Understanding where the p-value comes from.
Hi I'm Dr Nic and in this video
I'm going to tell you about where the p-value comes from.
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In other videos I explain what the p-value tells you and how to undertake a hypothesis test.
There are links to these videos in the description below.
In this video I'm going to work through an example to show you
where the p-value comes from and how you could calculate one for yourself
The p-value or sig can be found in the
output to statistical inference.
You can have p-values for testing if two means are different
or if the slope of a line is statistically significant
or if there is evidence of differences between categorical data.
All of the tests shown in the video "Choosing which statistical test to use"
produce a p-value
Normally you just get a computer to find the p-value.
You put in the data, tell a computer what you want to test for
and out comes the p-value
However it is good to have an understanding of what a p-value actually is.
A p-value is a probability. That is what the P stands for.
In SPSS it is called sig.
We want to know if an effect that shows up in a sample is evidence of
an effect in the population from which the sample is drawn.
The p-value tells us
the probability that we would get the sample result by chance if there is no
effect in the population.
We will use a test for a population mean to illustrate this.
We will use the same example as is shown in hypothesis test for a mean in Excel.
In this case the orchard owner is comparing the mean weight of her apples
with the export standard of a 152 grams per apple.
The orchard owner can see that the mean weight of the apples in the sample is
about a 149 grams. Could that just be by chance that the
apples in the sample are mostly lighter than the ones in the whole orchard
and in reality the average weight of the apples in the orchard is 152 grams or above?
We can use probability theory to work out how likely it is to get a mean
of a 149 grams or less if the mean weight of the apples in the population is 152 grams.
This builds on the central limit theorem so you
might like to watch that video now if you're not familiar with it.
The central limit theorem talks about the nature of the sampling distribution of the mean.
If we were to take a whole lot of samples of size 15 from an orchard that really
did have a mean weight of 152 grams, we would get a variety of sample means.
This is known as the sampling distribution of the mean.
The mean of those sample means would be
the same as the mean of the population.
The spread of the sample means is given as the standard error,
the formula for which is sigma (the population standard deviation)
over the square root of n the sample size.
The central limit theorem states that we can use a normal distribution to model the
sampling distribution.
If we knew the population standard deviation we could
proceed from here however we do not usually know the population standard deviation.
It's actually pretty unlikely to know the population standard
deviation when we do not know the population mean,
but we do have a sample standard deviation, s, which we can use
as an approximation to the population standard deviation Sigma
With small samples we need to use the students T or
T distribution instead of the normal distribution.
It is always okay to use the T distribution
as it becomes the normal distribution for large samples.
The T distribution is like a standard normal distribution.
It has a mean of 0 and a standard deviation of 1.
To use the T distribution we need to find out how
many standard errors the sample mean is from the hypothesized population mean.
We calculate the sample mean minus the hypothesized mean and we find that it is
149 .2667 - 152
which is -2.7333
The standard error is 4.75795
the sample standard deviation, divided by the square root of
15 which gives 1.228497.
This is a measure of the spread of sample means.
We divide the distance that the sample mean is from
the hypothesized mean by 1.228497 and we get -2.22494.
That is saying that the sample mean
is a bit more than two standard errors below the hypothesized population mean.
Now we need to find out how likely that is in the appropriate T distribution
We use a T distribution with n minus 1 or 14 degrees of freedom.
Using Excel we use the function =t.dist(-2.22494,14,true)
or we can use a calculator to find the probability.
The probability is 0.021519 which we will round to 0.022.
For the T distribution with 14 degrees of freedom, the area under the
graph to the left of -2.22494 is 0.022.
This is the p-value we are looking for.
This p-value tells us that if the
population mean for all the apples in the orchard is 152 g,
then the probability of getting a result this much smaller than 152
or worse is point 0.022. About 2%.
This video explained where a p-value comes from.
There are links in the description to other videos to help you understand this concept
and what you would do with the p-value now you've got it
Even if you are not using Excel the video
"Hypothesis test for a mean in Excel" will help you with your understanding.
Do let me know what more you would like in the comments below.
Please like this video, subscribe, but most of all join the channel
especially if you're using our videos in your teaching.
Help the channel grow and help me help more and more
people like you.