Introduction to Statistics

answer the following discussion questions:

1. When our test statistic does NOT fall into the critical region, why do we say that we "Fail to reject the Ho" rather than just saying we accept the Ho?

2. Explain the difference between a one- and a two-tail test. When is each used?

3. What is a "P" value and why do we need it? (Hint: What does it give us in addition to our alpha?)

4. Did we show beyond a reasonable doubt that Dusty was not telling the truth?

5. Why do statisticians shudder at the word "prove" and use words like support, accept the H1, reject the Ho, etc.?

Reopened:

1-

When our test statistic does NOT fall into the critical region, why do we say that we "Fail to reject the Ho" rather than just saying we accept the Ho? 

Proving the null case is usually quite difficult so we take the position of it having the possibility as a solution since our data didn’t outright disprove it.

Explain the difference between a one- and a two-tail test. When is each used?

When looking for any extreme values apart from the mean’s 2-3 standard deviations one uses a two-tail test as you can have extreme values on either side of the bell curve. If however you are looking solely for a result that is further lower or further higher than the given hypothesis only one tail needs to be used effectively halving your probability. i.e. if I have a bunch of athletes race in a 100m dash and my mean is say 9.8 seconds, to check out the hypothesis of an 8.8 second 100m run wouldn’t bother with the right tail data. We don’t care in this case about times slower than 9.8 just the ones faster at the lowest extremes hence a one-tail test versus two.

What is a "P" value and why do we need it? (Hint: What does it give us in addition to our alpha?)

P is simply the area under the curve to the right or the left of a given test statistic (z – value). It is the “smallest level of significance for which we reject Ho” or the null case. If the P-value is £ a we reject the null case and if P is > a we “fail to reject” the null case.

Did we show beyond a reasonable doubt that Dusty was not telling the truth?

I think reasonable doubt needs to be defined. Because we had enough information to reject the null case it appears statistically that the alternative of a mean time >60min holds true. I’m overthinking it, but we aren’t given information on the timing device used in the initial reporter/baker claim. Was it digital? Analog? An atomic clock? We can usually assume small to no bias in a reporter with no ties to the person involved in the story right? So were Dusty and the reporter friends or was the reporter truly unbiased and an eyewitness? It seems evident the narrator of our story is biased against Dusty so could he/she have been less accurate than someone with no predispositions to cheesecake production times? Not to mention that someone who knowingly cheated wouldn’t reproduce the effort for a third party once much less 18times, so the inference is Dusty believed he was telling the truth but I suppose that can’t be statistically measured.

Why do statisticians shudder at the word "prove" and use words like support, accept the H1, reject the Ho, etc.?

Statistics are about categorizing probabilities not in uncovering certainties. With enough evidence it becomes highly likely that a specific hypothesis is correct but it’s probably from statisticians that claimed absolutes that Mark Twain quotes “There are three kinds of lies: lies, damned lies and statistics.” This is a study of trends and of logical expectations but outliers and chance, factors not considered etc. all play a role.

2-

When our test statistic does NOT fall into the critical region, why do we say that we "Fail to reject the Ho" rather than just saying we accept the Ho?

The term “Fail to reject the Ho” is used in place of the word “prove” (and sometimes “accept”) because it implies that the null hypothesis has been proven, which would be incorrect.  You can never prove a null hypothesis.  “Fail to reject” is more accurate because it is basically stating the evidence is not strong enough evidence to reject the Ho. (Triola, 2014)

 

Explain the difference between a one- and a two-tail test. When is each used?

Right-tailed test: Critical region falls to the far right tail of the curve. Use when H1 > (greater than) is present.

Left-tailed test: Critical region falls the far left tail of the curve. Use when H1 < (less than) is present.

Two-tailed test: Critical region is in both the far left and far right tails of the curve. Use when H1 ≠ is present, it means there are two directions where the null hypothesis can fall.

Note: If the null hypothesis falls in any of these regions it is to be rejected. 

 

What is a "P" value and why do we need it? (Hint: What does it give us in addition to our alpha?)

A P Value is the smallest level of significance that would reject the null hypothesis.  (It is the area under the curve, to the right or left of the test statistic.)  It is a Level of Significance.  If P= .082, if an α value greater than .08 is chosen you are guaranteed to reject Ho.  If α=.55 you definitely cause a rejection, but if α=. 08, you would fail to reject Ho.  The smallest level of significance should be determined at the start of the experiment, and once chosen cannot be changed, as it could be used to manipulate results. (Regis University, n.d.)

 

Did we show beyond a reasonable doubt that Dusty was not telling the truth?

Our test statistic (t) 2.778 is larger than the critical value (tc) 1.74, which means if fall the far right and inside of the critical region.  Therefore, we can reject Ho. Conclusion: The mean time to make a cheesecake is greater than 60 minutes.

 

Why do statisticians shudder at the word "prove" and use words like support, accept the H1, reject the Ho, etc.?

The word prove implies that the subject was true or false beyond a shadow of a doubt.  Having something so concrete does not leave room for any other solution.  There are times when a statement cannot be “proven” true, but can be “proven” false.  The simplest example I have seen is if someone claimed all swans are white, which is something that is easily confirmed by constantly seeing white swans, but could quickly be disproven by seeing a single black swan.   A quote I came along that helped me with this is:  “Remember we can never prove the null hypothesis. All we can prove is that there is a relationship or effect (H1) between two or more variables." (Huck, Disproving the Null Hypothesis)

References

Huck, S. W. (Disproving the Null Hypothesis, n.d.). Statistical Misconceptions. Retrieved from Statistical Misconceptions: http://www.statisticalmisconceptions.com/sample2.html

Regis University. (n.d.). MT270 - Elementray Statistics, Content, 7.4 The P Value in Hypothesis Testing. Retrieved from Regis University World Class: https://worldclass.regis.edu/d2l/le/content/185848/viewContent/2126705/View

Triola, M. F. (2014). Elementary Statistics. Boston: Pearson.