After completing this section you should be able to do the following:

Example

For instance consider an experiment where tree growth rates were increased by 25 percent following a forest thinning operation on 10 sites compared to tree growth rates on 10 sites which were not thinned. Inferential statistics allows us to decide if the increased growth rates are due to chance or are real.

There are primarily two ways to use inferential statistics:

1. The first is to estimate a parameter about a population.
2. The second is to test a hypothesis.

The average (mean) of our sample can be used as an estimator of the population mean. We could have chosen to use the median or mode to estimate the population mean as well. However, for the remainder of this section we will deal with estimating the mean of a population only.

The certainty in which a population mean lies within the range is typically expressed as 95% confidence interval, or a 99% confidence interval. As you add more certainty the width of the interval will increase.

Example

Our hypothesis is that tree mortality after a particular forest fire will be greater then 60%. In other words average tree mortality > 60%.

A two-tailed test is used when a research hypothesis is stated as the following:

Example

Tree mortality following fire will be equal to 60%. Where as our null hypothesis would read tree mortality following fire is not equal to 60%.

Under this scenario we could reject our research hypothesis if tree mortality was much larger then 60 or much smaller than 60.

Example

Lets say we collected the abundance of grass species 1 and 2 one year after a wildfire. We have determined that the two populations had burned under the same exact conditions at the same time etc… Now lets propose that species number 1 will be in higher abundance then species number 2 one year after the fire. This is our research hypothesis. Therefore our null hypothesis will be that there will be no difference in abundance of species 1 and 2

Once we have established our research and null hypothesis we run a test statistic for a given level (assigned as the alpha level). From this point, we can either reject our null hypothesis or not, and then draw any conclusions from the test.

Often times a t-test is used to compare the difference between two means.

Additional Information

Example

We may want to compare regeneration rates for three different tree species in northern Idaho. We would begin by taking samples from each population and then calculate the means from the three samples and make an inference about the population means from this.

It is common since that these three mean regeneration rates would all be different numbers however, this does not mean that there is a difference between the population means for the three tree types.

To answer that question we can use a statistical test called an analysis of variance or ANOVA. This test is widely used in natural resources, and you are bound to come across it when reading scientific literature.

The use of an ANOVA implies the following:

1. all the populations are normally distributed (follow a bell shaped curve)
2. all the population variances are equal,
3. and all the samples were taken independently of each other and are randomly collected from their population

Generally, our null hypothesis when conducting an ANOVA is that all the population means are equal and our research hypothesis will be that at least one of the population means is not equal.

Although an ANOVA is widely used and it does indicate that a population mean is different than others, it does not tell us which one is different from the others.

Additional Information

The following is a list of commonly used multiple comparison procedures:

1. Tukey’s Procedure
2. Student-Newman-Keuls procedure
3. Dunnett’s procedure
4. Scheffe’s Method

Remember: These tests are only used after we have rejected the null hypothesis using an ANOVA.

Additional Information

Example

We can use a regression model to predict the mortality of a ponderosa pine tree based on the amount of crown scorch.

Regression models are very common and you have all probably seen them used in the scientific literature. Often times regressions models are associated with a scatter plot, as shown below.

Figure 1. Shows the relationship between tree height and tree age for species X

We will now add a simple linear regression equation to our example and calculate the correlation of determination (R²):

You'll notice on the figure we have identified the statistical equation which describes the line of best fit and the correlation of determination identified as R². Although we will not go over the details of the correlation of determination you should be aware that it is essentially a measure of the strength of the linear relationship between the two variables.

We should note that just because there appears to be a strong relationship between tree height and tree age the model does not imply that tree height depends upon tree age, in other words the model does not show a cause and effect relationship. Great care should be used when drawing conclusions based on a regression analysis.

The scatter plot above could be used to build a simple linear regression model, however we could add more factors to help us better predict the outcome, these are called multiple regression models, or we could use some other form of a model that is not linear in nature, called a non-linear regression model.

Additional Information

Regression Models