S3 June 2011 Hypothesis Testing

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EL77
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#1
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For 7a, I don't understand why you're able to to perform a hypothesis test because the standard deviation given is the standard deviation of the sample not for the population, nor is it S (the unbiased estimator of the standard deviation of the sample). The mark scheme just uses the standard deviation given, 5.4, as the standard deviation for the hypotheses test but this can't be right because its the standard deviation of the sample? Unless i'm missing something? Thanks
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Mr Gum
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n is large enough (90) that a z test will be reasonably accurate (for smaller samples you'd need a t test). It would be usual to use the unbiased estimator. Notations and terminology vary here: some authors may refer to the unbiased estimator as the standard deviation of the sample. Again, since n is quite big, the difference between the biased and unbiased values will be small.
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EL77
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Sorry I'm not sure what you mean by a t test? So basically because n is large you can approximate the sample standard deviation to the standard deviation of the population? Is that what you mean?
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Mr Gum
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(Original post by EL77)
Sorry I'm not sure what you mean by a t test? So basically because n is large you can approximate the sample standard deviation to the standard deviation of the population? Is that what you mean?
Yes, that is correct.

The t distribution is a distribution that takes account of the effect of using the sample standard deviation rather than the population, and it enables us to do hypothesis tests in cases like yours, but where the sample size is too small for the z test to be an acceptable approximation - by a z test I mean a test using the Normal distribution (the standard normal distribution being that of the z statistic)
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