STATS - DOANE - Chapter 15 Chi-Square Tests

Chapter 15 Chi-Square Tests

True / False Questions

1.

In a chi chi-s -squ quar are e tes testt of of a 5 × 5 con conti ting ngen ency cy tabl table e at at α = .05, the critical alue is !".#5. FALSE

$%.05 = %#.!0 for d.f . = &5 - 1'&5 - 1' = 1#.

%.

If t(o t(o ariab ariables les are are in)ep in)epen) en)ent ent,, (e (oul) (oul) antici anticipat pate e a chi-s chi-squa quare retes testt statis statistic tic close to *ero. TRUE

The )i+erence )i+erence bet(een obsere) an) epecte) epecte) shoul) be near near *ero.

!.

The null null hypo hypothe thesi sis s for for a chichi-squ square are test test on a cont conting ingenc ency y table table is that that the the ariables are )epen)ent. FALSE

The null hypothesis hypothesis is in)epen)ence in)epen)ence ¬ )epen)ence'.

.

The shape shape of of the the chi-s chi-squa quare re )istri )istribut bution ion )epen) )epen)s s only only on on its its )egr )egrees ees of free free)o )o.. TRUE

The chi-square )istribution )istribution has only one paraeter paraeter &calle) )egrees )egrees of free)o'.

5.

In a chichi-squ squar are e test test for in)e in)epen pen)en )ence, ce, epe epecte cte) ) frequ frequenc encies ies ust ust be inte integer gers s &or roun)e) to the nearest integer'. FALSE

/pecte) frequencies frequencies are integers only in unusual situations &if frequencies frequencies are nice'.

#.

In a chi-sq chi-squar uare e test test for for in)ep in)epen) en)enc ence, e, obse obsere re) ) frequ frequenc encies ies ust ust be at leas leastt 5 in in eery cell. FALSE

Sall expected ¬ observed' frequencies are to be aoi)e).

".

In a chi-sq chi-squar uare e test test for for in)ep in)epen) en)enc ence, e, obse obsere re) ) an) an) epec epecte) te) frequ frequenc encies ies ust ust su across to the sae ro( totals an) )o(n to the sae colun totals. TRUE

/pecte) frequencies frequencies reallocate the ro( &or colun' total, so they ust su to the total.

.

In sap saple les s )ra(n )ra(n fro fro a popula populatio tion n in in (hich (hich the ro( an) colun colun catego categorie ries s are are in)epen)ent, the alue of the chi-square test statistic (ill be *ero. FALSE

Sapling ariation eists een if the null hypothesis is true for the population.

2.

In a hypoth hypothesi esis s test test usin using g chi-s chi-squa quare re,, if the the null null hypoth hypothesi esis s is true, true, the the sap saple le alue of the saple chi-square test statistic (ill be eactly *ero. FALSE

Sapling ariation eists een if the null hypothesis is true for the population.

10.

The chi-squar chi-square e test test for for in)epe in)epen)en n)ence ce is is a nonparaet nonparaetric ric test test &no parae paraeters ters are estiate)'. TRUE

The chi-square test test )oes not estiate a paraeter. paraeter.

11.

Cochran3s Cochran3s 4ule 4ule requir requires es obser obsere) e) frequen frequencies cies of 5 or ore ore in each cell of a contingency table. FALSE


1%.

 large large negati negatie e chi-squ chi-square are test statistic statistic (oul) in)icate in)icate that the the null null hypothe hypothesis sis shoul) be re6ecte). FALSE

It is a su of squares )ii)e) by a positie epecte) frequency so it cannot be negatie.

1!.

The )egrees )egrees of free)o free)o  in a ! ×  chi-squar chi-square e conting contingency ency table (oul) equal 11. FALSE d.f. = &! - 1'& - 1' = #.

1.

The null hypothesi hypothesis s for a chi-squ chi-square are contingen contingency cy test test of in)epen)e in)epen)ence nce for t(o t(o ariables al(ays assues that the ariables are in)epen)ent. TRUE

The null hypothesis hypothesis ust be phrase) li7e li7e this so there is only only one (ay it can be true.

15.

The chi-s chi-squar quare e test test is unreliab unreliable le (hen (hen there there are any cells cells (ith sall sall obsere) obsere) frequency counts. FALSE


1#.

The chi-squar chi-square e test test can can only only be be use) use) to assess assess in)ep in)epen)en en)ence ce bet(ee bet(een n t(o ariables. FALSE

Chi-square tests can be use) to test for goo)ness of 8t, for eaple.

1".

The chi-sq chi-squar uare e test test is is base) base) on the the analys analysis is of frequ frequenc encies ies.. TRUE

Its attraction is that the test can be perfore) on categorical )ata &counts'.

1.

 chichi-squ square are )istri )istribut bution ion is al(ays al(ays s7e(e s7e(e) ) right right.. TRUE

/specially for sall )egrees of free)o, f ree)o, the chi-square )istribution is right-s7e(e).

12.

 chi-squ chi-square are test test for for in)epen in)epen)enc )ence e is calle) calle) a )istri )istributi bution-fr on-free ee test test since since the test test is is base) on categorical )ata rather than on populations that follo( any particular )istribution. TRUE

The lac7 of assue) population population shape is an attraction of this test.

