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Hypothesis Testing L. Practice

Null and Alternative Hypotheses For the items below, write a null and an alternative hypothesis. Mark the claim with an*

.,u... ::=: 'J:---1. The average age of a CCC studen is ~-_-I-,~ uJ

VO --1-,o * Ho: ~=2.8'

2. The average SAT score for a scholarship athlete is less tQIDl.-450.

p., .A,,(__~ ,Lc_n L_ l.-. "1:_fj-Ho : _ -r; ~-.r-" Y'-, \

-¥Hi : ,),,L L 450

3. They put an averag~of ~t lea4,t 12 oz of beer in the cup. XC - f ,1 r w:.n. yi"

·fHo: _M_?.:: \2.

H1: .f{_,L-- \2.

4. A majority (J:.:U.QJ:e than 50%) of people favor the tax cut.

~ / ~ O~ -r¼'s _/ Ho : 'if' ~,'fl>

' vJ~ y.. \ i.l( H1 : 7T' ? "So

5. The mew is at most 5. ovvt-tk.:,. w:h. J,(t; _,(,,\, .t:. ----- r Ho: ,u_ > <;

;\c. H1: A~ S

6. 15% of the students who take the placement test don' t need any basic math classes. --~ = J; -Ho: TJ-=:-- . 1( - ~x.L ~ H1: I'~ ,S ~ 1,vJ)__ y{ 1

7. The amount is .at.kas.t 3.fil:L. ~ ·- "?' 1'

~ylc, * Ho: II~ ·• SO w,J:l---

~ llOA)~ ~ ~ wv\1-. X1

Types of Hypothesis Tests Which test should be completed? 2: ttest, 1:ztest, 5:lpropztest,

Claim about a mean, cr known, n ~ 30

Claim about a mean, cr unknown, n < 30

Claim about a percentage, proportion, or probability

Critical Values (Table 5, t - table) For the information below, find the critical value(s)

1. Claim about a mean, ~ , two tails, a= .05

ic-o t"tf)M ~w<EY 2. Claim about a mean, n = 15, two tails, a =.05

3. Claim about a mean, n = 100, left tail, a =.05

b6t\oM \ ~ k, L rovJ

4. Claim about a pro ortion, two tails, a =.0l

6ur\,,M (1>W ~ 6\\wW')'> ~

5. Claim about a proportion, right tail, a= .10 - \ ~--tt;M ~

ii\ l1 t

USL beifc;,..,.__ row1

n... ~ 30 VSR- Jt:;: vt.;I J V\.. L 3o

Sample problem Z test with two tails

In a newspaper article it states that the average time for a full-time student to complete a BA is . .five years . A researcher wants to test this claim for the students at her college. Using the graduates of the college for 2015-2019 as the population, the researcher randomly selected a group of J.D,O. students. For that sample, the mean length of time it took to graduate was.ilyears, with a population standard deviation of _lliears. At the 0.05 level of significance, does the newspaper's statement seem reasonable?

Step One:

Step Two:

Step Three:

Step Four:

Step Five:

-1 .96

Identify the claim The mean is 5

Put the claim in symbols µ=5

What is true if the claim is not true?

OffDs~t t- _:=. ~5 ~ H 0 µ = 5 (Null)

Identify the Null and Alternative Hypotheses: ('Q.M 1.,-L-- ),{ Ii 0, ::::. ~ '?. H1 µ * 5 (Alternative)

vl\ - "!s; --, £ Use the significance evel to find the critical value(s) and rejection region Two tails and 0.05 has critical values of ± 1.96

µ=5

/,JOf: P..N ; c,L- -(2,,v l cki I.(_ "ti> I { jl ct"

Fail to reject the Null

0

-4--No 11A ,' ls ol. = \ () <; Y\ ~ 3 0 6v tfo11--- (1) W

1.96

Rejection region

Use the z test formula to determine the z score for the sample data: +-~,s f ~ vJ\,-.~ ~r ciJe-~,..-- ~

5TAT -)TZ:ST

z = x - µ z = 4.8 - 5 z = -0.2 z = -4 ) CY 0.5 0.05

l ;z-~

Tn MO--~ Step Seven:

)( -=-4, ~ ~'.::-0 , -;

V\:::-101)

&\ o • •

t V\

~s CS .::.0.,~ i" :::. Lf t 1, I'\::. I 01J

~ )

Step Eight:

Make a decision t reject fail to reject the null, based on the values in step five and six: Since -4 excee s -1.96 it is in the rejection region

-Y ·,s w-,\~ ~ tk le-~{: (P ''sY tk- CV c{ -).1Cc, Answer the question about the claim: t does not seem like the statement that it takes five years to complete a degree is accurate.

