How to read ArtOfStat Displays for the exam. PDF

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ArtOfStat Displays...

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How to read ArtofStat to get P-values or Critical Values ArtofStat can provide the specific Critical Value where the Rejection Region starts for a particular alpha, or evaluate a p-value for a particular Test Statistic (note: Chi-sq shown, but concepts apply to all probability distributions).

Part 1. Critical Values and Rejection Regions

A) Our interest is in a curve with 3 df

B) This tab provides “the number-line value (or percentile) for a particular probability”

D) So in this case: What is X so that P(chi-sq > X) = 0.05?

G) So we see now that P(chi-sq > 7.815) = 0.05 Rejection Region for alpha = 0.05 starts when chi-sq = 7.815

E) Let’s set these two boxes to specify a right-tailed probability to be 5% C) The x-axis number-line shows the values of the test statistic (or the percentiles) – in this case, Chi-sq values

F) As we set the right tail 5% probability to be 5%, we see that 5% of the area is to the right of chi-sq = 7.815.

Remember the significance level is the threshold of probability that will mean we decide to Reject or Not Reject a Null hypothesis. So let’s say we have a sample to compare to a Null hypothesis, how do we determine to reject? We reject when the probability of a sample statistic being a likely outcome if the Null was true, is low. How low? When the probability of such a sample occurring by chance if the Null was true is less than alpha. How do we find these probabilities? By converting the difference between our sample statistic and the hypothesised population parameter into a Test Statistic (which will be specific for this sample). If that Test Statistic is big, it will indicate an unlikely sample statistic. The probability distribution tells us the specific probability of this specific Test Statistic (or one more extreme) occurring if the Null is True. This is the p-value (the probability-value). Maybe all we need is to know is the value of the Test Statistic that marks the boundary between likely and unlikely Test Statistics, or, in other words, where on the number line does the part of the curve with the probability of alpha start? This value of the Test Statistic is the Critical Value, because we can compare the Test Statistic based on our sample with this Critical Value, and if our Test Statistic is more extreme, that will mean the resulting p-value will be less than alpha. That means the Critical Value is the boundary that separates the Rejection Region from the rest of the probability distribution. So to revisit the chart above, alpha is 0.05. The Critical Value of Chi-sq (with 3 df) = 7.815. If my Test Statistic is greater than 7.815 the p-value must be < 0.05, because the Rejection Region starts at chi-Sq = 7.815.

Part 2. Test Statistic and p-values Say we had a Test Statistic based on our sample that was chi-sq = 8.2, and we wanted to evaluate the actual pvalue. We use the other tab in ArtofStat to find the probability of a Test Statistic this extreme (or more extreme). The diagram now shows, that if we plug in 8.2 on the number line, the chart shows the area to the right (the pvalue) is 0.042. This is the P(X>8.2).

B) This tab provides “the probability for a particular number-line value (or “x”) A) Our interest is in a curve with 3 df

D) So in this case: What is the probability of a chi-sq >= 8.2?

G) So we see now that P(chi-sq > 8.2) = 0.042

Specific p-value when Chi-sq = 8.2 is 0.042

E) Let’s set these two boxes so we can find a right-tailed probability when the number-line value = 8.2

C) The x-axis number-line still shows the values of the test statistic (or the percentiles) – in this case, Chi-sq values

F) As we set the number-line value to be 8.2, we see that the area to the right when chi-sq is 8.2 = 0.042...