%0.

9bsere) 9bsere) freq frequenc uencies ies in in a chi-s chi-squar quare e goo)nessgoo)ness-of of-8t -8t test test for for noralit norality y ay be be less less than 5 or een 0 in soe cells, as long as the epecte) frequencies are large enough. TRUE


%1.

In a chichi-squ squar are e goo) goo)nes ness-o s-off-8t test, test, a sal salll p-alue (oul) in)icate a goo) 8t to the hypothesi*e) )istribution. FALSE

:hen the p-alue is sall, (e are incline) to reject the the hypothesi*e) )istribution.

%%.

;or a chi-squa chi-square re goo)ne goo)ness-of ss-of-8t -8t test test for for a unifor unifor )istribut )istribution ion (ith (ith 5 catego categories ries,, (e (oul) use the critical alue for  )egrees of free)o. TRUE

- 1 - m = 5 - 1 - 0 =  (here m = 0 paraeters are estiate) an) k = = 5 d.f. = k categories.

%!.

;or a chi-squa chi-square re goo)ne goo)ness-of ss-of-8t -8t test test for for a unifor unifor )istribut )istribution ion (ith (ith " catego categories ries,, (e (oul) use the critical alue for # )egrees of free)o. TRUE

- 1 - m = " - 1 - 0 = # (here m = 0 paraeters are estiate) an) k = = " d.f. = k categories.

%.

;or a chi-squa chi-square re goo)ne goo)ness-of ss-of-8t -8t test test for for a noral noral )istribut )istribution ion using using  categori categories es (ith estiate) ean an) stan)ar) )eiation, (e ( oul) use the critical alue for " )egrees of free)o. FALSE

- 1 - m =  - 1 - % = 5 (here m = % paraeters are estiate) an) k = =  d.f. = k categories.

%5.

;or a chi-squa chi-square re goo)ne goo)ness-of ss-of-8t -8t test test for for a noral noral )istribut )istribution ion using using " categori categories es (ith estiate) ean an) stan)ar) )eiation, (e ( oul) use the critical alue for  )egrees of free)o. TRUE

- 1 - m = " - 1 - % =  (here m = % paraeters are estiate) an) k = = " d.f. = k categories.

%#.

 proba probabil bility ity plot plot usua usually lly allo(s allo(s outl outlier iers s to be be )etec )etecte) te).. TRUE

9utliers (ill be seen as unusual &tail' points far fro the ain bo)y of )ata.

%".

In a goo)ne goo)ness-of ss-of-8t -8t test, test, a linear linear probabi probability lity plot sugge suggests sts that that the the null null hypothes hypothesis is shoul) be re6ecte). FALSE

%.

In a chi-squar chi-square e goo)ness goo)ness-of -of-8t -8t test, test, (e lose lose one one )egree )egree of of free)o free)o for each each paraeter estiate). TRUE

- 1 - m (here m = nuber of paraeters that are estiate) an) k = = d.f. = k nuber of categories.

%2.

In a chi-squar chi-square e goo)nes goo)ness-of s-of-8t -8t test, test, (e gain one )egree )egree of of free)o free)o  if n increases by 1. FALSE

- 1 - m for m paraeters an) k categories categories & n )oes not enter this forula'. d.f. = k -

!0.

In a chichi-squ squar are e goo) goo)nes ness-o s-off-8t test, test, a sap saple le of n obserations has n - 1 )egrees of free)o. FALSE

- 1 - m for m paraeters an) k categories categories & n )oes not enter this forula'. d.f. = k -

!1.

The ois oisson son goo)n goo)ness ess-of -of-8t -8t test test is inapp inapprop ropriat riate e for contin continuou uous s )ata. )ata. TRUE

oisson )ata are integers.

!%.

The >ologor >ologoro-S o-Sirn irno o an) an) n)erso n)erson-
The /C<; proi)es proi)es the basis for seeral such such tests.

!!.

?oo)ness-of ?oo)ness-of-8t -8t tests tests using using the /C<; &/pirica &/piricall Cuulati Cuulatie e
@es, @es, (ith each )ata alue consi)ere) consi)ere) separately &no &no grouping into categories'. categories'.

!.

n attracti attraction on of the >olog >ologoro oro-Sir -Sirno no test test is that it it is fairly fairly easy easy to )o )o (ithout (ithout a coputer. FALSE

The >-S >-S test is )one (ith a coputer. coputer.

!5.

n attracti attraction on of the n)ersonn)erson-
The -< test is )one (ith a coputer &it requires requires an inerse )istribution )istribution function'.

!#.

/C<; tests tests hae hae an an a)antag a)antage e oer oer the the chi-squar chi-square e goo)nes goo)ness-of s-of-8t -8t test on frequencies because an /C<; test treats obserations in)ii)ually. TRUE

/ach )ata alue is consi)ere) separately &no grouping into categories' so ore po(er.

!". !".

In an an /C< /C<; ; test test for for goo goo)n )nes esss-of of-8 -8t, t, the the n obserations are groupe) into categories rather than being treate) in)ii)ually. FALSE

In /C<; tests, each )ata alue is consi)ere) separately &no grouping into categories'.