\ V\ ,e. >J ~ "') -f'L<_. nvJ..A we..,. (J( r ~

S 'j l &A ) W L- cL,~ fh1 A \c. ~ ~ , (\,' v,, ll~ to k t-c~

Practice problem Z test with two tails

In a newspaper article it states that the average time for a full-time student to complete an AA js M, =- 3 three years. A researche/ wants to test this claim for the students at her community college. Using the gradua tes of the college for 2015-2019 as the population, the researcher randomly selected a group offutudents. For that sample, the mean length oftime it took to graduate was 3.2 years, with a standard deviation of..Ql..xears. At the 0.10 level of significance, does the newspaper's ')'C\ ~ statement seem reasonable? , .,...__ h:)

_L.1- fA_A , L \ <' 5 (l:::-5~ Step One: Identify the claim 'C.1--JL. L--"\ "' ' x -;..s,1. Step Two: Put the claim in symbols A :::. 3 S -;:.. O, 1

Step Three:

Step Four:

Step Five:

What is true if the claim is not true? A,:S. 3

Identify the Null and Alternative Hypotheses:

Xo M-==- 3 ){ 1 M~3

Use the significance level to find the critical value(s) and rejection region

(\. -::. "3 s- s µ= ~ofto fV\ f7Jw

Rejection region

Fail to reject the Null Rejection region

Step Six:

Step Seven:

Step Eight:

-i '4

z = x:µ :::_ 1 3 / - -)T~S\ X:::.},'Z-

✓n ) ~ i-,k,s.x- n-==- s~ (£)

Mak~ a decision to r. eject or fail to reject the null, based on ,the values in step five '.\

ands1x: (~ycJ- ~ . nJJ. (3l 1 \S t.'\-CJ~ ih- (<G'i~) Answer the question about the claim:

~ ~',"" ~ net h D\f\,Z eN i clw u_ +v

~ 1€..'\SOv'\ q,ble.,,

rttcJ. -vt

· e problem th one tail

A teacher claims that it takes less than 90 minutes to complete her final ... on average. A sample of 20 studen"°k an average of 85 minutes with a standard deviation of 5 minutes. With a - - -significance level of .05, is there evidence to support her claim?

Step One:

Step Two:

Step Three:

Step Four:

Step Five:

Identify the claim tu- ti-\ e.tn.... 1 t, \ l S S ~""' °l ()

Put the claim in symbols _)A.. L 9 0 ( r w ~ \A I )

What is true if the claim is not true? .M. ;J °1 D (~ w.A1- Uo) Identify the Null and Alternative Hypotheses: J,i

O .u_ ~ ~ o

Kt ..Lt'- czo Use the significance level to find the critical value(s) and rejection region

J I J ,M.. L '10 Q\.(qw Y'v t

µ=

Rejection region

Fail to reject the Null

- \olZJ o Step S'x: Use the t test formula to determine the t score for the sample dat ~Y

- --- · STAT .u..-= °lo t=x~µ -'-LY7 T~l x=-% ✓n ._. -z,,,: l 1-i~ S ~~

~ Mak-e a decision to reject or fail to reject the null, based on the valu~ (n step five

and six: rt j cX 1L& Vl vJJ._ S-1' V'I c.e - l-f. '-fl ~ le.; s. ·thew - I. 7Z.fj

Step Eight: Answer the question about the claim:

Jq ;/\A w Pl ~ '\ le Vi ~~ h °IO ~ ~ ; s o J-cutVU..

s. WA -fi.J- i l- t~ k,J' kss 4 "to-:- .

