Estimation.DOC

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9.0 Estimation of a Random Variables possible value

When we have collected data from an experiment, we can nowuse statistical estimation to determine the likely value of such variablewhen a future observation is to be performed.

Statistical inferenceconsists of using methods by which onemakes inferences or generalizations about a population.

Point Estimator = a rule or formula that tells us how to calculatean estimate based on the measurements containedin a sample. The number that results from thecalculation is called a point estimate.

Interval Estimator = a formula that tells us how to use sampledata to calculate an interval that estimates a

population parameter.

Estimation Formulas and applied problems

A. Estimating the mean

A.1 Case 1: x known X Zn

x

2

(! The "uality control manager at a light bulb factory needs toestimate the average life of a large shipment of light bulbs. The

process standard deviation is known to be ## hours. $ randomsample of %# light bulbs indicated a sample average life of &%#hours.

a. 'et up a #) con*dence interval estimate of the true average lifeof light bulbs in this shipment.

b. 'et up a %) con*dence interval estimate of the true average lifeof light bulbs in this shipment.

c. Tell why an observed value of &+# hours would not be unusual eventhough it is outside the con*dence interval you calculated.

A.2 Case 2: x not known X t s

n

2

(+!The irector of -uality of a large health maintenanceorganization wants to evaluate patient waiting time at a localfacility. $ random sample of +% patients selected from the

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appointment book. The waiting time was de*ned as the timefrom when a patient signed in to when he or she was seen bythe doctor. The following data represent the waiting time (inminutes!

.% .% 3%.4 &.2 +.4+%.3 +.2 +2.4 %+.# +%.3+4. &. 3&. 3. +.5#.5 +. . 3%. 3+.%3.& &.2 5.3 &.# &4.4

'et up a % ) con*dence interval estimate of the populationaverage waiting time.

B. Estimating the dierence beteen to means !"#$%

B.1 Case 1: Large indeendent samles ( )x x Zn n

1 22

1

2

1

2

2

2

+

(&! 6t is desired to estimate the di7erence between the mean startingsalaries for all bachelor8s degrees graduates of 9' in the:olleges of 1ngineering and :omputer 'cience during the pastyear. The following information is available from the :areerevelopment ;+,%## and astandard deviation of >&2,%%.

$ random sample of starting salaries for :omp'ci graduatesproduced a sample mean of >+,### and a standarddeviation of >&%,3%+.

:onstruct a %) con*dence interval for the di7erence betweenmean starting salaries for graduates of the two colleges. 6nterpretthe interval.

B.2 Case 2: Small indeendent samles wit! e"#al variances

( )x x t Sn n

p1 22

1 2

1 1 +

where

2

11

21

2

22

2

11

++

=nn

S)n(S)n(Sp

value of t/2is based on (n?n+@+! degrees of

freedom

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(3! The Aarm has received claims from its customers thatthe weight ofpoint-of-sale adult pigs from its two farms are different by at least 5 kgs. The residentveterinarian deided to hek the validity of this laim. !e took 1" pigs from eah farm#savailable pigs for sale$ and the following data was found.

Aarm

Aarm

+sample size #

pigs#

pigs0ean weight #.% 2%.#

standard deviationof wts

&.+ &.3

a. 6s the claim valid at %) level of signi*cance Bb. 1stimate a % ) con*dence interval for mean di7erence in pig

weights between those of farm and those of farm +.

B.$ Case $: Small indeendent samles wit! #ne"#al variances

( )x x ts

n

s

n1 2

2

1

2

1

2

2

2

+

where the distribution oft/2has degrees of

freedom

=+

+

sn

sn

s

n

n

s

n

n

1

2

1

2

2

2

2

1

2

1

2

1

2

2

2

2

21 1

The value of should be rounded downto nearest

integer.

(%! Cou8re trying to determine if a new route from your house toschool would save you at least # minutes of travelling time. Courecorded 3 weeks8 travelling time using the two di7erent routes andyour data showed

0ean traveltime

'tddeviation

;ld Doute (&times!

%%.+ minutes %.+minutes

/ew Doute (5 3+.5 minutes #.&

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times! minutes

1stimate a # ) con*dence interval of the di7erence in travellingtimes if you took the new route instead of the old one.

B.% Case %: &atc!ed airs '(eendent samles)

9et d, d+.. dnrepresent the di7erences between pairwise observationsin a random sample of n matched pairs. Then the small sample

con*dence interval ford%(1-2) is

d t S

n

d

2 where d and 'd are the mean and std.dev. of the n

sample di7erences.

(4! $ new diet program called *+ive It ,* claims to be e7ective intaking out the unwanted pounds o7 of obese people. $s abenchmark to compare against, you used the >ritikin program.Cou randomly selected people8s and ac"uired their body weightsbefore and after each program. The following table shows the

data.:onstruct a %) con*dence interval for the mean weight lossunder each program.

EFive 6tpE

person

Gefore $fter >ritikinperson

Gefore $fter

##kgs

5# kgs +3 kgs 5# kgs

+ +3 2% + % 2& % 2# & +% 42

3 +% 23 3 % 2#% % 2 % 2% 234 + 5% 4 23 25 #% 2% 5 5% 5%2 + 2% 2 +% 2% #2 + % +# % 5% # #% 5% 2% 5# + 5#

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+ 23 2 + #2 2& 5% 42 & % 423 2# 43 3 2% 43% 4 5% % 4 5%

&. Estimating the variance( )

( ) ( )n s n s

1 12

2

2

22

12

2

(5! :onstruct a % ) :6 on the variance based on the following set ofdata (the amount of Hrypton gas (in milliliters! that leaked out eachtime its container was dropped from 3 ft above ground!