!.

:hen ra( )ata )ata are are aailab aailable, le, /C<; tests usually usually surpass surpass the the chi-s chi-squar quare e test test in in their ability to )etect )epartures fro the )istribution speci8e) in the null hypothesis. TRUE

/ach )ata alue is consi)ere) separately &no grouping into categories' so ore po(er.

!2.

The n)erson-< n)erson-
Aost soft(are pac7ages hae the -< norality test because norality tests are popular.

0.

robab robabili ility ty plots plots are are use) use) to test test the assu assupti ption on of nora noralit lity y. TRUE

Aost soft(are pac7ages hae the noral - because norality tests are popular.

1. 1.

In a test test for for a uni unifo for r )ist )istri ribu buti tion on (it (ith h k categories, categories, the epecte) frequency frequency is nBk in each cell. TRUE

;or unifority (e epect nBk in in each category.

Multiple Choice Questions

%.

If saples saples are )ra(n fro a populatio population n that that is noral, noral, a goo)ness-o goo)ness-off-8t test test for norality coul) yiel)

A.

Type I error but not Type II error.

D.

Type II error but not Type I error.

C.

/ither Type I error or Type II error.

<.

Doth Type I an) Type II errors.

If the hypothesis & H0 population is noral' is true, (e cannot coit Type II error &failing to re6ect a false hypothesis'. Dut in reality, (e (oul) not 7no( that H0 is true.

!.

The nuber nuber of cars (aiting (aiting at a certain certain resi)en resi)ential tial neighborh neighborhoo) oo) stop stop light light is obsere) at #00 a.. on 1#0 )i+erent )ays. The obsere) saple frequencies are sho(n here

En)er the null hypothesis of a unifor )istribution, the epecte) nuber of )ays (e (oul) see 0 cars is

.

10.

D.

%0.

C.

!0.

D.

0.

= 1#0B = 0. nBk =

.

 chi-squ chi-square are goo)ness goo)ness of 8t 8t test test for a noral noral )istri )istributi bution on use) use) 0 0 obserati obserations, ons, an) the ean an) stan)ar) )eiation (ere estiate) fro the saple. The test use) si categories. :e (oul) use ho( any )egrees of free)o in loo7ing up the critical alue for the testF

.

!2

D.

!"

C.

5

D.

!

- 1 - m = # - 1 - % = ! (here m = % paraeters are estiate) an) k = = # d.f. = k categories &n is not in the forula'.

5.

 chi-squ chi-square are goo)ness goo)ness of 8t 8t test test for a noral noral )istri )istributi bution on use) use) #0 #0 obserati obserations, ons, an) the ean an) stan)ar) )eiation (ere estiate) fro the saple. The test use) seen categories. :e (oul) use ho( any )egrees of free)o in loo7ing up the critical alue for the testF

.

#

B.



C.

52

<.

5"

- 1 - m = " - 1 - % =  (here m = % paraeters are estiate) an) k = = " d.f. = k categories &n is not in the forula'.

#.

:hich :hich of these these statee stateents nts concer concerning ning a chi-squa chi-square re goo)ness-o goo)ness-off-8t 8t test test is correct F

A.

D.

opulation ust be norally )istribute).

C.

ll the epecte) frequencies ust be equal.

". ".

:hic :hich h of of th the fol follo lo(i (ing ng is not a a potential solution to the proble that arises (hen not all epecte) frequencies are 5 or ore in a chi-square test for in)epen)enceF

.

Cobine soe of the coluns

D.

Cobine soe of the ro(s

C.

Increase the saple si*e

D.

)) ore ro(s or coluns

Sub)ii)ing ro(s or coluns (oul) a7e the epecte) frequencies saller.

.

:hich :hich of these these statee stateents nts concer concerning ning a chi-squa chi-square re goo)ness-o goo)ness-off-8t 8t test test is correct F

. It is inapplicable to to test for a noral )istribution (ith open-en)e) open-en)e) top an) botto classes. D.

It is generally generally a better better test than than the chi-square chi-square test test of in)epen)ence. in)epen)ence.

C. There is no (ay to get the )egrees )egrees of free)o free)o since the right tail goes to to in8nity. speci8e) )istribution. )istribution. D. It can be use) to test (hether a saple follo(s a speci8e) The ?9; test as7s (hether the the saple contra)icts a propose) propose) population )istribution.

2.

 proofr proofrea)er ea)er chec7e chec7e) ) 1#0 a)s for graati graatical cal erro errors. rs. The The saple saple frequenc frequency y )istribution is sho(n

En)er the null hypothesis of a unifor )istribution, the epecte) nuber of ties (e (oul) get 0 errors is

.

10.

D.

%0.

C.

!0.

D.

0.

= 1#0B = 0. nBk =

50.

 proofr proofrea)er ea)er chec7e chec7e) ) 1#0 a)s for graati graatical cal erro errors. rs. The The saple saple frequenc frequency y )istribution is sho(n

Esing a goo)ness-of-8t test to )eterine (hether this )istribution is unifor (oul) result in a chi-square test statistic of approiately

.

