Given the heat evolved in calories per gram of a cement mixture is normally distributed with mean 99.8 and standard deviation 5.2.

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Given the heat evolved in calories per gram of a cement mixture is normally distributed with mean 99.8 and standard deviation 5.2.

1 thought on “Given the heat evolved in calories per gram of a cement mixture is normally distributed with mean 99.8 and standard deviation 5.2.”

    • (A) If the acceptance region is defined as 98.5 ≤ x¯ ≤ 101.5, find the type I error probability α is 0.09296.
    • (B) For the case in which the true mean heat evolved is 103, β is 0.04648.

    What is heat?

    • The transfer of kinetic energy from one medium or object to another, or from an energy source to a medium or object, is referred to as heat.
    • Energy can be transferred in three ways: radiation, conduction, and convection.
    • When kings weren’t hiding in oak trees or being beheaded in the 1600s, they enjoyed a game of horse racing.
    • And when they took their horse for a gallop to prepare it for a race, they called it heat because they were obviously warming the horse up.
    To find the answers to the given situation:
    Use n = 5, everything else held constant:
    (A) To find the type I error probability α:
    • P( X ≤ 98.5) + P( X >101.5)
    • P(X-100/(2/√5) ≤ 98.5-100/(2/√5)) + P(X-100/(2/√5)>101.5-100/(2/√5))
    • P(Z ≤ −1.68) + P(Z > 1.68)
    • P(Z ≤ −1.68) + (1 − P(Z ≤ 1.68))
    • = 0.04648 + (1 − 0.95352) = 0.09296
    So, if the acceptance region is defined as 98.5 ≤ x¯ ≤ 101.5, find the type I error probability α has is 0.09296.
    (B) To find β:
    • β = P(98.5 ≤ X ≤ 101.5 when µ = 103)
    • P(98.5-103/(2/√5) ≤ X-103/(2/√5) ≤ 101.5-103/(2/√5))
    • P(−5.03 ≤ Z ≤ −1.68) = P(Z ≤ −1.68) − P(Z ≤ −5.03)
    • = 0.04648 − 0 = 0.04648
    So, for the case in which the true mean heat evolved is 103, β is 0.04648.
    Therefore, answers to both the questions are shown.
    Know more about the heat here:
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    The correct question is given below:
    The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100, and the standard deviation is 2. You wish to test H0 : µ = 100 versus H1 : µ 6= 100 with a sample size of n = 5 specimens.
    (a) If the acceptance region is defined as 98.5 ≤ x¯ ≤ 101.5, find the type I error probability α.
    (b) Find β for the case in which the true mean heat evolved is 103.

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