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Math Help - Sampling

  1. #1
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    Sampling

    The head of a business school claims that the average MBA graduate is offered a starting salary of $55,000. The standard deviation of the offers is $4,600.

    a) What is the probability that in a random sample of 38 MBA graduates, the mean starting salary is less than $53,000?

    I am saying that

    P( \bar{X}< 53,000) = P\left(Z<\frac{53,000-55,000}{\frac{4,600}{\sqrt{38}}}\right)= \dots

    b) What if only 16 students had been sampled?

    This is confusing, can I say

    P(\bar{X}<53,000) = P\left(Z<\frac{53,000-55,000}{\frac{4,600}{\sqrt{16}}}\right)= \dots ??

    as \sigma is known, can Z be used for n=16 ?
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  2. #2
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    Quote Originally Posted by Bushy View Post
    The head of a business school claims that the average MBA graduate is offered a starting salary of $55,000. The standard deviation of the offers is $4,600.

    a) What is the probability that in a random sample of 38 MBA graduates, the mean starting salary is less than $53,000?

    I am saying that

    P( \bar{X}< 53,000) = P\left(Z<\frac{53,000-55,000}{\frac{4,600}{\sqrt{38}}}\right)= \dots

    b) What if only 16 students had been sampled?

    This is confusing, can I say

    P(\bar{X}<53,000) = P\left(Z<\frac{53,000-55,000}{\frac{4,600}{\sqrt{16}}}\right)= \dots ??

    as \sigma is known, can Z be used for n=16 ?
    Hint: You are given population mean and s.d. So what pdf does the sample mean follow?
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  3. #3
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    Quote Originally Posted by aman_cc View Post
    Hint: You are given population mean and s.d. So what pdf does the sample mean follow?

    Will the mean be approximately normal? If so both my answers will be correct?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Bushy View Post
    Will the mean be approximately normal? If so both my answers will be correct?
    You have to make an underlying assumption of normality at least for the second as you have a small sample and you want to use the t-distribution.

    For the first you may get away with relying on the CLT to justify normality, but I would still say I am assuming such (and for that matter also use the t-distribution).

    CB
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