Anonymous
Anonymous asked in Science & MathematicsMathematics · 11 months ago

Statistics - test statistics & p-values?

Update:

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

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  • Alan
    Lv 7
    11 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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  • Anonymous
    11 months ago

    Ok. Ask a question please so we can help.

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