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Math Help - 2-sided alternative hypothesis

  1. #1
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    2-sided alternative hypothesis

    Assuming the rest of my work is correct for this question, I am having trouble on part c. I don't know if I have the right formula and, if I do, I don't know how to calculate t_{\frac{0.025}{2}}.

    Let X equal the length of wood blocks manufactured. Assume distribution of X is N(\mu,\sigma^{2}). The greatest lenth is 7.5 inches. We shall test the null hypothesis H_{0}: \mu=7.5against a 2-sided alternative hypothesis using 10 observations.

    a) Define test statistic and critical region for an  \alpha=0.05 significance level.

    test statistic-  t=\frac{\bar{x}-7.5}{\frac{s}{\sqrt{10}}}

    critical region- |t|=\frac{|\bar{x}-7.5|}{\frac{s}{\sqrt{10}}} \ge t_{\frac{\alpha}{2}}(10-1)=2.262

    Calculate the value of the test statistic and give your decision using the following data (n=10)
    7.65
    7.60
    7.65
    7.70
    7.55
    7.55
    7.40
    7.40
    7.50
    7.50

    \bar{x}=\frac{75.5}{10}=7.55
    s^{2}=0.01056
    s=0.10274

    t=1.539
    1.539 is not greater than 2.262, therefore, it fails to reject H_{0}: \mu=7.5

    c) Is \mu=7.50 contained in a 95% confidence interval for \mu?

    \bar{x}+/- t_{\frac{0.025}{2}}(n-1)(\frac{s}{\sqrt{n}})

    As of now, I am using my values of \bar{x}=7.55, s=0.10274 and n=10

    Thank you for helping!
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  2. #2
    Super Member Random Variable's Avatar
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    You would only need to know  t_{0.025/2} if you were calculating a 97.5% confidence interval for  \mu .
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  3. #3
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    I didn't understand that either. My book seems to use that for the two sided hypotheses when finding the confidence interval. Is there a different formula I should use? or the same without dividing \alpha by 2.

    The critical region equation for hypotheses one mean and variance unknown written in the book for H1: \mu does not equal \mu_{0} is |\bar{x}-\mu_{0}| \ge t_{\frac{\alpha}{2}}(n-1)\frac{s}{\sqrt{n}}
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  4. #4
    Super Member Random Variable's Avatar
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    Quote Originally Posted by larz View Post
    I didn't understand that either. My book seems to use that for the two sided hypotheses when finding the confidence interval. Is there a different formula I should use? or the same without dividing \alpha by 2.

    The critical region equation for hypotheses one mean and variance unknown written in the book for H1: \mu does not equal \mu_{0} is |\bar{x}-\mu_{0}| \ge t_{\frac{\alpha}{2}}(n-1)\frac{s}{\sqrt{n}}
     \alpha = 0.05 in your problem
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