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# Statistics - test statistics & p-values?

I'm kind of confused with these testbook questions and was wondering if someone could walk me through them:

1. Find the approciate rejection regions for the large-sample test statistic z in these cases:

a. A left-tailed test at the 1% significance level.

b. A two-tailed test with α = 0.01

2. Find the p-value for the following large-sample z tests:

a. A right-tailed test with observed z = 1.15

b. A two-tailed test with observed z=-2.78

c. A left-tailed test with observed z=0.81

### 2 Answers

- AlanLv 711 months ago
This is a hard question if we don't have guidance from your textbook.

Because some textbooks do it different. We have the actual p-value , then some textbook ask you to change

to compare it directly to alpha

so let's start with the easier 1st question

1. .

a. left tail

so the left tail is at the bottom

0 to alpha

Here 0 to 1% or p =0.00 to 0.01 in p

in p values

(0.00 < p < 0.01)

in z-values

-infinity <z< P(z< Z) = 0.01

P(z< -2.33) = .00990

p(z< -2.32 ) = .01017

if you interpolate of use a calculator

P(z< -2.326347874) = 0.01

so for z region

(-infinity< z < -2.326 )

b.

so a two-tail test with alpha = 1 % or alpha = 0,01

must have 1- alpha = 1 - 1 % = 0.99 in the middle as the acceptance range

so that mean 0.5% to 99.5 % is the acceptance range

0 to 0.5 % and 99.5 % to 100 %

which means in goes from

0 < p < alpha/2

and

1 - alpha/2 < p < 1

so

0< p < 0.01/2 = 0.005

0 < p < 0.005 ( lower rejection range )

1 - 0.01/2 < p < 1

0.995 < p < 1

so in terms of p

rejection ranges are

0 < p < 0.005

and

0.995< p < 1

p = 0 is Z = -infinity

p = 1 is Z = + infinity

p = 0.005 if you look it up gives

P(z< -2.57) = .00508

p(z< -2.58) = 0.00494

if you interpolation

P(z< -2.576 ) = 0.005

and due to symmetry

P(z< 2.576) = 0.995

so in terms of z, the rejection range is

(-infinity< z< -2.576) and (2.576 < z< +infinity)

since you and your question didn't ask for the rejection range in z or p ,

I gave you both

2.

This is where it gets tricky at times.

a,

1st read directly out of a z-table

P(z< 1,15 ) = .87493

p = 0.87493 is the actual p value.

but some textbook change it so it can be compare directly to alpha

will p = 1 - original p = 1 - 0.87493 = 0.12507

so if alpha = 0.10

then 0.12507 is compares to < 0.10

Instead of comparing 0.87493 against 1- alpha = 1 - 0.10 =0.90

so depending on your text book

p = 0.87493 or

p = 0.12507

b. z = -2.78

so read from z-table

P(z< -2.78) = .00272

so some text books

say p = 0.00272

other text books say since the reject range is

which means in goes from

0 < p < alpha/2

and

1 - alpha/2 < p < 1

we will multiply p by 2 so we can compare it directly to alpha

so p = 2*original p = 2* 0.00272 = 0.00544

so some textbooks

say p = 0.00272

others will say p = 0.00544

c, z = 1.18 left tail

P(z< 1.18 ) = .88100

p = 0,881 is the answer period, there are no two ways

in a one tailed test

for a left tail, we can compare against alpha

in a right tail, we can compare original p against 1 - alpha or

or subtract p from 1 and compare directly against alpha.

say here alpha was 0.10 , the lower z = -2.326

you can reject the NULL hypothesis if 0< p < 0.10

but p = 0.881 is much greater than 0.011

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- Anonymous11 months ago
Ok. Ask a question please so we can help.

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