Statistics

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The probability that a radish seed will germinate is 0.6. Estimate the probability that of 130 randomly selected seeds, exactly 90 will germinate.

Note: I keep getting the answer wrong.

  • Statistics -

    This is a binomial distribution with parameters N=130, p=0.6, r=90
    so
    P(X=90)=C(N,r)p^r(1-p)^(N-r)
    =C(130,90)*0.6^90*0.4^40
    =5.3347282074*10^33*1.0804695562359849*10^-20 * 1.2089258196146345*10^-16
    =0.00697

  • Statistics -

    But, I'm suppose to use the normal distribution approximation to the binomial distribution.

  • Statistics -

    Glad that you mentioned that an approximation is required. The question asks for exactly 90 seeds, which is discrete.

    Here's how I would proceed to approximate a discrete random variable from a continuous distribution.

    "Exactly 90" is approximately equal to the random variable X=89.5 to 90.5.
    We can generally approximate a binomial distribution by a normal distribution when np>5. Here np=130*0.6=78 > 5, so approximation will be reasonable.

    The equivalent μ=np=78
    σ
    =√(npq)
    =√(130*.6*(1-0.6))
    =√(31.2)
    =5.585696

    Z(X=90.5)=(90.5-78)/5.585696=2.237859
    Z(X=89.5)=(89.5-78)/5.585696=2.058830

    Here, we are dealing with a small difference of two probabilities, so normal tables (on paper) by interpolation may or may not be adequate. I suggest you use a calculator with a Z function, or use a normal distribution calculator online, such as:
    http://stattrek.com/online-calculator/normal.aspx

    Using 5 digits, I get
    P(X=90.5)=0.98738, and
    P(X=89.5)=0.98024
    (remember to use the respective Z-values when looking up probabilities)

    Thus
    P(89.5≤X≤90.5)
    =0.98738-0.98024
    =0.00714
    (approximated using normal distribution)

    (compared with value of 0.00697 using the binomial distribution).

  • Statistics -

    No wonder I keep getting the wrong answer. Thank you so much for helping me how to get the answer. Now I know how to do the next problem which is similar to this problem.

  • Statistics :) -

    You're most welcome.
    I am glad things are working out.
    Post if you have difficulties.

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