55.

B.

"2.

C.

5.

<. <.

1#1.

&10 - 0'%B0 G - 0' %B0 G &"1 - 0' %B0 G &1 - 0' %B0 = "2.05.

51.

 proofr proofrea)er ea)er chec7e chec7e) ) 1#0 a)s for for graatic graatical al errors errors.. The )istri )istributio bution n obtaine) obtaine) is sho(n

t α = .01, (hat )ecision (oul) (e reach in a goo)ness-of-8t goo)ness-of-8t test to see (hether this saple cae fro a unifor )istributionF

A.

4e6ect the null null an) conclu)e the )istribution )istribution is not unifor.

D.

Conclu)e that there there is insuHcient ei)ence to re6ect re6ect the null.

C.

o conclusion conclusion can be a)e )ue to to sall epecte) frequencies. frequencies.

<.

o conclusion conclusion can be a)e )ue to ina)equate ina)equate saple si*e.

&10 - 0'%B0 G - 0' %B0 G &"1 - 0' %B0 G &1 - 0' %B0 = "2.05 J $%.01 = 11.! for d.f. = !.

5%.

 chi-squ chi-square are test of in)epe in)epen)enc n)ence e is a one-tai one-taile) le) test. test. The reason reason is that that

. (e are testing (hether (hether the frequencies frequencies ecee) ecee) their epecte) epecte) alues. alues. square the )eiations )eiations so the the test statistic lies lies at or aboe *ero. *ero. B. (e square C. hypothesis tests are are one-taile) one-taile) tests (hen )ealing )ealing (ith saple )ata. )ata. <.

the chi-square )istribution is positiely s7e(e).

The chi-square test test statistic contains &Obs - Exp'%, so )i+erences )i+erences in either )irection are positie.

5!.

:e soetie soeties s cobine cobine t(o ro( ro( or colun colun categori categories es in in a chi-squar chi-square e test test (hen (hen

.

obsere) frequencies frequencies are ore than 5.

D.

obsere) frequencies frequencies are less than 5.

C.

epecte) frequencies are ore than 5.

D.

epecte) frequencies are less than 5.

Consoli)ating t(o ro(s &or coluns' (oul) increase epecte) frequencies frequencies &but fe(er d.f.'.

5.

To )eterin )eterine e ho( (ell (ell an an obsere) obsere) set of of frequen frequencies cies 8ts an epecte epecte) ) set set of frequencies frequencies fro a oisson )istribution (e ust estiate

.

no paraeters.

B.

one paraeter & λ'.

C.

t(o paraeters & μ, σ '. '.

<.

three paraeters & μ, σ , n'.

:e lose one etra )egree of free)o (hen (e estiate the oisson ean λ.

55.

The crit critica icall alue alue in a chi-s chi-squa quare re test test for for in)epen in)epen)en )ence ce )epen )epen)s )s on on

.

the norality of the )ata.

D.

the ariance of the )ata.

C.

the nuber of categories.

<.

the epecte) frequencies.

- 1 - m for m estiate) paraeters an) k categories. categories. χ %α )epen)s on d.f. = k -

5#.

In a chi-squar chi-square e test test of in)epen)en in)epen)ence, ce, the nube nuberr of )egrees )egrees of free)o free)o equals equals the

.

nuber of obserations inus one.

D.

nuber of categories inus one.

ro(s inus one one ties the nuber of coluns inus inus one. C. nuber of ro(s <.

nuber of saple saple obserations inus the issing obserations.

- 1'&c - 1'. χ %α )epen)s on d.f. = &r -

5".

In or)er or)er to apply apply the chi-squar chi-square e test test of in)ep in)epen)e en)ence, nce, (e prefer prefer to to hae hae

.

at least 5 obsere) frequencies frequencies in each cell.

B.

at least 5 epecte) obserations in each cell.

C.

at least 5 percent of the obserations in each cell.

<.

not ore than 5 obserations in each cell.

Karger epecte) frequencies are )esirable &at least 5 accor)ing to Cochran3s 4ule'.

5.

>ortholt >ortholts s that that fail to eet eet certain certain precise precise speci8ca speci8cations tions ust be be re(or7 re(or7e) e) on the net )ay until they are (ithin the )esire) speci8cations.  saple of one )ay3s output of 7ortholts fro the Aelo)ic >ortholt Copany sho(e) the follo(ing frequencies

;in) the chi-square test statistic for a hypothesis of in)epen)ence. in)epen)ence.

. .

".%%

B.

.1"

C. C.

5.1!

<. <.

#.0

The test statistic is χ %calc = L&Obs - Exp'%BExp (here Exp = M&ro( su' × &col su'NBn.

52.

>ortholt >ortholts s that that fail to eet eet certain certain precise precise speci8ca speci8cations tions ust be be re(or7 re(or7e) e) on the net )ay until they are (ithin the )esire) speci8cations.  saple of one )ay3s output of 7ortholts fro the Aelo)ic >ortholt Copany sho(e) the follo(ing frequencies

;in) the p-alue for the chi-square test statistic for a hypothesis of in)epen)ence. in)epen)ence.

.

Kess than .01

D.

