Final Handouts - Stat220

8/17/2019 Final Handouts - Stat220

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Probability Distributions

Random Variable – a function whose value is a real number determined by each element in the sample space

Probability Distribution – a table or formula or a graph listing all possible values that a rv can take on, along with

the associated probabilities

Remark : We shall use an uppercase letter, say X, to denote a random variable and its corresponding lowercaseletters, x in this case, for one of its values.

Examples

1. n a random experiment of tossing a coin ! times, let X be a random variable that pertains to the number of heads

in the outcome.

"he possible outcomes are

# x

$$$ !

$$" %

$"$ %

"$$ %$"" 1

"$" 1

""$ 1""" &

"herefore, the probability distribution of X is given by

x & 1 % !

'(X)x* ) f(x* 1+ !+ !+ 1+

"his means that the f(&* ) '(X)&* ) 1+ is the probability of obtaining no head in the random experimentand so on.

Notes:

1. f(x* ) '(X)x*

%. f(x*)1

%. - fair die is thrown. -ssign random variable X ) of spots on any throw.

"he probability distribution of X in tabular form is:

'robability /istribution in 0raphical 'resentation


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x p( x

)

1/6

145623

∑ = xall

1'(x*

!. What is the probability distribution of number of girls in families with two children

"he random variable here (denoted by X* refers to the number of girls.

"he possible outcomes are:

X 2utcomes% 00

1 30

1 03

& 33

0 1 2 3

0

0.1

0.2

0.3

0.4

0.5

How about family of three?

X child 1 child % child !

& 3 3 3

1 3 3 0

1 3 0 3

1 0 3 3

% 3 0 0

% 0 3 0% 0 0 3

! 0 0 0


3/22

0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

0.25

0.3

How about a family of 1?

!s family si"e in#reases$ the distribution looks more and more normal%

Number of Successes

3.02.01.00.0


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Number of Successes

10987654321-0

Des#ripti&e 'hara#teristi#s of a RV: 4ean and 5ariance

Expe#ted Values ()ean* of a RV

6et X be a discrete rv with prob. /istribution

x x1 x% 7 xn

'(X ) x* f(x1* f(x%* 7 f(xn*

1

( * ( *n

i i

i

E X x f x µ =

= = ∑

"he mean or expected value of X is

Varian#e of a Random Variable

6et X be a discrete random variable with mean 8 and probability distribution given by the above table

% % %

1

var( * ( * ( * ( *n

i i

i

X E X x f xσ µ µ =

= = − = −∑

"he variance of X is

[ ] %%var( * ( * ( * X E X E X = −

9omputational ormula for ;%:


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Probability

Distribution

DiscreteProbabilityDistribution

UniformDistribution

BinomialDistribution

Hypergeometric

Distribution

PoissonDistribution

ContinuousProbabilityDistribution

ormalDistribution

Where :

1 &.?

! &.%%@ &.&!

? &.&1

4ean)&A&.&>B1A&.?B%A&.%%B!A&.1B@A&.&!B?A&.&1 ) 1.@C

5ariance ) (&D1.@C*%A&.&>B(1D1.@C*%A&.? B(%D1.@C*%A&.%%B(!D1.@C*%A&.1 B(@D1.@C*%A&.&!B(?D1.@C*%A&.&1

)&.>CC

2r using the alternative formula

5ariance ) &%A&.&>B1%A&.?B%%A&.%%B!%A&.1B@%A&.&!B?%A&.&1 – 1.@C% ) &.>CC

#tandard /eviation) sEuare root of the variance ) &.C!%>@

+,)E +PE'-!. PR,/!/-.-0 D-+0R-/20-,N+

Dis#rete Probability Distributions

- dis#rete random &ariable is a variable that can assume only a countable number of values

)any possible out#omes:

› number of complaints per day

› number of "5Fs in a household


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› number of rings before the phone is answered

,nly two possible out#omes:

› gender: male or female

› defective: yes or no

› spreads peanut butter first vs. spreads Gelly first

'ontinuous Probability Distributions

- #ontinuous random &ariable is a variable that can assume any value on a continuum (can assume an uncountable

number of values*

› thickness of an item

› time reEuired to complete a task

› temperature of a solution

› height, in inches

"hese can potentially take on any value, depending only on the ability to measure accurately.

