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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(

8/4/2019 Students Tutorial Answers Tutorial 7

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5328..01915.03413..0

)5.00()10(

)5.01(

10

5055

10

5040)5540(

=+=

8/4/2019 Students Tutorial Answers Tutorial 7

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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(

==

8/4/2019 Students Tutorial Answers Tutorial 7

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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.