o    Message expanded. Message read Intro to null & alternative hypotheses

o    posted by Jerome Tuttle , Aug 02, 2017, 5:11 AM

The null hypothesis is a statement that the population parameter equals some claimed value.  The alternative hypothesis is that the population parameter is > (greater than), < (less than), or ≠ (not equal) to that same claimed value; the alternative hypothesis is not allowed to contain an equal sign.

     If the alternative hypothesis contains ≠, then it is a two-tailed test, and you will reject Ho for very large or very small test statistics of your sample.  If the alternative hypothesis contains >, then it is a one-tailed test, and you will reject Ho for very large test statistics of your sample. If the alternative hypothesis contains <, then it is a one-tailed test, and you will reject Ho for very small test statistics of your sample. 

     The conclusion of the test should state the decision, addressing the original assertion, in nontechnical terms.

     Take some time reading steps 1, 2 and 3 of Figure 8-1 on page 385 and the example of these steps on page 386.  An important part of these problems is choosing the proper null and alternative hypothesis - if you choose the wrong hypotheses, you will get the wrong answer.

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o    Message expanded. Message read Innocent until proven guilty

o    posted by Jerome Tuttle , Aug 02, 2017, 5:15 AM

In some sense the prior chapters of the course have been leading up to chapter 8 where we test hypotheses about a population parameter.  We test a hypothesis, and we will reject the hypothesis only if the evidence is strong enough. 

     This is similar to our legal system, where the null hypothesis is innocence - we begin by assuming innocence, and then we reject the hypothesis of innocence only if the evidence of guilt is beyond a reasonable doubt.

     Note the word reasonable in the sentence above.  Our legal system does not demand certainty.  Consequently, it expects some errors.  And it gets them!

     We will reject a hypothesis statistically when the probability of a particular observed event is extremely small;  in other words, when it is highly unlikely to occur by chance.

     Note that a legal verdict of not guilty does not say the accused is innocent.  Why do you think that is?  What is the equivalent statistical statement when we do not reject a hypothesis statistically?

     (Please do not use this discussion area to discuss recent current events court cases that you think are unfair, unless you can relate them to statistics.)

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o    Message expanded. Message read Intro to P-value & Critical value methods

o    posted by Jerome Tuttle , Aug 02, 2017, 5:13 AM

Hypothesis testing is usually done with either the P-value method or the Critical Value method.

     The P-value method asks what is the tail probability P of getting a particular calculated test statistic if the null hypothesis is true, and then compares that P to a pre-determined value of alpha such as 5%. Here we are comparing two probabilities.  To find the P-value, you calculate the tail probability of your test statistic.

     The Critical Value method compares a calculated test statistic against the critical value corresponding to a pre-determined value of alpha such as 5%.  For the Normal probability we are comparing two z's. To find the critical value, you calculate the z value or t value corresponding to a tail value of alpha.

     In either case, the conclusion is determined by the values in the comparison.  We will reject the null hypothesis if the calculated value is in the unlikely region of the probability distribution.  The hypothesis testing file I gave you yesterday tries to illustrate this.

     Before we get into the specifics of how this is done with numbers - does the above seem reasonable, or at least plausible?  I think it is helpful to gain some agreement on the plausibility of this, before we get into the specifics.

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o    Message expanded. Message read Chapter 8 formulas

o    posted by Jerome Tuttle , Aug 03, 2017, 5:23 AM

There are several formulas for test statistics in chapter 8 and it is easy to get them confused.  Here is a quick summary about the tests in chapter 8: 

     Testing a claim about a population proportion p:

Let p_hat be the sample proportion, equal to x/n.  Let p be the population proportion, and  q = 1 - p.  The test statistic is

z = (p_hat - p) / √(p*q/n) , and the P-values and critical values use a z or Normal Distribution.

     Testing a claim about a population mean μ, when σ is unknown:

Let xbar be the sample mean.  The test statistic is t = (xbar - μ ) / (s /√n ), and the P-values and critical values use a t Distribution with n-1 degrees of freedom.

     Testing a claim about a population mean μ, when σ is known:

Let xbar be the sample mean.  The test statistic is z = (xbar - μ  ) / (σ /√n ), and the P-values and critical values use a z or Normal Distribution.

    Testing a claim about a population SD σ or population variance σ² is in section 8.5, and is not in our syllabus, but MyStatLab may ask a question about this anyway because it appears in Table 8-2 on page 387.  The test statistic equals (n-1)s²/σ², and hopefully that is all they will ask.  I don't think you need to study the chi-square distribution to answer what they ask.

     Statdisk will actually do all of this within ANALYSIS > HYPOTHESIS TESTING > choose the correct test.

     To my knowledge, Excel does not have a useable built-in function for testing one population proportion or one mean.  There is an Excel =Z.TEST function, but the Internet contains many warnings not to use it. There is an Excel =T.TEST function, but it is for testing two population means, not one.

     Any questions?

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o    Message expanded. Message read Does anyone ever use this stuff?

o    posted by Jerome Tuttle , Aug 03, 2017, 5:25 AM

The language of null and alternative hypotheses is not confined to statistics books.  Medical researchers, manufacturers, psychologists, educators, and others who use statistics also use these terms.