Det(een .01 an) .0%5

C.

Det(een .0%5 an) .05

<.

?reater than .05

χ %calc = .1#" (ith d.f. = 1 is bet(een χ %.05 = !.1 an) χ %.0%5 = 5.0%. Esing /cel, the p-alue is =COISP.
#0.

n operati operations ons analyst analyst counte) counte) the the nuber nuber of of arrial arrials s per inute inute at a ban7 TA TA in each of !0 ran)oly chosen inutes. The results (ere (ere 0, !, !, %, 1, 0, 1, 0, 0, 1, 1, 1, %, 1, 0, 1, 0, 1, %, 1, 1, %, 1, 0, 1, %, 0, 1, 0, 1. :hich goo)ness-of-8t test (oul) you recoen)F

. .

Enifor.

B.

oisson.

C. C. <. <.

oral. Dinoial.

rrials per unit of tie (ith a sall ean (oul) reseble a oisson oisson )istribution.

#1.

n operati operations ons analyst analyst counte) counte) the the nuber nuber of of arrial arrials s per inute inute at an TA in each of !0 ran)oly chosen inutes. The results (ere (ere 0, !, !, %, 1, 0, 1, 0, 0, 1, 1, 1, %, 1, 0, 1, 0, 1, %, 1, 1, %, 1, 0, 1, %, 0, 1, 0, 1. ;or the oisson goo)ness-of-8t goo)ness-of-8t test, (hat is the epecte) frequency of the )ata alue X = = 1F

. B.

Ipossible to )eterine. 11.0

C. C.

1.00

<. <.

%."

The saple ean is 1.00 so n × P& X = = 1 Q λ = 1.00' = &!0'&.!#"2' = 11.0!".

#%.

The table belo( is a tabulation tabulation of opinion opinions s of eployee eployees s of olotl olotl Corporation Corporation,, (ho (ere saple) at ran)o fro pay recor)s an) as7e) to coplete an anonyous 6ob satisfaction surey.

;or a chi-square test of in)epen)ence, )egrees of free)o (oul) be

A.

%

D.

!

C.



<.

#

;ee)bac7: d.f. = &% - 1'&! - 1' = %.

#!.


;or a chi-square test of in)epen)ence, the critical alue for α = .01 is

A.

2.%10.

D. D.

.#05.

C. C.

11.!.

<. <.

1#.1.

χ %.01 = 2.%10 for d.f. = &% - 1'&! - 1' = %.

#.


ssuing in)epen)ence, in)epen)ence, the epecte) frequency frequency of satis8e) hourly eployees is

.

0.

B.

20.

C.

"5.

<.

#0.

e%1 = &R%'&C1'Bn = &10'&1%0'B%0 = 20.

#5.

To carry carry out a chi-squa chi-square re goo)ne goo)ness-of ss-of-8t -8t test test for for norali norality ty you nee) at least least

. D. C.

<.

5 categories categor ies altogether altoget her.. 5 obserations obserat ions in each category categor y. 5 epecte) obserations obserati ons in each category categor y. 50 saples or ore.

Decause d.f. = k - 1 - m = k - ! since m = %, (e nee) at least k = =  groups each (ith e R 5.

##.

Stu)ents Stu)ents in in an intro intro)ucto )uctory ry colleg college e econoic econoics s class class (ere (ere as7e) as7e) ho( ho( any cre)its cre)its they ha) earne) in college, an) ho( certain they (ere about their choice of a6or. a6or. Their replies are are suari*e) belo(. belo(.

En)er the assuption of in)epen)ence, in)epen)ence, the epecte) frequency frequency in the upper left cell is

A.

15.02.

D. D.

%.00.

C. C.

12."%.

<. <.

%0.%%.

e11 = &R1'&C1'Bn = &#'&%'B1% = 15.02.

#".



.

%.

D.

2.

C.

.

<. <. d.f. = &! - 1'&! - 1' = .

1%".

#.



. .

5.221.

D. D.

".15.

C.

2..

<. <.

1#.2%.

χ %.05 = 2. for d.f. = &! - 1'&! - 1' = .

#2.


ssuing in)epen)ence, in)epen)ence, the epecte) frequency frequency of ery uncertain stu)ents (ith #0 cre)its or ore is

A.

1%.".

D. D.

%.00

C. C.

1.5#.

<. <.

11.02.

e!1 = &R!'&C1'Bn = &!'&%'B1% = 1%.".

"0.


:hich stateent is ost nearly correctF

. D.

The contingency table iolates Cochran3s 4ule. 4ule. isual inspection of colun colun frequencies suggests in)epen)ence. in)epen)ence.

C. t α = .05 (e (oul) easily re6ect the null hypothesis of in)epen)ence.

<.

t α = .05 (e cannot re6ect he null hypothesis of in)epen)ence.

χ %calc = %2.5% J χ %.05 = 2. for d.f. = &! - 1'&! - 1' = .

"1.

s an an in)epe in)epen)ent n)ent pro6ect, pro6ect, a tea tea of statistic statistics s stu)ent stu)ents s tabulate tabulate) ) the the types types of ehicles that (ere par7e) in four )i+erent suburban shopping alls.


