Wk6-HW6.docx - WEEK 6 HOMEWORK 6 LANE CHAPTERS 11 12 AND 13...

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WEEK 6 – HOMEWORK 6: LANE CHAPTERS, 11, 12, AND 13; ILLOWSKY CHAPTERS 9, 10 INTRODUCTION TO HYPOTHESIS TESTING WHAT IS A HYPOTHESIS TEST? Here we are testing claims about the TRUE POPULATION’S STATISTICS based on SAMPLES we have taken. The most common statistic of interest is of course the POPULATION MEAN ( µ ). But, we can also test its VARIANCE and its STANDARD DEVIATION. (We can also compare TWO or more means to see if there are significant differences. We must have a basic hypothesis, referred to as the NULL Hypothesis (Ho) and an ALTERNATE Hypothesis (Ha). Our NULL ( and ALTERNATE) Hypotheses can take three forms: (1) Ho: µ < some number; Ha: µ > that number (< is “less than or equal to” and > is “greater than or equal to” ), (2) Ho: µ > some number; Ha: µ < that number , or (3) Ho: µ = some number; Ha µ ≠ that number ; ( means “not equal NOTE THAT Ho MUST HAVE THE “EQUALS” IN IT WHEREAS Ha NEVER DOES. (1) Is referred to as a “ONE-TAILED TEST TO THE LEFT” (2) Is a “ONE-TAILED TEST TO THE RIGHT” (3) Is a “TWO-TAILED TEST” NEXT , we need to decide what level of significance, i.e.(how sure we want to be about our hypothesis. This is where comes in again. Do we want to test at the 10%, 5% or 1% level of significance? Another wrinkle is that for the TWO- TAILED test, since our value could be greater OR less than some number, we use α /2 for each extreme, so for 10% it’s 5% (0.050) at each end (tail of the curve), for 5% it’s 2.5% (0.0250) at each end, and for 1% it’s 0.5% (0.0050) at the ends. You have heard about this kind of split before with confidence intervals, but think about it. Here is a graphical display of all this: to”) α
As you can see, there is a CRITICAL z-VALUE for each of these test depending on the significance level alpha ( α ) or α/2. In HW4 questions 1 and 2, you found the critical z-values for alpha’s of 1%, 5% and 10%, which would work for the one- tailed tests. For the two tailed tests we need to split these alphas (α/2) and find the critical z-values (at the positive and negative tails of the graph) So, for an α of 1% (0.0100) it would be α/2 or 0.005 in the left tail (negative z-value) = -2.575 and for the far right tail (0.005 in that tail) we would have to find the z-value for an area to the LEFT of 99.5% (0.9950) and this is +z = +2.575
Continuing on, for an α of 5% for a two-tailed test the z-values for α/2 would correspond to areas under the curve of 0.0250 at each end. The far left tail would have a negative z-value of -1.96 (see picture above) and the far right tail would have a positive z-value of +1.96 that in the Table represented an area of 97.5% (0.9750) to the LEFT. Lastly, for an alpha of 10% , hence an α/2 at both ends of 5% (the two-tailed test), the negative z-value would be -1.645 . The positive z-value marking the upper 5% (Table value from 95% to the left) is +1.645 . SO, FOR YOUR USE IN ALL HYPOTHESIS TEST (AND WORKS FOR CONFIDENCE INTERVALS TOO) HERE IS A TABLE OF THE CRITICAL Z-VALUES FOR THE VARIOUS LEVELS OF SIGNIFICANCE (ALPHA’s) MOST COMMONLY USED. ALPHA (α) -Z-value (LEFT tail) +Z-value (RIGHT tail) ALPHA/2 (α/2) -Z-value (LEFT tail) +Z-value (RIGHT tail) 1% (0.0100) -2.33 +2.33 0.5% (0.0050) -2.575 +2.575 5% (0.0500) -1.645 +1.645 2.5% (0.0250) -1.96 +1.96 10% (0.1000) -1.28 +1.28 5% (0.0500) -1.645 +1.645
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