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Math Help - Statistic practice question help

  1. #1
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    Talking Statistic practice question help

    I have difficulties in solving this question. if possible, please help me out! thanks!!

    The time in minutes that an adult male takes to burn off 5 grams of fat if walking at a steady pace is normally distributed with a mean, μ, equal to 25 minutes and a standard deviation, σ, equal to 5 minutes. Remember to draw diagrams.
    a. What proportion of adult males walking at a steady pace take between 26 minutes and 29 minutes to burn off 5 grams of fat?
    b. In a random sample of 30 adult males walking at a steady pace what is the probability that the mean time to burn off 5 grams of fat is less than 23 minutes?
    c. An adult male is classified as having a slow metabolism if the time taken to burn off 5 grams of fat while walking at a steady pace is in the top 5% of times. What is the cut‐off time for an adult male walking at a steady pace to be classified as having a slow metabolism? Give your answer to the nearest minute.
    d. A sport physiologist suggests that 10% of adult males walking at a steady pace can burn off 5 grams of fat in less than 20 minutes. What is the probability that in a random sample of 100 adult males walking at a steady pace less than 8 of them can burn off 5 grams of fat in less than 20 minutes?
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  2. #2
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    Re: Statistic practice question help

    You have a random variable with normal distribution X\sim {\cal N}(25,5).
    Its density function is f_X(t)=\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(t-\mu)^2}{2 \sigma^2}}=\frac{1}{5 \sqrt{2\pi}} e^{-\frac{(t-25)^2}{50}}.

    Recall that P(X<x)=F_X(x)=\int\limits_{-\infty}^{x}f(t)dt and P(a<X<b)=\int\limits_{a}^{b}f(t)dt=F(b)-F(a).
    And you should know that any normally distributed function can be transformed to have STANDARD normal distribution: X\sim {\cal N}(\mu,\sigma) \Rightarrow Z=\frac{X-\mu}{\sigma}\sim{\cal N}(0,1) meaning that F_X(x)=F_Z\left(\frac{x-\mu}{\sigma}\right) which is great since values of standard normal distribution function are calculated and given in a table so you never have to solve the integrals above.

    Here with X\sim {\cal N}(25,5) using transformation as above will give you Z=\frac{X-25}{5}\sim{\cal N}(0,1) and F_X(x)=F_Z\left(\frac{x-25}{5}\right).

    a. P(26<X<29)=F_X(29)-F_X(26)=F_Z\left(\frac{29-25}{5}\right)-F_Z\left(\frac{26-25}{5}\right)=F_Z\left(\frac{4}{5}\right)-F_Z\left(\frac{1}{5}\right)=0.7881-0.5793=0.2088

    b. P(X<23)=F_X(23)=F_Z\left(\frac{23-25}{5}\right)=F_Z\left(-\frac{2}{5}\right)=1-F_Z\left(\frac{2}{5}\right)=1-0.6554=0.3446

    c. One has to find the cutoff time T by solving the inequality P(X<T)= 0.95. Now, F_X(T)=0.95 leads to F_Z\left(\frac{T-25}{5}\right)=0.95. Looking at the table you will find that \frac{T-25}{5}\approx 1.645. From here you find T=32.225 minutes, or rounded to nearest minute T=32 minutes.
    Last edited by MathoMan; April 22nd 2012 at 02:31 PM. Reason: Error in subtraction 29-25 :
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  3. #3
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    Re: Statistic practice question help

    As for d:

    A sport physiologist suggests that 10% of adult males walking at a steady pace can burn off 5 grams of fat in less than 20 minutes.
    That means that for an arbitrarily chosen male there is 10% chance that walking at steady pace he will be able to burn off 5 grams of fat in less than 20 minutes.
    What is the probability that in a random sample of 100 adult males walking at a steady pace less than 8 of them can burn off 5 grams of fat in less than 20 minutes?
    Here you consider having a random variable with Binomial distribution X\sim {\cal B}(100,0.1). That random variable can be approximated with normally distributed random variable Ywith distribution parameters \mu=n\cdot p = 10 and \sigma=\sqrt{np(1-p}=3, Y\sim {\cal N}(10,3).
    The answer to the question is
    P(Y<8)=F_Y(8)=F_Z\left(\frac{8-10}{3}\right)=F_Z\left(-\frac{2}{3}\right)=1-F_Z\left(\frac{2}{3}\right)=1-0.7486=0.2514
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  4. #4
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    Re: Statistic practice question help

    thank you for helping!I got almost all correct except for C and D when I did the question.
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  5. #5
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    Re: Statistic practice question help

    You're welcome mate.
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