Null Hypothesis Exam Question - Please help!

creamcracker30

Hi guys,

My younger brother is struggling with his college work and needs help on a particular topic which he cant get his head around. As his older sister I'm trying to help him out but unfortunately as its not my background I dont really understand it either
Here is the question he was given in his sample exam paper and he doesnt know how to do it. Would any kind soul out there be willing to give me a simple explanation of how it would be done. Thanks so much in advance, it would mean alot to my brother if he could get his head around this one!

The number of fracking walls (in thousands) in a sample if ten US states is as follows:

6, 14, 40, 40, 80, 39, 29, 1, 0, 28, 53

Test the Null Hypothesis that the mean value per state is 30 based on this sample.

Black Swan

creamcracker30 wrote: »

H
The number of fracking walls (in thousands) in a sample if ten US states is as follows:

6, 14, 40, 40, 80, 39, 29, 1, 0, 28, 53

Test the Null Hypothesis that the mean value per state is 30 based on this sample.

The question is not clear when it specifies "in a sample if ten US states is as follows," but n=11 in the data set. Is the sample size n=10 or n=11? This needs to be clarified first.

Another problem is that the sample size is small for hypothesis testing.

bnt

Because the sample is so small, it sounds like a job for Student's t-distribution, using confidence intervals. I don't remember exactly how to do it, though ...

creamcracker30

Apologies Black Swan, that was my typing error.

The question states "in a sample of ten US states".

Thank you in advance.

Pythia

creamcracker30 wrote: »

Apologies Black Swan, that was my typing error.

The question states "in a sample of ten US states".

Thank you in advance.

You say 10 but there are 11 numbers. You also need to state at what level of confidence you will reject at. He needs to review hypothesis testing in his notes, this is very basic stuff. I'm sure the lecturer already gave some examples so he should study them first. You can google to find the way to do it. Google 'hypothesis testing'.

Basically find the mean, st dev and t value at the required alpha (which isn't stated in your question so you will have to assume a value and state the assumption) and you have your n is 11. Use the formula to see if 30 is contained in the confidence interval. If it is, you can accept the null hypothesis.

**Timbuk2**

What your question is asking is to the test the hypothesis
[latex]H_0: \mu = 30[/latex]
[latex]H_1: \mu \neq 30[/latex]
where [latex]\mu[/latex] is the population mean number of fracking walls.

It's a small sample size so it's the Students t-distribution that you use.

He'll find all this in his notes, but the easiest way to do it is to calculate a t test statistic

[latex]T = \displaystyle{\frac{\bar{x}-\mu}{s/\sqrt{n}} }[/latex]
[latex]\bar{x}[/latex] is the sample mean, s is the sample standard deviation - work these out using a calculator, or excel!
T follows a student's t distribution with n-1 degrees of freedom.

Now you can either use this T statistic to look up a p-value (using statistical tables, or an online calculator, or using a program like R/excel), or look up a critical value using the level of alpha (it's not given in the question, he'll have to assume one - alpha=0.05 is a common one).
He only needs to do one of these, they'll always give the same result.

If p-value < alpha, then reject the null hypothesis.
If p-value > alpha, then fail to reject the null hypothesis (i.e. insufficient evidence to say the true mean differs significantly from 30).

or equivalently
if the T statistic lies between [latex]\pm[/latex]Critical Value then fail to reject the null hypothesis.
Otherwise, reject the null hypothesis.

He can then go on to calculate confidence intervals for the true mean if he wants, but there's no need to in this question - you can also test whether the sample mean lies within the confidence interval or not, but as this test is a two tailed test, this will always give the same result as the two-tailed hypothesis test

PaulieBoy

A quick look at Amazon and you should be able to pick up a copy of Probability and Statistics for Engineers and Scientists by Walpole & Myers. An old edition will set him back small change and is well worth it.
I found it great when doing my stats module, it's got lot's of great questions that will get you through.

MathsManiac

I'm sure the intention of the question is as described by Timbuk2, but I would have a slight worry about whther the assumptions required for the t-test are really met here.

The sample size is pretty small, and the population size is only 52, and we know nothing about the population distribution other than what we can infer from the sample. (The Shapiro-Wilk test doesn't throw up any major cause for concern, as the test statistic is a healthy 0.935, but the issue is at least worth mentioning.)

At the very least, one should consider applying the finite population correction sqrt((Np - N)/(Np -1)) to the standard error. In this case, sqrt((52 - 11)/(52 -1)) =~ 0.897