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Math Help - Confidence Interval

  1. #1
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    Confidence Interval

    A math expert kept track of how many questions he answered over one week: 10, 20, 30, 20, 20, 70, 40 What is the 95% confidence interval for the mean number of questions answered each day? (to the nearest question)

    a-(10, 70)
    b-(12, 48)
    c-(26, 34)
    d-(15, 45)
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  2. #2
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    Quote Originally Posted by magentarita View Post
    A math expert kept track of how many questions he answered over one week: 10, 20, 30, 20, 20, 70, 40 What is the 95% confidence interval for the mean number of questions answered each day? (to the nearest question)

    a-(10, 70)
    b-(12, 48)
    c-(26, 34)
    d-(15, 45)
    Assuming that the number of questions answered is a gaussian random variable \mathcal{N}(m,\sigma^2), you have to use the confidence interval for the Student distribution: you know that T=\sqrt{n}\frac{\overline{X}-m}{\overline{\sigma}} is distributed according to a Student distribution with n-1=7-1=6 degrees of freedom, where n is the size of the sample, and \overline{X} and \overline{\sigma} are the empirical mean and standard deviation.
    By looking up in tables (or such as), you find a confidence interval for T, i.e. a number t such that P(-t<T<t)=0.95 or equivalently P(T>t)=0.025. This should be t\simeq 2.45.
    This gives you a confidence interval for m: P(\overline{X}-\frac{t\overline{\sigma}}{\sqrt{n}}<m<\overline{X}  +\frac{t\overline{\sigma}}{\sqrt{n}})=0.95.
    All that remains is to compute \overline{X}, \overline{\sigma} from the data to deduce the bounds of this interval.
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