An apple juice producer buys all his apples from a conglomerate of apple growers in one northwestern state. The amount of juice obtained fro

Question

An apple juice producer buys all his apples from a conglomerate of apple growers in one northwestern state. The amount of juice obtained from each of these apples is approximately normally distributed with a mean of 2.25 ounces and a standard deviation of 0.15 ounce. 77 percent of the apples will contain at least how many ounces of juice?

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Genesis 2 weeks 2022-01-08T07:18:07+00:00 1 Answer 0 views 0

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    2022-01-08T07:20:06+00:00

    Answer:

    a=2.25 -0.739*0.15=2.139

    So the value of height that separates the bottom 23% of data from the top 77% is 2.139.

    So we can conclude that 77% of the apples will contain at least 2.139 ounces  

    Step-by-step explanation:

    Previous concepts

    Normal distribution, is a “probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean”.

    The Z-score is “a numerical measurement used in statistics of a value’s relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean”.  

    Solution to the problem

    Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:

    X \sim N(2.25,0.15)  

    Where \mu=2.25 and \sigma=0.15

    The z score formula is given by:

    z=\frac{x-\mu}{\sigma}

    For this part we want to find a value a, such that we satisfy this condition:

    P(X>a)=0.77   (a)

    P(X<a)=0.23   (b)

    Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

    As we can see on the figure attached the z value that satisfy the condition with 0.23 of the area on the left and 0.77 of the area on the right it’s z=-0.739. On this case P(Z<-0.739)=0.23 and P(z>-0.739)=0.7
    7

    If we use condition (b) from previous we have this:

    P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.23  

    P(z<\frac{a-\mu}{\sigma})=0.23

    But we know which value of z satisfy the previous equation so then we can do this:

    z=-0.739<\frac{a-2.25}{0.15}

    And if we solve for a we got

    a=2.25 -0.739*0.15=2.139

    So the value of height that separates the bottom 23% of data from the top 77% is 2.139.

    So we can conclude that 77% of the apples will contain at least 2.139 ounces  

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