In this presentation, we will be going over the subject of Hypothesis Testing in Statistics.

Specifically, we will be going over one of the most important topics:

We will first define some terms as well as the steps, then show a couple examples.

Let’s start with stating our null hypothesis. Our null hypothesis is represented by the symbol:

Note: If all of the Parameter Estimates were zero, we wouldn’t be able to make these practical interpretations.

Our alternative hypothesis is represented by the symbol:

\[Ha\]

The alternative hypothesis in the Global \(F\)-Test reads as follows:

\(Ha\): At least one \(B_{i}\) differs from zero.

This doesn’t mean all of the \(B\)s differ from zero, but it means that at least one of the independent variables in the model has shown some effect on the dependent variable.

(The model can be further fit via other tests, but we will stick to this one for now)

We are left with:

• \(Ho: B_{1} = B_{2} = ... = B_{i} = 0\)

• \(Ha\): At least one \(B_{i}\) differs from zero

Here are examples of other types of Hypothesis Tests:

\[Ho: B_{1} \geq 0\] \[Ha: B_{1} < 0\]

Setting your significance level, or Alpha \(\alpha\), can help determine whether we should accept or reject \(Ho\).

Typically, statisticians will set \(\alpha = 0.05\), meaning that there is a \(5\)% chance that their conclusion is wrong.

Still, that leaves a \(95\)% chance that it is right.

This also means that, if you want to reject your null hypothesis, you’re going to want to end up with \(p-value < 0.05\).

Whenever \(p-value < \alpha\), you reject your null hypothesis.

Subsequently, whenever \(p-value > \alpha\), you do not reject your null hypothesis.

Here is a table summarizing Type I and Type II Errors in Hypothesis Testing:

Note : Both error types are bad, but the severity of each can vary depending on the situation.

This is where the researcher really gets to set the tone for their hypothesis testing.

Pay attention to:

• Whether data is collected via experimental design or observational means.

• The nature of the data

• What type of questions you want to ask about the data

• What you expect from the data

There are many different formulas for calculating different types of test statistics, but they all serve the same purpose.

Test statistics are used to help determine whether or not to reject \(Ho\).

They are compared to the Critical Value(s) based on the \(\alpha\) you set.

Reject \(Ho\) if:

Right-Tailed

\(t-statistic > t-critical\)

Left-Tailed

\(t-statistic < t-critical\)

Two-Tailed \(- t.statistic < - t.critical\) or \(t.statistic > t.critical\)

The Rejection Region represents an interval that leads to the rejection of \(Ho\). Even though we are looking at a Two-Tailed F-Test , here is a summary of the rejection regions of one-tailed tests as well:

After you have completed Steps 1-6, you have plenty of information to help you decide whether or not to reject \(Ho\).

The actual conclusions would be unique to every test, but in general, conclusions read as follows:

If \(Ho\) is Not Rejected:

• There is not sufficient evidence to reject \(Ho\).

If \(Ho\) is Rejected:

• There is sufficient evidence to reject \(Ho\).

Let’s look at another hypothesis test. This one involves correlation. For this example, let’s look at the data set “Trees.”

Let’s first graph each of these and see if we can identify any correlation between the independent and dependent variables.

Here is the head of the “Trees” data set.

head(trees)

##   Girth Height Volume
## 1   8.3     70   10.3
## 2   8.6     65   10.3
## 3   8.8     63   10.2
## 4  10.5     72   16.4
## 5  10.7     81   18.8
## 6  10.8     83   19.7

We see that there are three variables: Girth, Height, and Volume.

Since it makes the most sense to look at Girth and Height as functions of Volume, let’s see if there’s any type of relationship between them by graphing the data.

Both Girth and Height seem to have a positive linear correlation to Volume.

To make things easier, let’s assume we’ve already calculated the correlation coefficients, \(r\). First, we need to define our Alpha.

Let’s set \(\alpha = 0.05\).

The formula for the test statistic is:

\[t = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}\] The \(p\)-value is then calculated using the \(t\)-distribution and \(n - 2\) degrees of freedom.

You can then compare the \(t\) test statistic to the \(p\)-value to see if it falls in a rejection region.

Luckily, we live in the 21st century and have technology. There are many ways to compute these test statistics and \(p\)-values, and there are many different types of statistical software out there (R included) that can help us.

This is especially useful when we start adding more variables to our analysis, as well as transforming them.

If you look at the correlation matrix on the next slide, you will notice that the two independent variables have fairly strong positive correlations to the dependent variable.

(That is, the correlation coefficients are close to or greater than \(0.6\), as seen in the first matrix)

The correlation matrix also runs its own hypothesis tests, which would read as follows:

\(\rho = 0\) (No correlation)

\(\rho \neq 0\) (Some correlation)

The second matrix entitled “P” shows the \(p\)-values for each correlation. Luckily, all of the ones we want have \(p\)-values less than \(0.05\), so we can say that there is sufficient evidence to reject the claim that there is no relationship between our \(x\)’s and \(y\).

Correlation Matrix previously mentioned:

pcorr = rcorr(as.matrix(trees))
pcorr ## Correlation Matrix using R

##        Girth Height Volume
## Girth   1.00   0.52   0.97
## Height  0.52   1.00   0.60
## Volume  0.97   0.60   1.00
## 
## n= 31 
## 
## 
## P
##        Girth  Height Volume
## Girth         0.0028 0.0000
## Height 0.0028        0.0004
## Volume 0.0000 0.0004