Question

Ugh STATISTICS sucks.

Answer 1

You have the correct P value, but you have the wrong conclusion. If you get a P value smaller than alpha, then you reject the null hypothesis. You can think of the P value as the chances of getting the null correct (though not entirely accurate, this kind of logic is applicable in a way). The smaller the P value, the less chances the null is correct. The alpha value is the threshold on when to make a decision on rejection vs failing to reject.

The answer is choice D

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One way to get the P value is to enter the data into excel and use the TTest function to do a paired T test on the data. Look at the one tail P value and you should get 0.009947481 which is fairly close to 0.00991. A screenshot is attached to show you what I mean.

We're doing a one tailed test because of the alternative hypothesis having a "greater than" symbol. If it was a "not equal" symbol, then we would be doing a two tailed test.

There are many free online tools that will do paired T tests as well. GeoGebra is one such tool that I use all the time.

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The longer method to finding the P value is described as such:

Step 1) Subtract each of the data values in each row. This forms a column of differences d

Step 2) Compute the mean of this new column of values. I'll call this dbar. The "bar" indicates there is a horizontal line over the "d" to indicate we're talking about a sample mean. You should get dbar = 2.5

Step 3) Compute the sample standard deviation of the column of d values. I'll call this sd which stands for "standard deviation of the d values". You should get roughly sd = 2.798809271

Step 4) Compute the t test statistic to get

t = (dbar)/(sd/sqrt(n))

t = (2.5)/(2.798809271/sqrt(10))

t = 2.82466341395113

Step 5) Use a calculator to find P(T > 2.82466341395113) = 0.0099474806 approximately (degrees of freedom df = n-1 = 10-1 = 9). This is the P value. Again its fairly close to 0.00991.