%.% 4.2 4.5 %.3 4.3 5.%5.2 5.% 2.& 3.% 2. %.5

.% 2.4 .5 %.2 2.+ 4.2

'. Estimating the ratio of to variances

( ) ( )( )s

s F

s

s F

1

2

2

2

21 2

1

2

2

2

1

2

2

2

12

1 2

1 1

$ $

( )

( )s

s F

s

sF

1

2

2

2

2 1 2

1

2

2

2

1

2

2

22

2 1

1

$

$

(2! $n investor wants to compare the risks associated with twodi7erent computer stocks, 6G0 and >hil:om, where the risk of agiven stock is measured by the variation of daily prices. Theinvestor obtains random samples of daily price changes for 6G0and >hil:om. The sample results are summarized in theaccompanying table. :ompare the risks associated with bothstocks by forming a % ) c.i. for the ratio of the true populationvariances.

6G0 >hil:omn =+ n+ = +

I = #.%2% I+ = #.%5+' = #.#+& '+ = #.#3

E. Estimating a proportion p Z p q

n

2

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(! 6n a recent employee satisfaction survey made by J.Dizal6ndustries, &%4 out of 3## employees stataed that they were verysatis*ed or moderately satis*ed with their Kobs. :reate a %) c.i.estimate of the population proportion of the employees who were

satis*ed with their Kobs.

F. Estimating the dierence beteen to proportions

( )p p Z p q

n

p q

n1 2

2

1 1

1

2 2

2

+

(#! 6n a controversial survey of dating preferences made at theniversity of the >hilippines in +, % out of %## studentssurveyed claimed LFood :onversationalistM as the one of the mostpreferred trait of a dating partner. Aurthermore, 5% out of %##claimed Lsexual attractivenessM as the most important trait.1stimate the di7erence between the proportions of the >studentry who preferred good talk over good looks at a #)con*dence level.

(. )est for (oodness of Fit beteen a h*pothesi+ed'istribution and actual data

Nypothesis ata follows the hypothesized value.

Test statistic formulaWhere

( )=

=

k

i i

ii

e

eo

1

2

2

where oi = observed fre"uency of the ith interval cell (i= tok!

ei = expected fre"uency according to the hypothesizeddistribution

= Total / x (>robability > ifor ith interval.!= / >i

ecision DeKect the hypothesis if I+ O I+,k@ value on chi@s"uared

distribution table with probability of error and degrees of

freedom = k@.

1xample

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$ die is to be tested if each side occurs as e"ually fre"uent asthe others. To do this, ## throws of the die was made and thenumber of occurrences per side were recorded.

I + & 3 % 4Are"uenc

y + # % 4 ++

+

%

:an we say with a %) probability of error (or %) con*dence! that thedie results follow a uniform distributionB

,RA&)-&E ,RB/E12

. The production records for an automobile manufacturer show thefollowing *gures for production per shift.

422 5## 4 454%4 5# 43 5#&5 5#+ 443 5#2455 422 4 4224+% 422 422 45#& 445 %35 45

a. Five a %) con*dence interval for the production output (cars, inthis case!

b. Five an %) con*dence interval for the standard deviation ofproduction output that should be known to occur.

c. What proportion of the shifts should you expect to produce 42#

cars or moreB Five a %) con*dence interval for this proportion.

+. $ single leaf was taken from each of 9uciano Tang8s tobacco plants.1ach was divided in halfP one half was chosen at random andtreated with preparation 6 and the other received preparation 66.The obKect of the experiment was to compare the e7ects of the twopreparations of mosaic virus on the number of lesions of halfleaves after a *xed period of time. Aor a %) level of signi*cance,examine the research hypothesis that the lesions that occurredfrom di7erent preparations are signi*cantly di7erent. Cou can dothis by making a con*dence interval of the di7erences between

each prep.

>lant + & 3 % 4 5 2 #

>rep6

2

+#

3 &2 +4 % #

+% 5 &

>rep66

4 + &+ + 2 & 4

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3 %

&. J: Qideo 'tores wants to know how long it takes for its customersto check@out a video rental. 'uppose that you are to obtain a

random sample of +# video checkout times (in minutes! . Thefollowing table showed the ordered se"uence of data collected(read the data by row!

ay

.+ +.54 &.2 3. .+2 %.#4 %.45 4.##

ay+

&.5 3.%3 #.+2

+.3%

5.4 2.+

ay&

. #.2% +.% %.5 .5% %.+

Treat each problem below independently. sing a #.#% level ofsigni*cance

a. $ssuming normality, give an interval estimate of the time ittakes for a customer to check out videos.b. $ssuming normality, give an interval estimate the proportion

of customers who check out within & minutes.

3. 9ouie wore II9 sized clothes in June +##+. Today, he canconsider himself normal 9arge sized ('ize 9!. Ne shows you amonth@by@month record of his body weight for the past year. Nealways weighed himself at the beginning of each month. Ne wantsyour opinion on certain statistical claims. Ne started on a dietprogram 'eptember .

&ont! -#ne -#l A#g Set /ct 0ov (ec -an e &ar Ar &a

Weight ++# +4 +% +# +#4 +## + 25 2 5+ 4+ %%a. Five a #) con*dence interval for the month@to@month

di7erences in his weight since he started on the diet program.b. Nas his average monthly weight signi*cantly changed since

he started with the programB :ompare his average weightbefore the program and his average weight after theprogram. se a=#.#% 6s there a signi*cant reductionB

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