.

%0.

B.

1%.

C. C. <. d.f. = &5 - 1'& - 1' = 1%.

!22. #.

"%.



. .

10.#.

D. D.

1.#.

C. C.

%.1.

D.

1.55.

χ %.10 = 1.55 for d.f. = &5 - 1'& - 1' = 1%.

"!.


ssuing in)epen)ence, in)epen)ence, the epecte) frequency frequency of SEs in aesto(n is

.

1%.

B.

%1.

C.

"5.

<.

#0.

e = &R'&C'Bn = &'&100'B00 = %1.

".

/ployee /ployees s of 9Co Afg. (ere sureye) sureye) to ealuat ealuate e the the copany3s copany3s pension pension plan. plan. The table belo( )isplays )isplays soe of the results of the the surey. surey.

The epecte) frequency frequency for the sha)e) cell cell in the table (oul) be

. .

1#!.

D. D.

15.

C. C.

1#5.

D.

1#0.

e%% = &R%'&C%'Bn = &00'&!%0'B00 = 1#0.

"5.


A.

#.

D.

".

C. C. <. d.f. = &! - 1'& - 1' = #.

"22. 1%.

"#.


The appropriate conclusion conclusion (oul) be

A.

)o not re6ect H0.

D. D.

re6ect H0 at α = .10.

C. C.


<. <.


χ %calc = #.%0# )oes not een ecee) χ %.10 = 10.# for d.f. = &! - 1'& - 1' = #.

"".

@ou test test a hypothes hypothesis is of in)epen)e in)epen)ence nce of t(o ariables ariables.. The nuber nuber of obserations is 500 an) you hae classi8e) the )ata into a  by  contingency table. The test statistic has UUUUUUUUUU )egrees of free)o.

.

1# 2

B.

C.

22

<.

2

Saple si*e )oes not enter into the calculation d.f. = & - 1'& - 1' = 2.

".

@ou (ant to test test the the hypothes hypothesis is that that the prie rate an) inVatio inVation n are in)epen)en in)epen)ent. t. The follo(ing table is prepare) prepare) for the test on the basis basis of the results of a ran)o saple, collecte) in arious countries an) arious tie perio)s

Esing α = .05, (hat is the critical alue of the test statistic that you (oul) useF

. .

!.1

D. D.

1%.52

C.

5.221

<. <.

".15

χ %.05 = 5.221 for d.f. = &% - 1'&! - 1' = %.

"2.


The epecte) frequency frequency for the sha)e) cell cell is

A.

%%.5.

D.

!0.

C.

0.

<. <.

0.5.

e%! = &R%'&C!'Bn = &5'&"5'B150 = %%.5.

0.


:hat is the alue of the test statisticF

. . D. D.

!0#.%5 0.00

C.

5.

<. <.

1!.#1

χ %calc = &0 - %%.5'%B%%.5 G &!0 - !0'%B!0 G &5 - %%.5'%B%%.5 G &5 - %%.5'%B%%.5 G &!0 !0'%B!0 G &0 - %%.5'%B%%.5 = 5..

1.

@ou (ant to test test the the hypothes hypothesis is that that the prie rate an) inVatio inVation n are in)epen)en in)epen)ent. t. The follo(ing table of frequencies frequencies is prepare) prepare) fro a ran)o saple, saple, collecte) in arious countries an) arious tie perio)s

Dase) on an analysis of the )ata in this table, (hich conclusion can be a)e at α = .01F

.

The prie rate an) inVation rate are in)epen)ent.

B.

The prie rate an) inVation rate are not in)epen)ent.

C. Sall obsere) frequencies frequencies in soe cells suggest that no reliable conclusion conclusion can be a)e. <. Sall epecte) frequencies frequencies in soe cells suggest that no reliable conclusion can be a)e. χ %calc = 5. greatly ecee)s χ %.01 = 2.%10 for d.f. = &% - 1'&! - 1' = %.

%.

@ou (ant to sell sell your your house, house, an) an) you )eci)e )eci)e to obtain obtain an apprai appraisal sal on it. Koo7in Koo7ing g at past )ata you )iscoer that actual prices obtaine) for houses an) the appraisals gien for the prior to their sale (ere as follo(s

Dase) on these )ata (e can say that

.

no conclusion is possible (ithout 7no(ing α.

D.

appraisal an) actual price are not in)epen)ent at α = .05.

C.

appraisal an) actual price are in)epen)ent at any α.

<.

the )egrees of free)o are insuHcient for a )ecision.

Colun frequencies are all in the sae ratio !% so perfect in)epen)ence eists eists & e = f '. '.

!.

refer reference ences s for the the type type of )iet )rin7 fro a ran)o ran)o saple saple of of 1%1 shoppers shoppers are are in the table belo(.  researcher is intereste) in )eterining if there is a relationship bet(een the type of )iet )rin7 preferre) preferre) an) the age of the shoppers.

In perforing a chi-square test of in)epen)ence on these )ata, ho( any )egrees of free)o (ill the test statistic haeF

.

1

B.

%

C.



<.

#

d.f. = &! - 1'&% - 1' = %.

.


