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Confidence Intervals for Population Proportion

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					         Confidence Intervals for Population
                     Proportion
                                  Section 9.3
                                Objectives
• Obtain a point estimate of the mean
• Construct and interpret a confidence interval for the
  population proportion
• Determine the sample size necessary for estimating a
  population proportion within a specified margin of error.
                               Information
• Most frequently reported type of problem.
• Point estimate = p hat = x/n
• Formula for confidence interval
    • p hat ± zα/2 (square root (p hat(1- phat)/n)
    • Remember: nphat(1-phat)≥10 and n≤0.05*N
    • Note: we have used phat instead of p to approximate the value of p, like
      we use s to approximate σ
• Formula for sampling distribution of phat
    • σphat = square root(p(1-p)/n)
                                Procedure
•    Find p hat
•    Find zα/2
•    Find n = sample size
•    Plug in information in formula.
•    We are [confidence level] confident that the proportion of
     [whatever] is between [lower bound] and [upper bound].
                             #20 page 442
•    A study of 74 patients with ulcers was conducted in which
     they were prescribed 40 mg of Pepcid. After 8 weeks, 58
     reported ulcer healing.
    – Obtain a point estimate
    – Verify requirements
    – Construct and interpret a 99% confidence interval for the
      proportion of patients with ulcers receiving Pepcid who will
      report ulcer healing.
                         #20 continued
•   Point estimate = phat = 58/74
•   74(58/74(1-58/74)= 12.54 >10 and 74≤0.05*N (74/.05 <
    N)
•   n = 74, x = 58, so phat = 58/74
•   Alpha = 1 - .99 = .01
    α/2 = .005
     zα/2 = z.005 = invnorm(1 - .005) = 2.58
•   58/74 ± 2.58 (square root (58/74(1 – 58/74)/74)
                         #20 continued
•   58/74 ± 2.58 (square root (58/74(1 – 58/74)/74)
•   58/74 ± .12347
•   0.660 < p < 0.907
•   We are 99% confident that the proportion of patients with
    ulcers receiving Pepcid who will report ulcer healing is
    between 0.660 and 0.907.
            Determining the Sample Size
•
  n = phat(1-phat)(zα/2/E)2
•
  If phat is unavailable, then use
  n = 0.25(zα/2/E)2
• E = margin of error in decimal form
                         #25 page 443
• A researcher wishes to estimate the proportion of adults
  who have high-speed internet access. What size sample
  should be obtained if she wishes the estimate to be within
  0.03 with 99% confidence if she uses a 2007 estimate of
  0.69 obtained from Neilsen Net Ratings?
•
  n = 0.69(1-0.69)(2.58/.03)2 = 1583

				
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