In the modern era, this test is greatly simplified by either using dedicated sta

Question

In the modern era, this test is greatly simplified by either using dedicated statistical software (PAST, Matlab, SAS, R) or by carrying out the steps in a spreadsheet software such as Excel or Google Sheets. In this example we will be comparing the heights (in meters) of two species of maple trees, sugar maple and red maple, both common to the ravine system surrounding LFC. The observed heights for the sugar maples are the following: 13.24, 14.24, 13.90, 13.73, 14.44, 16.26, 15.60, 13.10, 14.55, 15.46. The observed heights for the red maples are: 18.15, 13.45, 17.75, 16.04, 15.55, 18.84, 18.88, 14.67, 19.34, 16.98. First enter these data into a new spreadsheet, each species in its own column. Using the methods from the previous homework, calculate the mean heights for each species and the dataset as a whole.

Then calculate the sampling variance for each group and the observation set as a whole by using the =VARA() function, entering the range of cells you want it to use inside the parentheses. Unconnected ranges of cells can be included together by separating each group with a comma inside the parentheses: =VARA(B2:B11, E2:E11). The specific ranges used are purely for example and are based on my own spreadsheet, yours will very likely be different. The pooled standard deviation is calculated by averaging the sampling variances for the two groups and then taking the square root of the mean variance: =Sqrt(Average(B13, D13)). The t-value is calculated by subtracting the smaller group mean from the larger, and then dividing this value by the product of the pooled standard deviation and the square root of 2 divided by the number of observations: =(Mean1-Mean2)/(PStev * sqrt(2/20)). The t-value is then compared to the t-distribution for the number of degrees of freedom in this analysis to return the probability of seeing this t-value. This is done by using the =tdist() function which requires three entries separated by commas: the first is the t-value, the second is the degrees of freedom which in this case is n1-1+n2-1=18, and the third is the number of tails which in most cases will be 2. The value returned by this function is the probability of the null-hypothesis being true and if the p-value is less than 0.05, then the null-hypothesis (that both groups are the same) is rejected.

For our next worked example we will look at nitrogen concentrations in different types of soil. We took 8 samples from an abandoned farm field and another 8 from a prairie restoration. Back at the lab we measure the following concentrations for the farm samples: 10.0, 6.0, 1.4, 4.3, 6.9, 5.0, 6.7, 5.1, 3.9, 9.0 and the following for the prairie samples: 20.0, 4.7, 16.7, 10.5, 21.3, 20.2, 20.0, 6.2, 13.8, 15.4. Organize your data and calculate the mean and standard deviation for each group. Compare the standard deviations; are the two groups equally variable? Excel (and Sheets) allows you to directly carry out a t-test on your data using the =TTEST() function. The function requires four entries (arguments) to be input, separated by commas. The first and second are the cell ranges that include each group's data, the third is the number of tails (2 again), and the fourth tells the program what kind of t-test you want it to perform. Entering 1 tells Excel to do a paired test (not what we are doing), entering 2 tells it that you are comparing two groups but that they have the same variance structure or standard deviation (right comparison but incorrect structure), and entering three tells it that you are comparing two groups that may not have the same variance (this is the one we will usually want). What does comparing the two groups' standard variation tell you about this setting? The number this function returns is the p-value for this test, the probability that the null-hypothesis is true given the observed data.

Questions:

1: What are the means and variances for the groups in the maple dataset?

2: What is the t-value you calculated for the maple dataset?

3: What is the p-value for the t-test of the maple heights?

4: What are the means and standard deviations for the groups in the soil nitrogen dataset?

5: What is the p-value for the t-test of the nitrogen dataset?

Explanation / Answer

VARIANCE

SUGAR MAPLE = 1.086707

RED MAPLE= 3.924917

wHOLE = 3.834023

MEAN

SUGAR = 14.452

RED =16.965

WHOLE = 15.7085

2. T value = 2.00505

3. P value = 0.000911

4 mean

farm= 5.83

prairie = 14.88

standard deviation

farm = 2.500689

prairie = 6.007921

5. Pvalue = 0.000864

HIT LIKE IF I HELPED YOU. SCREEN SHOT OF THE CALCULATION ARE SHARED. FUNCTIONS OF EACH ANSWERS TO BE SOLVED ARE GIVEN IN THE PARAGRAPH.

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