Exer#ise

/etermine whether the following random variables are discrete or continuous. #tate possible values for the

random variable.

1. "he number of light bulbs that burn out in a room of 1& light bulbs in the next year.%. "he number of leaves on a randomly selected 2ak tree.

!. "he length of time between calls to C11.

Dis#rete Probability Distributions

1. Hniform distribution

%. 3inomial distribution

!. $ypergeometric distribution@. 'oisson distribution

1. Hniform /istribution

1( I * f x N

N =

1 % ,...,, N x a a a=

or

%

1( *1%

N V X −

= 1( *%

N E X +=

Example


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#uppose that an employee is selected at random from a staff of 1& to supervise a certain proGect. What is the

uniform distribution of X ) employee number selected

ANSWER:

6et X ) employee number

) J 1, %, 7., 1&K

"herefore, the probability distribution in formula form is f(x* ) 1+1&, x ) 1, %,7, 1&.

Exer#ises

1. "wo clergymen are to be chosen at random from @ clergymen -, 3, 9, / for a community mass. ind the

uniform distribution of X ) selected clergymen.

-nswer: f(x* ) 1+>, x ) 1, %, 7, >

%. 9onsider the random experiment of tossing a die where X ) outcome of the die. ind the distribution of X.

-nswer: f(x* ) 1+>, x ) 1, %, 7, >

3% /inomial Distribution

- binomial experiment is one that possesses the following properties:

1. "he experiment consists of n repeated trials.%.


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or f(%* ) !9%(1+%*%(1+%*1 ) !+

or f(!* ) !9!(1+%*!(1+%*& ) 1+

"his is the same answer from previous example.

Exer#ises

1. ind the probability of obtaining exactly three %Fs if an ordinary die is tossed ? times.-nswer: &.&!%%

%. - multipleDchoice EuiM has 1& Euestions, each with @ possible answers of which only 1 is correct. What is the

probability that sheer guesswork yields from ? to > correct answers-nswer: &.&N@>

4% Hyper5eometri# Distribution

9haracteristics of a $ypergeometric


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1. rom a class of @ girls and ! boys, % are to be selected at random. What is the probability that % girls are

selected

-nswer: &.%?N

%. - store has 1& iphones of which ! are defective. ive iphones will be delivered to their customer. What is the

probability that there will be % defective iphones on the delivery

-nswer: &.@1>N

6% Poisson Distribution

- random variable X , the number of successes in a fixed interval, follows a 'oisson process provided thefollowing conditions are met

1. "he probability of two or more successes in any sufficiently small subinterval is &.

%. "he probability of success is the same for any two intervals of eEual length.!. "he number of successes in any interval is independent of the number of successes in any other interval provided

the intervals are not overlapping.

( I *

R

xe f x

x

λ λ λ

−

=

( *V X λ = ( * E X λ =

&,1,%,.. x =for I where S is the average number of outcomes occurring in the given time interval

or specified region

Example

"he number of typographical errors in new editions of textbooks varies considerably from book to book.

-fter some analysis he concludes that the number of errors is 'oisson distributed with a mean of 1.? typos per 1&& pages. "he instructor randomly selects 1&& pages of a new book. What is the probability that there are no typos

"hat is, what is '(X)&* given that S ) 1.?

f (0 )=e−1.5

1.50

0! 7%3341 There is about a 22% chance of finding zero

errors”

Exer#ises

1. "he average number of days school is closed due to typhoon during rainy season in a certain city is @. What is the

probability that the schools in this city will close for > days during rainy season

-nswer: &.1&@%%. - certain bank receives an average of N bad checks per day. What is the probability that it will receive ? badchecks on any given day

-nswer: &.1%NN

+pe#ial 'ontinuous Probability Distribution

Normal Distribution


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"he most important probability distribution in the entire field of statistics is the normal distribution. ts

graph, called the normal curve, is a bellDshaped curve that describes many sets of data that occur in nature.

- normal random variable X is completely specified by two parameters : mean T and variance ;%.

