# Probability and negative mean

#### Shahab

##### New Member
Hi guys
I think I need some help.

We are performing a pre-post analysis. it is believed that the average of the variable Y is 60 in pre-situation (Y1) and it is believed to become 55 in post-situation (Y2).
we Define a new variable Y = Y2 - Y1
if Y is observed and known to be Y=-3.11, with a normal distribution and standard deviation of 1.53, what is the level of significant achieved?

I believe mu = Y2-Y1 = 55-60 = -5 is this correct?
if we calculate z = (Y-mu)/sigma = (-3.11-(-5))/1.53 = 1.23 and the corresponding probability from table is 89.07%
are my calculations and probabilities right?
What can I say in this situation? what can I claim? what does the "89.07%" show?

#### obh

##### Well-Known Member
Hi Shahab,

When you look at a table you need to understand what the value in the table's cell means, this depends on the table!
Usually, the Z table is p(X≤x), but there are also inverse tables and more.
Generally, with today's computers, there is no reason to use the historical tables...

If Y distributes Normal(mu=-3.11, sigma=1.53)
z = 0.8907 (aprox)
P( Y ≤ -3.11 ) = 0.8907.
P( Y > -3.11 ) = 0.103.

You may look at http://www.statskingdom.com/z_table.html,
When you hover over the cell, you will see also the probability on the top
Or simply use the calculator ...http://www.statskingdom.com/normal.html

#### Shahab

##### New Member
Dear obh
Thanks for your hint. Actually my problem is a little bit more confusing than that I could explain. I am reading a book, which is a double edged sword!
sometimes it is great, sometimes ....

I attached the part of the book that confused me! I would be glad to have some help on this.

My problem is on section 1.9.2.1 which claims that the null hypothesis is rejected, and it calculates the significance level to be 2%. However, to the best of my knowledge, it is not right. Not only the H0 hypothesis should not be rejected, but also it should be accepted since the significance level is 10.3%

I attached a few pages before the mentioned section in case it is needed. Thanks in advanced. If you just say that I am right or wrong would work. If I am right, then it is ok, and if I am wrong, I will try to find it out myself. Thanks again.

P.s: there exist a number of other misspellings in the text too.

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#### obh

##### Well-Known Member
Hi Shahab,

I won't read all the pages... but I can say their solution is to a different question, so you need to read again and see if you understand the question.

If you know from previous researches that Y1=60 and Y2=55 than the null mean is -5
If you don't know the means of Y1 and Y2 and you calculate the averages 60,55 based on a sample, the null assumption made before you have the data, and not based on the sample data, so it is mean=0.

Their null assumption is that Y2-Y1=0 or you may write Y1=Y2

The assumptions:
Y ~ Normal(mu=0, sigma=1.53)
H0: Y=Y2-Y1=0
Now if you get:
P( X > 3.11 ) = 0.0210424.
So if you use one tail the significant level is 0.02

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#### Shahab

##### New Member
Thanks a lot obh. I think you made it quite clear for me. thanks a lot again. wish you the best.