1

BES Tutorial Sample Solutions, S2 2010 It will be posted on BES website with one week delay.

WEEK 9 TUTORIAL EXERCISES (To be discussed in the week starting

September 20) 1. Perform the following hypothesis tests of the population mean. In each case,

illustrate the rejection regions on both the Z and X distributions, and calculate the p-value (prob-value) of the test. (a) H0:=50,H1:>50,n=100,=55,=10,=0.05

Rejectionregion:

5010 100 . 1.645Alternatively

. 50 1.645 10

100 51.645Since

55 5010 100 5 . 1.645Canreject 0H andconcludethatthepopulationmeanisgreaterthan50 .

0.05

X 50 51.645

reject

2

0.05

0 1.645 Z

reject

5 0.0000

(b) H0: = 25, H1: < 25, n = 100, = 24, = 5, = 0.1 Rejectionregion:

255 100 . 1.28Alternatively

. 25 1.28 5

100 24.36Since

24 255 100 2 . 1.28Canreject 0H andconcludethatthepopulationmeanislessthan25.

0.1

X 24.36 25

reject

3

0.1

1.28 0 Z

reject

2 0.0228

(c) H0: = 80, H1: 80, n = 100, = 80.5, = 4, = 0.05 Rejectionregion:

804 100 . 1.96 1.96Alternatively:

. 80 1.96 4

100 79.216or

. 80 1.96 4

100 80.784Since

80.5 804 100 1.25is not less than 1.96 or nor greater than 1.96 we do not reject 0H andconcludethatthepopulationmeanisequalto80.

4

0.025 0.025

X 79.216 80 80.784

reject reject

0.025 0.025

Z 1.96 0 1.96

reject reject

2 1.25 2 0.1056 0.2112

5

2. A real estate expert claims the current mean value of houses in a particular area is more than $250,000. A random sample of 150 recent sales prices in the area yields a sample mean of $265,000. It is known that house values in the area are approximately normally distributed with a standard deviation of $50,000. (a) Perform an upper tail test of the null hypothesis that the population mean

house value in the area is $250,000. Use a 5% level of significance and state the rejection (critical) region in terms of both and z.

Let X valueofahouseinthearea $265,000, $50,000, ~, Wewishtotest

: 250,000;: 250,000Rejectionregion:

250,00050,000 150 . 1.645or

. 250,000 1.645 50,000150 256,715.68

Since

265,000 250,00050,000 150 3.67 . 1.645Hencewe reject 0H and conclude that themeanhouse value in thearea ismorethan 000,250$ .

(b) Why is an upper tail test most appropriate in this case? Thenatureoftheresearchproblemdictatesanuppertailtest.Inthiscasewewill not believe the experts claim of unless there is significant sampleevidencetodoso.Thisimpliesanuppertailtest.

6

(c) What is the p-value associated with the test statistic used in the part (a) test? Interpret this value.

3.67 0.5 0.4999 0.0001

Thepvalue istheprobabilityofobtainingateststatisticmoreextremethantherealizedvalue,assumingthenullhypothesisistrue.Thelowerthepvalue,thegreateristheevidenceforrejectionofthenullhypothesis.Inthiscaseitisveryunlikelytofindasamplemeanasextremeas$265,000givenapopulationmeanof$250,000.

(d) Define the type I and II errors in the context of the part (a) test. TypeIError:Concludingthathousingpriceismorethan$250,000,whileitisreally$250,000.TypeIIError:Notbeingabletorejecttheclaimthathousingpriceis$250,000,whileitisreallymore. 3. What effect does increasing the sample size have on the outcome of a

hypothesis test? Explain your answer using the example of a one-tail test concerning the mean of a normally distributed population with known variance. (It is expected that students will find this question difficult. Hint; think what happens to the standard error of the mean as n increases and the effect this has on the test statistic if the sample mean remains unchanged and so does the true mean.)

Supposeanuppertailtest

: ;: Under:

~, ~ ,

~0,1

The point on N(0,1) corresponds to the point on thedistributionof .

7

Thedistributionof is:

X 0 nz

0 Butsupposethetrueistotherightof.Thenthetruedistributionof issay:

0 nz 0

The shaded area in the above diagram gives the probability of correctlyrejectingH0(i.e.thepower,1whichisgreaterthan)Nowsupposethesamplesizeisincreased.Asaresult: decreases&hence

decreases.

8

Supposethenewsamplesizeisn1>n.Thedistributionof willnowlooksomethinglike:

0 1

0 nz X

Notethatwithafixedtherejectionregioncutoffisnowsmaller.Again,ifthetrue is actually to the right of , the probability of rejecting the sameincorrect null hypothesis is higher than before. Diagrammatically the truedistributionof willbesay

0 X

10 n

z AgaintheshadedareaintheabovediagramgivestheprobabilityofcorrectlyrejectingH0.

9

Conclusion:TheprobabilityofcorrectlyrejectingafalseH0.(thepowerofthetest,asdiscussedinlecturesthisweek)increasesasnincreasesgivenwekeeptheTypeIerror()fixed.