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26 November, 01:23

1. In an experiment to measure the lifetimes of parts manufactured from a certain aluminum alloy, 73 parts were loaded cyclically until failure. The mean number of kilocycles to failure was 783, and the standard deviation was 120. Let μ represent the mean number of kilocycles to failure for parts of this type. A test is made of H0 : μ ≤ 750 versus H1 : μ > 750. a. Find the P-value. b. Either the mean number of kilocycles to failure is greater than 750, or the sample is in the

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  1. 26 November, 01:50
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    a. P value = 0.0094, b. 0.94% of the distribution

    Step-by-step explanation:

    From the given information, let X be the lifetimes of parts manufactured from a certain aluminum alloy

    Total No. of parts n = 73, Mean number of kilocycles to failures (x') = 783

    Standard deviation s = 120

    Let μ be the mean number of kilocycles to failure for parts

    a. Null hypothesis, The mean number of kilocycles to failure is less than or equal to 750, H₀ : μ ≥ 750

    Alternative hypothesis, The mean number of kilocycles to failure is greater than 750, H1 : μ > 750

    For large samples, the sample standard deviation (s) is approximate to population standard deviation (σ).

    Use the following formula to compute the test statistic

    z = (X' - μ₀) / (s/√n)

    z = (783 - 750) / (120/sq73) = 2.35

    Th test is one - tailed, therefore the p value is probability of observing a sample mean greater than to 783

    P = P (X'>783)

    = P (z > 2.35), (since, z = (X' - μ₀) / (s/√n))

    = 1 - 0.9906 = 0.0094

    b. From the above output, it is observed that 0.94% of samples are greater than 750. Therefore, either the mean number of kilocycles to failure is greater than 750, or the sample is in the most extreme 0.94% of its distribution
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