) 1.64*sqrt(p*(1-p)/n) <= 0.05
  
  i.e. n >= 0.16*(1.64/0.05)^2
  
  i.e. n >= 172.1344
  
  Thus the sample should be of size at least 173.
  
  Thus, for lower confidence, a smaller sample size suffices.
  
  Hope this helps!
  
  Earn Coins
  
  Coins can be redeemed for fabulous gifts.
metadata: {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}