Alpha Significance: Understanding & Analyzing Data Set

[gummz]

Homework Statement

What conclusions can be drawn from this data set? What assumptions are you making?

Homework Equations

http://i.imgur.com/M9YQGAF.png
I hope this is legible.

The Attempt at a Solution

The solution is what I'm having trouble with.
I just don't get how that test statistic has anything to do with whether that hypothesis is rejected or not, and what meaning it has for the test statistic to be larger than that number.

 

[RUber]

This is an interesting question that might require a lengthy discussion.
In general, the process of calculating a test statistic (Z value) is designed so that you can relate your data set to the normal curve. Remember that most normally distributed stuff will be observed in the biggest part of the curve, and it is less common to see something in the tails (left or right extremes) of the curve.
When you find $|Z_{0.025}| = 1.96$, that is giving you a value which says that 2.5% of observations from a population that has mean of zero and standard deviation 1 (the normal curve standard) will be greater that 1.96 and 2.5% will be less that -1.96. Those two tails account for 5% of the population.
Your calculated test statistic was greater than 1.96, which indicates that it would be an uncommon observation if your null hypothesis is true.
Your conclusion, then is that, accepting the small risk that your observation was a random chance (alpha), you can reject the null hypothesis. That is that these samples did not come from a population with the stated mean and standard deviation.