Students Tutorial Answers Tutorial 7

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  • 8/4/2019 Students Tutorial Answers Tutorial 7

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    BES Tutorial Sample Solutions, S2/11

    TUTORIAL 7

    WEEK 8 TUTORIAL EXERCISES (To be discussed in the week starting

    September 12)

    1.Suppose a normally distributed random variable X has a mean of 50 and avariance of 100. Also suppose a sample of size 16 is drawn from this

    population. Calculate the following probabilities:(a) P( X> 55)

    3085.01915.05.0)5.0(

    10

    5055)55(

    ==>=

    >=>

    ZP

    ZPXP

    (b) ( > 55)X ~ )16100,50(N

    0228.04772.05.0

    )20(5.0)2(

    410

    5055)55(

    ==

    =>

    ZPZP

    ZPXP

    (c) )5540(

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    5328..01915.03413..0

    )5.00()10(

    )5.01(

    10

    5055

    10

    5040)5540(

    =+=

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    2.A pet food manufacturer produces cans of cat food with a nominal contentweight of 400 grams per can, however the can-filling machine yields acontent weight standard deviation of 20 grams. The cans are supplied to

    wholesalers in boxes of 64 cans, and wholesalers require that the mean can

    weight per box be at least 400 grams. To reduce the probability of a box ofcat food not meeting a wholesalers requirements, the machine is set to

    produce a mean can content weight of 403 grams. Calculate the probability

    that a randomly selected box of cat food does not yield a mean can weightof at least 400 grams.

    Let =X weight of can in grams then X ))20(,403?( 2 Since n=64 is large by the central limit theorem

    X

    64

    )20(,403

    2

    N approximately & hence

    1151.03849.05.0

    )2.10(5.0

    )2.1(

    )820(

    403400)400(

    ==

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    )433,89,689,67(

    872,10561,78

    117

    000,6096.1561,78

    025.0

    =

    =

    =n

    zx

    The calculated interval is one of the possible realizations of the 95%

    confidence interval. In repeated sampling, 95% of intervals calculated in this

    way would contain the true .

    4.What would be the effects on the width of the confidence intervalcalculated in Question 5 above of:(a) a decrease in the level of confidence used?

    Decreases width(b) an increase in sample size?

    Decreases width(c) an increase in the population standard deviation?

    Increases width(d) an increase in the sample standard deviation?

    No effect on the width since we are told the population standarddeviation.

    (e) an increase in the value ofx found?No effect on the width

    5.Again referring to the statistical population in Question 3 above, determinethe sample size required to estimate the population mean to within 5,000kms with 90% confidence.

    000,60,000,5,645.105.02/

    ====

    Bzz

    Sample size required

    67.389000,5

    )000,60(645.122

    2/ =

    =

    =

    B

    zn

    A sample size of 390 would be required.