Properties of the Normal 'ur&e:

1. "he mode, which is the point on the horiMontal axis we)here the curve is a maximum, occurs at x ) T.%. "he curve is symmetric about a vertical axis through the mean T.

!. "he normal curve approaches the horiMontal axis asymptotically as we proceed in either direction away

from the mean.

@. "he total area under the curve and above the horiMontal axis is eEual to 1.

2nit8+tandard Normal Distribution

"o determine the probability distribution of a normal random variable, it is necessary that we perform achange of scale, which converts the units of measurements from the original scale, x scale, into standard units or MD

scores by means of the formula

z= x−µ

σ

"he distribution of the variable U is commonly known as standard normal distribution where the mean T )& and the standard deviation ; ) 1.

Note: We make use of standard normal table to evaluate probabilities involving normal random variables.

#hown below is a part of a standard normal table

M &.&& &.&1 &.&% &.&! &.&@ &.&? &.&> &.&N &.& &.&C

&.& &.&&&& &.&&@& &.&&& &.&1%& &.&1>& &.&1CC &.&%!C &.&%NC &.&!1C &.&!?C

&.1 &.&!C &.&@! &.&@N &.&?1N &.&??N &.&?C> &.&>!> &.&>N? &.&N1@ &.&N?!

&.% &.&NC! &.&!% &.&N1 &.&C1& &.&C@ &.&CN &.1&%> &.1&>@ &.11&! &.11@1

&.! &.11NC &.1%1N &.1%?? &.1%C! &.1!!1 &.1!> &.1@&> &.1@@! &.1@& &.1?1N

How to use:


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"o find the '(&VMV&.!@*, look at the first column and locate &.! then on the first row, locate .&@ (i.e.,

&.!B.&@ ) &.!@*, the point of intersection is &.1!!1. "his is the probability of interest.

"o find the '(D&.!@VMV&*, note that there is no negative value of M in the table but remember that a normal

random variable is symmetric, i.e., '(D&.!@VMV&* ) '(&VMV&.!@*. "herefore, the probability of interest is also eEual

to &.1!!1.

"o find '(D&.!@VMV&.!@*, this is eEual to '(D&.!@VMV&*B'(&VMV&.!@* ) &.1!!1B&.1!!1 ) &.%>>%.

"o find '(&.11VMV&.!@*, this is eEual to '(&VMV&.!@* – '(&VMV&.11*)&.1!!1D &.&@! ) &.&C!.

Example

f scores are normally distributed with a mean of 1&& and a standard deviation of %& then find the

probability of a personFs having an score of at least 1!&.

ANSWER:

"he random variable X follows a normal distribution with T ) 1&& and ; ) %&. We are asked to find '(X

≥ 1!&*. "o use the standard normal table, we transform this into MDscore.

'(X ≥

1!&* ) ' ( X −µσ ≥ 130−100

20 ) ) '(M ≥ 1.?&*

rom the normal table, '(&VMV1.?&*)&.@!!%.

"herefore, '(M ≥

1.?&* ) &.?&D&.@!!% ) &.&>>.

Note: V in normal distribution is eEuivalent to≤

. "his is also true for eEual to≥

.

Exer#ise

1. ind the following areas under standard normal curve:

a. to the left of M ) 1.N? h. to the right of M ) D1.%?

b. to the left of M ) %.!! i. between M ) & and M ) %.@&

c. to the right of M ) %.!! G. between M ) D%.!& and M ) &d. to the right of M ) 1.%! k. between M ) D&.? and M ) !.@

e. to the left of M ) D&.> l. between M ) 1.%! and M ) %.%

f. to the left of M ) D%.?

g. to the right of M ) D &.@?

%. ind U if a. the area between & and M is .@C%%

b. area to the left of M is .C1C%c. area to the right of M is .CCN@

d. area between DM and M is .>%>

e. area between –M and & is .@>@

!. Word 'roblem


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a. - certain type of storage battery lasts on the average !.? years with a standard deviation of &.? year.

