Step 3: Determine whether observed density differs greatly from the expected density.

Once you have a mean and standard deviation of historical tree density, you must determine whether the current tree density is extremely low or high, relative to the historical tree density. Either conclusion would support the hypothesis of a bottom-up effect of umbrella trees on boreblasters.

Let's consider how you would accomplish this task. First, compare the current tree density to the mean of the historical tree density. If the current density is less (or greater) than the mean, we can conclude that the current density is less (or greater) than expected. However, we still don't know if the current density is extremely unexpected.

To determine just how extreme the current tree density might be, you need to use the normdist function of Microsoft Excel. This function requires three pieces of data for a variable: 1) an observed value, 2) the mean, and 3) the standard deviation. The values are entered into Excel as follows:

=norm.dist(observered_value, mean, standard_deviation, TRUE)

The function returns the probability of observing a value less than (

For example, entering the following function in Excel

=norm.dist(255, 270, 16, TRUE),

would return 0.1743 (or 17.43%), which equals the probability of observing a value less than 255 when the mean equals 270 and the standard deviation equals 16.

In the normdist function, enter the current tree density as the observed value but enter the mean and standard deviation of the historical tree density. The function should return the probability of observing a density less than the current density. If this probability is less than 5%, we should conclude that the current density is extremely low compared to historical densities.

If you want to know the probability of observing a density greater than (>) the current density, recall that the following relationship:

P(y > x) = 1 - P(y x)

where P(y > x) equals the probability of observing a value y that is greater than the value x, and P(y x) equals the probability of observing a value y that is less than the value x.

Subtracting the value returned by the normdist function of Excel from 1.0 will yield the probability of observing a density greater than the current density. If this probability is less than 5%, we should conclude that the current density is extremely high compared to historical densities.

Directions: Use the normal probability distribution for the historical density of umbrella trees to answer questions 11-12.

1. If you were provided the current density of umbrella trees, which probability should you estimate to determine if that umbrella tree density is unusual enough to cause the boreblasters to disperse, the probability of observing a historical density of umbrella trees that is equal to, less than, or greater than the current density?

The probability should be of observing a historical density of umbrella trees is greater than the current density.

1. Calculate the probability of observing a historical density that is more extreme than the current density of 27,000 m²km^-2 (either less than or greater than the current density, depending on your answer to question 11). Express your answer as a percentage (%). Round all calculated values to the nearest whole number. For example, if you calculate the value as 213.8218, round to 214.

mean is 26994

standard deviation is 1008