Esing α = .0%5, (hat is the critical alue of the test statistic that you (oul) use in a )ecision rule to test an appropriate hypothesisF

. .

5.0%

D. D.

5.22

C.

".!

<. <.

1.5

χ %.0%5 = ".!" for d.f. = &! - 1'&% - 1' = %.

5.


:hat can you conclu)e for the )ata analysis at α = .05F

.

The eans are equal for all three groups.

D. There is insuHcient insuHcient ei)ence ei)ence to conclu)e that the type of )rin7 )rin7 an) age are )epen)ent. C.

Conclu)e that the type of )rin7 an) age are )epen)ent.

<.

o conclusion is possible (ithout 7no(ing the p-alue.

χ %calc = %1.%1 greatly ecee)s χ %.05 = 5.221 for d.f. = &! - 1'&% - 1' = %.

#.

 taste taste test test of ran)oly ran)oly selecte) selecte) stu)ents stu)ents (as con)uc con)ucte) te) to see see if there there (as a )i+erence in preferences preferences aong four popular )rin7s. The follo(ing table sho(s the frequency of responses

The epecte) nuber nuber of stu)ents preferring preferring
.

%5.

D.

0.

C.

50.

<.

#0.

= %00B n = 51 G ## G ! G 0 = %00 so, assuing a unifor )istribution, e = nBk = = 50.

".


Esing α = .0%5, the critical alue of the test you (oul) use in )eterining (hether the preferences preferences are the sae aong the )rin7s is

. .

5.221.

D. D.

".!".

C.

2.!.

<. <.

11.0".

-1 =  - 1 = !. χ %.0%5 = 2.! for d.f. = k -1

.


The alue of the chi-square chi-square test statistic you you (oul) use in testing (hether (hether the preferences preferences are the sae aong the )rin7s is

. .

".5.

B.

.1%.

C. C. <. <.

10."#. <.1%.5#.

&51 - 50'%B50 G &## - 50' %B50 G &! - 50' %B50 G &0 - 50' %B50 = .1%.

2.


Esing α = .0%5, (hat can you conclu)e fro your analysisF

. 4e6ect 4e6ect the null an) conclu)e soe )rin7s are preferre preferre) ) ore than others. others. B.

C. <.

There is not enough ei)ence ei)ence to say that a preferen preference ce eists. eists. epsi is the preferre) )rin7. ;or no conclusion because Cochran3s 4ule is iolate).

χ %calc = &51 - 50'%B50 G &## - 50' %B50 G &! - 50' %B50 G &0 - 50' %B50 = .1% )oes not ecee) χ %.0%5 = 2.! for d.f. = k -1 -1 =  - 1 = !.

20.

The 9nar) 9nar) 4etail 4etailers ers ntinti-Thef Theftt lliance lliance &94 &94T T' publishe publishe) ) a stu)y that that claie) claie) the causes of )isappearance of inentory in retail stores (ere !0 percent shoplifting, 50 percent eployee theft, theft, an) %0 percent faulty paper(or7. The anager of the Aelo)ic >ortholt 9utlet perfore) an au)it of the )isappearance of 0 ites an) foun) the frequencies sho(n belo(. She (oul) li7e to 7no( if her store3s eperience follo(s the sae pattern as other retailers.

En)er the null hypothesis that her store follo(s the publishe) pattern, the epecte) nuber of ites that )isappeare) )ue to shoplifting is

.

1#.

D.

0.

C.

%.

<.

%".

n = !% G ! G 11 = 0 so for shoplifting e = .!0 × 0 = %.

21.


Esing α = .05, the critical alue you (oul) use in )eterining (hether the Aelo)ic >ortholt3s pattern )i+ers fro the publishe) stu)y is

. .

".15.

B.

5.221.

C. C.

1.2#0.

<. <.

1.#5.

-1 = ! - 1 = %. χ %.05 = 5.221 for d.f. = k -1

2%.


The alue of the chi-square chi-square test statistic you you (oul) use in testing (hether (hether there is a )i+erence fro the publishe) pattern is

. .

".5.

B.

5.0%.

C. C.

2."#.

<. <.

2.%%.

&!% - %'%B% G &! - 0' %B0 G &10 - 1#' %B1# = 5.01#" (ith π 1 = .!0, π % = .50, π ! = . %0 an) n = 0.

2!.


Esing α = .05, (hat can you conclu)e fro your analysisF

. The store3s pattern is clearly clearly signi8cantly signi8cantly )i+erent )i+erent fro the publishe) publishe) )ata. )i+erent fro the B. The store3s pattern is alost, but not quite, signi8cantly )i+erent publishe) )ata. C. <.

The store3s pattern is ery close to the publishe) )ata. :e can for no conclusion conclusion because because Cochran3s 4ule is iolate). iolate).

χ %calc = &!% - %'%B% G &! - 0' %B0 G &10 - 1#' %B1# = 5.01#" (ith π 1 = .!0, π % = . 50, π ! = .%0 an) n = 0 an) χ %.05 = 5.221 for d.f. = k -1 -1 = ! - 1 = %, so (e cannot quite re6ect H0 π 1 = .!0, π % = .50, π ! = .%0.