     However, in a scholarly paper as you might find in the UOP library, the author may not provide the details of the statistical method but may merely present the P-value and a single test statistic like the t statistic.  For example, "The results show that students who pass MTH 231 earn significantly more income than students who do not (p = .015, t = 2.226)."  Sorry - I made that up.  

     Of course, scholarly papers may use more advanced statistical techniques, and may use more than the one independent variable than we do in this course.  But the wording of a statistical conclusion may be similar to what we are doing.

     If you have a scholarly paper handy such as one you found on the UOP Library, see if the conclusion is stated similarly to what I wrote above, and share the conclusion with the class.

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o    Message expanded. Message read A different way about thinking about hypotheses

o    posted by Jerome Tuttle , Aug 03, 2017, 5:41 AM

Here is a different way of thinking about null and alternative hypotheses.  <Please download file art16>

    Suppose I have a population of people, and suppose each person is either 100% blue or 100% red.  I think the population is mostly blue, with some red, but I don't know for sure and they are too many to count.  Of course blue and red can represent anything - how about people who earn most of the mastery points before taking the quiz, versus people who don't?  (Hint, hint.)

     I take a sample, and they are mostly red, with some blue.

     I think the population is mostly blue, but I draw a sample that is mostly red.  Hypothesis testing asks, if the population is mostly blue, how likely is it I will draw a sample that is mostly red?  It is possible, but unlikely.  So maybe, a better explanation is that the population is not mostly blue, and in fact the population is mostly red. 

      In this example the reason I am rejecting the null hypothesis is that I really had the wrong idea about the (first) population to begin with.

      I hope this helps, and I hope I have not confused you more. 

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o    Message expanded. Message read You need to be able to do these two calculations

o    posted by Jerome Tuttle , Aug 03, 2017, 5:47 AM

You need to be able to do these two calculations:  Given a z value, find its tail probability p value; and given a t value, find its tail probability p value.  We did exactly this in week 3.

     The P-value is the tail probability at the calculated value of the test statistic.  However, if your alternative hypothesis is "NOT EQUAL TO", then you need to use 2 times the one tail P-value.

     The z table contains two decimal point z values.  Given a z value to three decimal places, how do you find the P-value?  Suppose you had z = 1.057, and you want the P-value to its right.  The table gives for z = 1.05, left probability = .8531; for z = 1.06, left probability = .8554; so you can approximate for z = 1.057, left probability = (.8531 + .8554)/2 = .8543, and so right probability = 1 - .8543 = .1457.  Excel's =NORMSDIST(z) will give you an exact answer:  =NORMSDIST(1.057) = .8547, 1 - .8547 = .1453

     Tables of the t distribution provide only a few pre-determined probability levels and do not directly allow you to calculate the P-value without approximating it. However, since you are comparing a P-value to a pre-determined alpha value like 5%, it may be enough to know your P-value is either less than alpha or greater than alpha, rather than knowing the exact value.

     For example, with a 1-tail left test with n = 11 so df = 10, and a test statistic of t = -.486, Table A-3 shows in the df = 10 row that the t values go down as tail probability areas get higher, and if t = .486 were in the table its 1-tail probability would be greater than .10.  So we would fail to reject Ho for alpha = .10, or for smaller alphas. (By symmetry of the t curve, if we reject Ho for t = +.486 on a 1 tail right test, we would reject Ho for t = -.486 for a 1 tail left test.)

     Excel functions T.DIST.2T and T.DIST.RT will calculate the P-value directly: =T.DIST.2T(x, degrees of freedom) gives the 2 tailed probability; note x must be  >0. In older versions of Excel this function was =TDIST(x, degrees of freedom,2). =T.DIST.RT(x, degrees of freedom) gives the 1 tailed probability; note x may be <0.  =T.DIST.RT(.486,10) gives .31872 as the P-value.

     Or, you may use Statdisk to calculate a P-value.  ANALYSIS > PROBABILITY DISTRIBUTIONS> choose a distribution such as T, enter the degrees of freedom, and then either enter a T value or a probability, click EVALUATE.  You should test one of these against a textbook example to convince yourself you are doing it correctly.

     I had trouble understanding the P-value the first time I encountered it.  If you're unsure, please ask me.

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o    Message expanded. Message read The two methods compared

o    posted by Jerome Tuttle , Aug 03, 2017, 5:51 AM

Hypothesis testing is usually done with either the P-value method or the Critical Value method.

     The P-value method asks what is the tail probability P of getting a particular calculated test statistic, and then compares that P to a pre-determined value of alpha such as 5%. Here we are comparing two probabilities.

     The Critical Value method compares a calculated test statistic against the critical value corresponding to a pre-determined value of alpha such as 5%.  For the Normal probability we are comparing two z's, and for the t distribution we are comparing two t's. 

     In either case, the conclusion is determined by the values in the comparison.  We will reject the null hypothesis if the calculated value is in the unlikely region of the probability distribution.

     The two methods give identical results.

     Is the distinction clear?

     However, choose one method or the other, but don't take a mixture.  Don't take a test statistic and compare it to alpha.

 

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