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    Soft-drink volume in test

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    A soft-drink bottler sells "one-liter" bottles of ginger ale. The production manager
    wishes to monitor the content of the bottles. He takes a random sample of 30
    bottles, and obtains the following contents, in milliliters:

    1025 977 1018 975 977 990 959 957 1031 964
    986 914 1010 988 1028 989 1001 984 974 1017
    1060 1030 991 999 997 996 1014 946 995 987

    (a) Find the mean and standard deviation of this sample, and compute the 95%
    con fidence interval for the true mean content. State in words what this interval
    means.

    (b) Find the 95% tolerance interval for 99% of the production. State in words
    what this interval means.

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    https://brainmass.com/statistics/normal-distribution/soft-drink-volume-test-541291

    Solution Preview

    a.
    quantity
    q[1:n] = 1025 977 1018 975 977 990 959 957 1031 964 986 914 1010 988 1028 989 1001 984 974 1017 1060 1030 991 999 997 996 1014 946 995 987

    Here, n = 30 (sample size)

    mean (q_bar) = sum(q[i])/30 = 992.63

    SD = sqrt ( sum((q - mean)^2)/n) = 29.22

    Let us assume true mean is mu.
    Using t-distribution:
    degree of freedom, d = n - 1 = 30 - 1 = 29,
    Because,
    1-alpha = 0.95
    => alpha = 0.05
    => alpha/2 - 0.025
    => 1 - alpha/2 = 0.975

    Hence,
    P(-t(n-1,1-alpha/2) < (mu-q_bar)/(SD/sqrt(n)) < t(n-1, 1-alpha/2) = 0.95
    P(-t(29, 0.975) < (mu - ...

    Solution Summary

    Using given data, in first problem an interval with 95% confidence is estimated in which volume of soft-drink is expected to be. In second part 95% tolerance interval for given percentage of production is estimated.

    $2.19

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