Practice problem Proportion test with one tail v>p,~

A dean claims tha- e than 75%, af students at her co Hege are in counti) In a sample of 3 3 students, 27 were in county. With a significance level of .01, is there evidence to support her----....

claim? S'-+ i ~ ~ StepOne: Identifytheclaim Mc>r-1L.~1~ 7r;rl, Q\("'Q.. ,·.,., c_u u/\f'_J x.-;..Z,7

( - 1 ) y') '.:---~?. Step Two: Put the claim in symbols 1J > ,, 7S ~0"2? w l ~ I '

Step Three:

Step Four:

Step Five:

Step Six:

Step Seven:

Step Eight:

11 L:r , (~~~Kc) What is true if the claim is not true? I - '::, · O

Identify the Null and Alternative Hypotheses: ),{ o !' II~ JS' H1?1T-., .. 1s

Use the significance level to find the critical value(s) and rejection region

fA-=-.,o \ ) CN\Sl.- ~~l )

Fail to reject the Null Rejection region

0 ). 2,7,7..(; Use the proportion test formula to determi e the z score for the sample data:

~

Z = p-po

✓P~% - STAT

TE-S' s-~ IMc-~v:-

~ =., 7S"

Make a decision to reject or fail to reject the null, based on the values in step five

and six: p ~, l tu r Q_ ~ J ~ n vJ1. J 'te~ Ve\ l ~ ~ ,Al o,I:. G'°' s-R- ~ C "v.. Answer the question about the claim:

/\Jo.-b .QtJoucfL 0V\' Jwc.e +o 5ttW°rt ~ Mort. ·--tklN\J 7 r;'Z, IA.rt..- I Y' co v V\ ~

Practice Problems ·

1. The National Center for Educational Statistics surveyed 400 college graduates about the length of time required to earn their Bachelor' s degree. The mean is 4.75 years, and the population standard deviation is 1.68 years. Test the .£!aim that it takes less . . \ than 5 years to complete a BA. (a=.05) ("vJ \A, ){ ~ S'° ( r wn j.,1, I

o\! ' "PfcsL'k ,l,{ .,,. r /:; loh :~KG)

- ---~ -t-_j..:~-- ~C:A'f- fl,V~ J_µI..(__ t-o tel\\Jl,v51M I s .irt ~j-,,r

t'\:::- ttOlJ 1_ ~ 1 Jib~') b(jitofY\ row i 9 o~1 ~ "ti'\~ l

~ )I.I\-I µ,~YI l, i u

/Est x-:::. ':>7Lf(,ll} '5 -= -z.. 51YO 2 ~LIA,~ -. 1..tcv' I"-:::. l ':>

-------n I " owS On.

-----d-,o/'.e. ~ 1~

to ,t- ~ tk JlC\ \ -'V'

~ M-J"lllow

3. A university ~ ould like to claim that students participating in its study abroad programs will significantly improve their language skills. Researchers tested the claim by comparing a group of study abroad students to the typical results of students who take the language proficiency exam. In the study abroad group 36 of 45 students passed. Typically, 75% of students pass. Can the University tout the benefits of its

• i ~p.)~

7T / ~7~ eif67,v s : \.e

\T~ .. 7~ V\~

STAT TE-s,

study abroad program? (a=.01) J..rv, w y{ 1

~b 1'f' ~ ,7~ , ~ \l-0( lT > ., -);

,~rod

o(::.01 ~~/\

~: I pro rt k<;t-

bri -+tv{Y\ n"" T..J~ ,=: 1,,1-i (,

~ J},v5~.,"Y\ ~ ~(l, IS V\Jt VlOL'f ~\J ~:.Jl -h:, . s '-\,) ~ h"'~ ~~ru,r(.,

Template for a Hypothesis Test

Identify the Null and Alternative Hypotheses:

Use the significance level to find the critical value(s) on the chart and identify rejection region(s)

0

Use the calculator to determine the test value for the sample data:

STAT TEST 1: ztest OR 2:ttest OR 5:lpropztest

Make a decision to reject or fail to reject the null:

Answer the question about the claim: