Normal probabilities for jar masses, a shaded upper tail with its boundary, and an unknown mean from a lower percentage

Normal probabilities for jar masses, a shaded upper tail with its boundary, and an unknown mean from a lower percentage

13 marks

The masses of honey in jars filled at a farm are normally distributed with mean \(340\) g and standard deviation \(6\) g.

  1. A jar is chosen at random. Find the probability that the mass of honey in it is
    1. less than \(350\) g;
    2. between \(330\) g and \(350\) g.[4]

It is found that \(8\%\) of the jars contain more than \(d\) grams of honey.

    1. On the diagram above, mark \(d\) and shade the region that represents these jars.
    2. Find the value of \(d\).[5]
  1. A second farm fills jars whose masses of honey are normally distributed with mean \(\mu\) g and standard deviation \(6\) g. Of these jars, \(10\%\) contain less than \(330\) g. Find the value of \(\mu\).[4]

Solution:

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Answer:(a)(i) \(0.952\) (a)(ii) \(0.904\) (b)(i) region to the right of \(d\) shaded, with \(d>340\) (b)(ii) \(d=348\) (c) \(\mu=338\)

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