Stat's Help.. Please.

[pb2009]

If we are performing a one-tailed hypothesis test using a z-test and the level of significance of .05, the critical value is 1.645. How did we find this?

Can you help to explain this answer for me in simplier terms. I do not understand a single word of it. I have no clue what to do with statistics.

Answer:
As the hypothesis test is one-tailed, the level of significance is not split in half and only the positive half of the curve is considered. The complete curve has a value of 1, so half of the curve is .5. We are not looking at that entire half so we must subtract the .05 level of significance, which gives us .450. With this number, it is a matter of reviewing the table of standard normal areas (appendix C-1) by searching the table for the number closest to this one. Once the number is found, the numbers on both axis that correspond with this number are added together to reach the critical value level. Because our number falls between two numbers listed, the critical value falls between the two points (1.64 and 1.65).

 

[chrisr]

If you are a statistics beginner, let's see if we can clarify the graph.

Suppose the average height of 18 year olds is 5'10".
We call that the mean value.

You can imagine there are many 18 year olds less tall and more tall than the average,
however, the heights will cluster around that mean.

As we go down below 5' or above 6'8", there are extremely few 18 year olds there.
Even less at further extremes.
That explains the shape of the bell-shaped curve, called the Normal Distribution.

There are all kinds of measurements possible, with all kinds of means,
therefore the next step to make statistics manageable when dealing with this graph was to normalise.

Deviation refers to deviations away from the mean.
A "standard deviation" away from the mean is 1 z-unit.

The graph is "normalised" by converting to z-units.
This means you can analyze numerous different sets of data with the same "standardized" graph.

When using this graph, we do not calculate the probability that an 18 year old will be exactly 5'7".
We instead calculate probabilities of the nature
......... up to 5'7"
......... over 5'7"
......... between 5'5" and 5'9".

The area under the curve is 1, since all probabilities sum to 1.
It's quite a complex integral so don't bother with that yet.

Therefore, the area under part of the curve represents a certain probabilty,
in this case in relation to the heights of 18 year olds.

These probabilities are calculated using the z-values.

The parameter being measured is first converted to z-values, using the values of the mean and standard deviation.

Then you find it's probability using the area under the curve.

A one-tailed test would be something like....
What is the probability that an 18 year old is 6'5" or more in height.

A two-tailed test could be....
What is the probability that an 18 year old's height is at least 10 inches away from the average.

In your question, a level of significance of 0.05 corresponds to the very edge of the tail,
where only 5% of the area under the curve lies.
For a one-tailed test, this corresponds to z = 1.645.

You need to have a diagram of the standard normal curve to see this.

 

[pb2009]

ok, im sorry but I am very new to this side of statistics. I manage to get through the first course of it and now they are getting deaper into it in this class and I am just lost as a goose. I kind of understand it but I kind of don't if that makes any since. All the information they gave us to use is in the first part of my post.

 

[chrisr]

That's ok, it takes a while to get used to.

What they mean by......"because it's a one-tail test, it's not split in half" is the following..

0.05 level means 5% above 0% or below 95%, for a one-tail situation,
or 2.5% above 0% added to 2.5% below 100% for a two-tail situation.

If it was 2 tailed, we'd be looking for positive or negative bias,
which would be the 2.5% at the left edge of the bell-shaped normal curve
and the other 2.5% at the right edge.

Those two 2.5% parts sum to 5%.
The remaining 95% of the graph lies between those extremes.

Also the graph can be split in two symmetrical parts at the mean value, corresponding to z=0.
50% of the graph lies below the mean (centre) and 50% lies above it.

For a one-tailed test on the other hand, the 5% is at the right edge of the graph (as explained in your example),
though it could also be the 5% at the left edge.

For your question, then, just take the right edge, the 95% to 100% region of the graph.
z=0 corresponds to 50%, so since z=1.645 corresponds to 45% or 0.450,
this is 45% beyond the 50% level, or 95%.

Have a think about that anyway.