1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Determining a sample size

  1. Nov 21, 2015 #1
    1. The problem statement, all variables and given/known data

    An investigator, interested in estimating a population mean, wants to be sure that the length of the 95% confidence interval does not exceed 5. What sample size should she use if σ = 18?


    3. The attempt at a solution
    the formula I found in my book is n = [(z_(α/2) σ)/E]^2

    z_(α/2) = z.025 = 1.96

    I am fairly certain if the length of the interval can't exceed 5, then 5 will be the max error so
    E = 5
    n = [1.96(18)/5]^2 = 49.8

    Am I doing this correctly?
     
  2. jcsd
  3. Nov 21, 2015 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Avoid using canned formulas; rather, work things out from first principles. So ask yourself: if ##X_1, X_2, \ldots, X_n## are iid random variables from the distribution ##N(\mu,\sigma^2)##, what is the distribution of the sample mean
    [tex] \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \: ? [/tex]
    Now you need to know how large to make ##n## in order to have
    [tex] P(-2.5 \leq \bar{X} - \mu \leq 2.5 ) = 0.95, [/tex]
    assuming that you know ##\sigma = 18##. At that point you are ready to state with absolute confidence the appropriate test to use. (And no, I will not tell you if you are correct or not!)
     
  4. Nov 21, 2015 #3
    since

    P(-z_(α/2) ≤ (X-μ)/(σ/sqrt(n)) ≤ z_(α/2)) = .95

    P(-z.025 ≤ (X-μ)/(18/sqrt(n)) ≤ z.025) = .95

    -1.96 ≤ sqrt(n)(X-μ)/18 ≤ 1.96

    -1.96(18)/(X-μ) ≤ sqrt(n) ≤ 1.96(18)/(X-μ)

    [-1.96(18)/(X-μ)]^2 ≤ n ≤ [1.96(18)/(X-μ)]^2

    n = [1.96(18)/(X-μ)]^2

    which is what I got before if E = (X-μ)
    I think it does, but I am now not sure if it is equal to 5
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Determining a sample size
  1. Small Sample Size (Replies: 0)

Loading...