2. 2.

 cont contin inge genc ncy y tab table le sho( sho(s s

A.

frequency counts.

D.

eans of the )ata.

C.

eent probabilities. probabilities.

<.

chi-square alues.

Contingency tables contain count )ata.

25. 25.

:e (oul (oul) ) crea create te a con conti ting ngen ency cy tab table le by by

.

suing the probabilities of t(o ariables.

B.

cross-tabulating cross-tabulating frequencies frequencies of t(o ariables.

C.

applying the chi-square )istribution to a saple.

<.

using Cochran3s 4ule to estiate frequencies. frequencies.

 contingency table is a t(o-(ay t(o -(ay frequency )istribution.

2#.

:hich :hich )ata )ata set set is consis consisten tentt (ith the the hypot hypothes hesis is of a noral noral popu populat lation ionF F

A.

D.

C.

either )ata set.

<.

Doth )ata sets.

2".

:hich :hich state stateen entt is ost ost nearl nearly y corre correct ct rega regar) r)ing ing /C<; /C<; test testsF sF

. n attraction of the n)erson-ologoro-Sirno, >ologoro-Sirno, probability plot' generally gain po(er by consi)ering each )ata point separately. separately. Oo(eer, these tests are not easy (ithout a coputer.

Short Answer Questions

2.

Dase) on soe soe i)eas i)eas epres epresse) se) in his his psych psychology ology class, class, ohn ohn )eci)e )eci)e) ) to test a hypothesis about the possible relationship bet(een parent )oinance an) political ie(s. Oe use) a surey of 12 statistics stu)ents to prepare the cross-tabulation an) chi-square analysis sho(n belo(.
Aost cells hae obsere) frequencies that are quite close to (hat (oul) be epecte) un)er the hypothesis of in)epen)ence. in)epen)ence. The chi-square test statistic &5.51' is no(here near the critical alue &15.51' for α = .05. The p-alue &."01' says that such a saple coul) happen by chance about "0 ties in 100 saples if the t(o ariables (ere actually in)epen)ent, so the )ata )o not perit re6ection of the hypothesis of in)epen)ence. The lo(er left cell has a slightly sall epecte) frequency &.#', but the other epecte) frequencies are all at least 5 &Cochran3s 4ule' so a larger saple sees unli7ely to change the conclusion. ;ee)bac7 ;ee)bac7 Aost cells hae obsere) frequencies that are quite close to (hat (oul) be epecte) un)er the hypothesis of in)epen)ence. in)epen)ence. The chi-square test statistic &5.51' is no(here near the critical alue &15.51' for α = .05. The p-alue &."01' says that such a saple coul) happen by chance about "0 ties in 100 saples if the t(o ariables (ere actually in)epen)ent, so the )ata )o not perit re6ection of the hypothesis of in)epen)ence. The lo(er left cell has a slightly sall epecte) frequency &.#', but the other epecte) frequencies are all at least 5 &Cochran3s 4ule' so a larger saple sees unli7ely to change the conclusion.

22.

Dase) on soe soe i)eas i)eas epr epresse esse) ) in her her psychol psychology ogy class, class, ;rie) ;rie)a a )eci)e) )eci)e) to test test a hypothesis about the possible relationship bet(een political ie(s an) the nuber of traHc tic7ets receie). She use) a surey of 12 statistics stu)ents to prepare the cross-tabulation an) chi-square analysis sho(n belo(.
/cept in the 8rst colun, ost of the cells hae obsere) frequencies that are close to (hat (oul) be epecte) un)er the hypothesis of in)epen)ence. The chisquare test statistic &5."0' is (ell belo( the critical alue &2.' for α = .05. The p-alue &.%%%' says that such a saple coul) happen by chance about %% ties in 100 saples if the t(o ariables (ere actually in)epen)ent, in)epen)ent, so the )ata )o not perit re6ection re6ection of the hypothesis of in)epen)ence at the usual leels of signi8cance. The lo(er left cell has a sall epecte) frequency &!.%#', as )oes the lo(er right cell &.00'. The other epecte) frequencies frequencies are all at least 5 &Cochran3s 4ule'. The saple is fairly large, but if ;rie)a (ants to increase the epecte) frequencies, frequencies, she ight ta7e a larger saple. ;ee)bac7 ;ee)bac7 /cept in the 8rst colun, ost of the cells hae obsere) frequencies that are close to (hat (oul) be epecte) un)er the hypothesis of in)epen)ence. The chi-square test test statistic &5."0' is (ell (ell belo( the critical alue alue &2.' for α = . 05. The p-alue &.%%%' says that such a saple coul) happen by chance about %% ties in 100 saples if the t(o ariables (ere actually in)epen)ent, so the )ata )o not perit re6ection of the hypothesis of in)epen)ence at the usual leels of signi8cance. The lo(er left cell has a sall epecte) frequency &!.%#', as )oes the lo(er right cell &.00'. The other epecte) frequencies frequencies are all at least 5 &Cochran3s 4ule'. The saple is fairly large, but if ;rie)a (ants to increase the epecte) frequencies, frequencies, she ight ta7e a larger saple.

STATS - DOANE - Chapter 15 Chi-Square Tests

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