5.7 Normal Population with a unkno~n

Instead of z = ~/~We use t = ~/J;Developed by Gossett Guiness employee, a.k.a. Student

~/J; has extra variability due to using s as a guess at (]".

To be significant, t needs to be bigger than if we knew (]"and used z.bigger depends on how much information we have about a.

n = 1 =* 0 info about a...

degrees of freedom = df = n - 1

Example: Suppose, for a 3 minute egg timer,

n = 10, a = 0.05

Ho : J.t= 180 seconds

Ha : J.l,# 180q ttf

0,O~r o,t:t~({), 0 ~{

- 2 . 2.&, 1--

Reject Ho if ItI = I~/~ol > 2.26

t.12b7-

Say Y = 182.1 n= 10 BEy = .7fo = 1.01s = 3.2

t - 182.1-180= 2 08- 1.01 ..2 .og

Ha : J.l,# 180

Do not reject Ho : J.l,= 180 for a = 0.05Reject Ho : p, =1= 180 for a = 0.10

1'...~).Z~S 2.2'2

<i=-D.lb 0< ':: 0 ,0>"

t

0

How much

2 x 0.025 < p-value < 2 x 0.0500.05 < p-value < 0.10

tt~f

~(),OJ

0,0'].;.('

b I, ~.~?7

~,0 ctReject Ho if p-value < a.If a = 0.05,don't reject Ho.If a = 0.10, do reject Ho.

If we have Ha : J1.> 180

t = Ysl:o Reject Ho if t > 1.833

q ctf

.--'

1> \ .811

If t = 2.08, reject Ho if a = 0.05 but not if a = 0.025 (to.O25 = 2.262)

0.025 < p-value < 0.05

If t is on the reject side of Ho:I-sided p-value= lx 2-sided p-value

If t is so far from reject that it's on the other side of HoI-sided p-value = 1 -l x 2-sided p-value

Ha : J1.< 180 t = 2.08

0 ').6t

I-sided p-value = 1 - !x (2-sided p-value)0.95 < p-value < 0.975

o,o~

Let tcak = t calculated

Ho : p,= 180 s = 3.2 n= 10 Y = 182.1 BEy = ~o = 1.01

t - 182.1-180- 2 08calc- 1.01 -.

Ha : p, # 180 p-value = 2x P(t > Itealel)

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Ha : p, > 180 p-value = P(t > tcalc)

f~ ;.

O'

y(\.Q-Y'0 L

{. ~o-k:.. 0

Ha : p, < 180

--- ---~

Of e11't-j

tc" ic()

(

Ho : J.L= 180Ha : J.L=1=180

Reject Ho if ItI > to./2

Ha : J.L> 180Reject Ho if t > to.

--- 111/

Ha : J.L< 180Reject Ho if t < -to./2

Confidence Intervals

Y :i: to./2SEy Y = 182.1

182.1 :i: 2.262(1.01)182.1 :i: 2.28179.8 to 184.4

n= 10 s = 3.2

0 { ~/JI

\0

l>

SEy = ~o = 1.01

Important point: Finding 180 in 95% confidence interval is equivalent to notrejecting' Ho : J.L= 180 versus Ha : J.L=1=180 at the a = 5% level

Power and Sample Size

( + )2 u2 - (Zo</2+Z.8)2- (Zo</2+Z.8)n > Zo./2 Z{3 ~2 - ~2 - rP

~ = I J.La- J.LoI d = ~Gives us a first cut lower bound.

It also gives us a check on computer generated results.