-ssuming that the battery lives are normally distributed, find the probability that a given battery will last

less than %.> years.

b. "he Euality gradeDpoint averages of !&& college freshmen follow approximately the normal distributionwith a mean of %.1 and a standard deviation of &.?. $ow many of these freshmen would you expect to have

a grade between %.& and !.&, inclusive if the grade point average are computed to the nearest tenth

ᴥ+omethin5 to Read

/io5raphy of /laise Pas#al (1934 1993* +ol&er of the 'he&alier;s Dilemma

-s a precocious child growing up in rance, 3laise 'ascal showed an early inclination toward mathematics.

-lthough his father would not permit 'ascal to study mathematics before the age of 1? (removing all math texts

from his house*, at age 1% 3laise discovered on his own that the sum of the angles of a triangle are two right angles.

'ascal went on to become a distinguished mathematician, as well as a physicist, theologian, and the inventor of thefirst digit calculator. 4ost historians attribute the beginning of the study of probability to the correspondence

between 'ascal and 'ierre de ermat in 1>?@. "he two solved "he 9hevalierFs /ilemma – a gambling problem

related to 'ascal by his friend and 'aris gambler the 9hevalier de 4ere. "he problem involved determining the

expected number of times one could roll two dice without throwing a double >. ('ascal proved that the YbreakDevenZ point was %? rolls*

/io5raphy of 'arl 1@@* 0he


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!nferential"tatistics

#stimation

Point#stimation

Con$%ence!nter&al

#stimation

Hypot'esis (esting

Hypot'esisof o

%i)erence

Hypot'esisof o

*elations'ip

+ , parameter -un.non0

estimator , 3 -point estimate0

+ , parameter -un.non0

-25 40 con$%ence inter&al estimate

Example

Resear#h Problem: $ow effective is 'aracetamol in treating headache

+pe#ifi# ,bBe#ti&es:

1. to estimate the population average length of time headache is gone right after paracetamol intake

%. to determine whether treatment using paracetamol is better than the existing treatment that is known to

eliminate headache in 1? minutes after intake

Cuestion: $ow do we achieve these obGectives using inferential statistics

"o achieve obGective 1, we use estimation and for obGective % use hypothesis testing.Point -nter&al Estimation

1. Point Estimation – based on sample data, a single number is calculated to estimate the population parameter and

the resulting number is called point estimate

%. 'onfiden#e -nter&al Estimation – based on sample data, two numbers are calculated to form an interval,

consisting of the lower limit and an upper limit, within which the parameter is expected to lie with probability (1D[*1&& percent and the resulting pair of numbers is called an interval estimate or a confidence

interval


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or example, suppose we want to estimate the mean rating of students on the food Euality of a canteen. or n)%?

students,

´ x is calculated to be C\ out of 1&&\( perfect rating*.

´ x is called point estimator while its specific value, C\, is called point estimate.

-n alternative statement is:

"he mean rating is beteen ?\ and C!\.

(?\DC!\* is called interval estimate.

'hara#teristi#s of a 83 " 83

%F &.1& &.&? 1.>@?

%F@ &.&? &.&%? 1.C>

%F &.&% &.&1 %.!!

%FF &.&1 &.&&? %.?N?

Remarks:

1. "he unbiased estimator of the population mean T is the sample mean´ x

.

%.

x"he unbiased estimator of the population variance ;% is the sample variance s%.


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!. "he standard error #< is calculated as s+ .

Example

"he height in centimeters of a random sample of ? basil plants are 1@.>, 1%.?, 1?.!, 1>.1 and 1@.@. ind a

point estimate for the mean height of all basil plants.

#ince a good point estimate for the population mean height of basil plants is the sample mean height then

1@.> 1%.? 1?.! 1>.1 1@.@1@.?

? x

+ + + += =

"herefore, 1@.? is the estimated mean height of all basil plants.

'onfiden#e -nter&al Estimation

When the sampling distribution of a point estimator is approximately normal, an interval estimator or

confidence interval (9* can be constructed using the following reasoning: -ssuming that a C?\ 9 is to beconstructed.

1. or the standard normal variable U, C?\ of all values lie between 1.C>.

%. or an unbiased point estimator with a normal sampling distribution, C?\ of all point estimates lie within

1.C>A#< of the parameter of interest.

!. "he 9 is : point estimate margin of error where margin of error is M [+%A#< .

• n general, confidence interval can be established at any confidence coefficient (1D[*.

• 6ooking at the table below, if we want a C&\ 9 then M [+% ) 1.>@?.

• f we want a CC\ 9 then M [+% ) %.?N? .

0wo Desirable 'hara#teristi#s of a


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´ x∓ zα /2σ

√ n=( ́ x− zα /2 σ √ n , ´ x+ zα /2

σ

√ n )

%. - (1D [*1&&\ small sample 9 for a population mean T

´ x∓ t α 2 , n−1

s

√ n ¿( ´ x− t α

2 ,n−1

s

√ n, ´ x+t α

2 , n−1

s

√ n )

Where: t[+%,nD1 will come from a tDtable (see -ppendix "able %*

! part of t>table

df %@ %3@ %1 %@

1 >.!1 1%.N1 !1.% >!.>>

3 %.C% @.! >.C> C.C%

4 %.!? !.1 @.?@ ?.@

6 %.1! %.N !.N? @.>

@ %.&% %.?N !.!> @.&!

9 1.C@ %.@? !.1@ !.N1

Example: if [ ) .&? and n ) N then t[+%,nD1 ) t.&%?,> ) %.@?

Remarks:

1. 6arge sample 9 means n]!& while small sample 9 means nV!&.

%. f the population standard deviation ; is unknown and sample siMe n ] !&, use sample standard deviation s to

estimate ; and the formula for large sample 9.

!. t[+% value is found in tDtable with nD1 degrees of freedom

@. - larger confidence level produces a w i d e r confidence interval which provides little information.

?. 6arger values of ; produce w i d e r confidence intervals.

>. ncreasing the sample siMe decreases the width of the confidence interval while the confidence level can remainunchanged but this also increases the #ost of obtaining additional data.

Example

1. n order to use our confidence interval estimator, we need the following pieces of data:

!N&.1>

1.C>9alculated from

the data


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N?

n %?

6arge sample 9 is used because ; is known+given.

"herefore

´ x∓ zα /2σ

√ n=370.16∓1.96

75

√ 25=370.16∓29.40=(340.76,399.56)

4argin of error is %C.@&.

"he lower and upper confidence limits are !@&.N> and !CC.?>.

"o interpret, we are C?\ confident that the population mean T has values between !@&.N> and !CC.?>.

%. - simple random sample of ? months of food sales data provided the following information:

4onth: 1 % ! @ ?

Hnits #old: C@ 1&& ? C@ C%

"o construct a C&\ 9 for the population mean food sales per month, we need the following data:

C!

t[+% t&.&? ) %.1!%

s ?.!C

n ?

?.!C

?

÷

Hsing the formula for small sample 9 , we have C!%.1!% ) (N.>,C.1@*

"herefore, a C&\ 9 for the population mean food sales per month is between N.> and C.1@.

Exer#ises

1. ortyDseven sedentary men were studied over a oneDyear period. "hese men tried to lose weight through exercise

alone. "he results of the study showed that the mean weight loss of these men was &.N kilograms with a standard

deviation of @.. ind the C?\ confidence interval for the population mean and interpret the result.

%. -ssume that the monthly salary of a food technologist is normally distributed. ind a CC\ confidence interval

for the population mean monthly salary and interpret it if the observed salaries of ? food technologists are as

follows:%%,1!& 1,@>? %?,>1> %%,@@& 1C,>C

Hypothesis 0estin5

0oal is to make statement(s* regarding unknown population parameter values based on sample data

0iven


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- hypothesis is a claim (assumption* about a population parameter:

population mean

Example: 0he mean weekly expenditure on food in this #ity is µ 7 G163%

Re&iew of +tatisti#al Notations:

'arameters (population* #tatistics(sample*

4ean T x

5ariance ;% s%

#tandard /eviation ; s

'roportion p p

Elements of Hypothesis 0estin5

"he $ypotheses

Null hypothesis (Ho* the statement being testedI it represents what the experimenter doubts to be true

!lternati&e hypothesis (Ha* the operational statement of the theory that the experimenter believes to be true

and wishes to prove

Note: "he null hypothesis and alternative hypothesis must be nonDoverlapping statements about the population.

Example

9onsider a test by a light bulb manufacturer to examine the life of a new longDlife bulb it hopes to market.

"he leading brand of bulb in the market has a mean burning time of %&&& hours. or advertising purposes, the

manufacturer wishes to prove that the new bulb has a longer mean burning time.

6et µ ) mean burning time of the new longDlife bulb

#tatement of hypotheses: $o: µ ) %&&& vs $a: µ %&&&

0he 0est +tatisti#

D a statistic computed from the sample data that is especially sensitive to the differences between $o and $a

^ "he test statistic should tend to take on certain values when $o is true and different values when $a is true.

^ "he decision to reGect $o depends on the value of the test statistic

^ ! de#ision rule based on the &alue of the test statisti#: _eGect $o if the computed value of the test

statistic falls in the region of reGection.

Re5ion of ReBe#tion8ReBe#tion Re5ion

> the set of all values of the test statistic which will lead to the reGection of $o


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e&el of "igni$cance ,

Ho + , +o

Ha + 7 +oeft8taile% test

Ho + , +oHa + 9 +o*ig't8taile% test

Ho + , +o

Ha + : +o

(o8taile% test

Aa#tors that Determine the Re5ion of ReBe#tion

^ the behavior of the test statistic if the null hypotheses were true

^ the alternative hypothesis: the location of the region of reGection depends on the form of $a

^ level of significance (α*: the smaller α is, the smaller the region of reGection

.e&el of +i5nifi#an#e and the ReBe#tion Re5ion

.ower tail test also called leftDtailed test

2pper tail test – also called rightDtailed test

.e&el of +i5nifi#an#e (

*

D the siMe of the risk (& V α V 1* of erroneously reGecting $o that the researcher is willing to make

! +ummary of Possible De#isions in Hypothesis 0estin5 and their 'han#es of ,##urren#e

0rue +ituation

De#ision Ho is true Ho is false

_eGect $o "`'< error 92__


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^ "he choice of α usually depends on the conseEuences associated with making a "ype error.

9ommon 9hoices of [ 9onseEuences of "ype error

&.&1 or smaller very serious

&.&? moderately serious

&.1 not too serious

^ 3ecause of the inverse relationship of α and β, setting a very small α should also be avoided if the

researcher cannot afford a very large risk of committing a "ype error.

. 9onclusion+nterpretation, answering the problem.

Note: #teps ! and @ are sometimes interchange in some books.

0ests 'on#ernin5 ,ne Population )ean

$o: µ ) µo (where µo is a specified value of the population mean*

Ho 0est +tatisti# Ha De#ision Rule

8)8o

o x '

n

µ

σ

−=

(largeDsample test+MDtest*

88o MM[

8V8o MVDM[

88o MM[+%

8)8o

o x& "

n

µ −=

(smallDsample test+tDtest*

88o tt[,nD1

8V8o tVDt[,nD1

88o tt[+%,nD1

Example

"est the claim that the true mean of gas stoves in H# homes is less than !. (-ssume ; ) &.*

+olution


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1. ormulate the appropriate null and alternative hypotheses

$o: 8 ) ! $a: 8 V !

%. "his is a leftDtailed test

#uppose that α ) .&? is chosen for this test

!. /etermine the reGection region

"his is a oneDtailed test with α ) .&?.

#ince is known, the cutoff value is a M value:

_eGect $& if M V Mα ) D1.>@? I otherwise do not reGect $&

@. 9omputation of test statistic.

#uppose a sample is taken with the following results:

n ) 1&&, x ) %.@ (σ ) &. is assumed known*

"hen the test statistic is:

x8 %.@ ! .1>

M %.&; &. .&

n 1&&

− − −

= = = = −

? and >. _each a decision and interpret the result


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#ince M ) D%.& V D1.>@?, we reGect the null hypothesis and conclude that at ?\ level of significance, we have

sufficient evidence to say that the mean number of gas stoves in H# homes is less than to !.

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