Week 1 Lecture: The Normal Distribution (Chapter 6)

Transcription

1 Week 1 Lecture: The Normal Distributio (Chapter 6) This bell-shaped curved is probably the most commo ad well-recogized distributio i all of biology. However, ot all bell-shaped curves are ormal. Whe a radom variable is said to be ormally distributed, it will have a mea µ ad a stadard deviatio ; i.e., X ~ N(µ,). Whe X ~ N(µ,), the Z = X µ ~N(0,1), which is called the stadard ormal curve. This Z-values gives the umber of stadard deviatios away from µ = 0. Table B.2 gives the proportios of a ormal distributio which lie beyod a specified Z-value. Because the ormal distributio is symmetrical, we ca use the table to fid differet proportios for a radom variable. For ay ormal distributio, µ + cotais 68.3% of the populatio µ + 2 cotais 95.5% of the populatio µ + 3 cotais 99.7% of the populatio. Example: X = a ormal distributio of weights, with µ = 70 kg ad = 10 kg. What if we wat to fid the proportio of the distributio that is greater tha 80 kg; ie., P(X > 80)? First, we eed to X µ compute Z = = = The, we have 10 P(X > 80 kg) = P(Z > 1.0) = or 15.87%. This ca also be thought of as the probability of radomly 60 (1) 70 (0) 80 (1) 1

2 samplig a weight, X, greater tha 80 kg from a populatio with µ = 70 kg ad = 10 kg. What if we wat to determie: P(X > 60 kg)? Agai, calculate Z = = The, you 10 have: P(X>60 kg) = P(Z > -1.0) = = (you ca do this because the distributio is symmetrical). X = 60 (1) You ca also fid the proportio betwee two poits: P(55 < X < 65) = P(-1.5 < Z < -0.5) = ( ) ( ) = What if we wat the value of X that represets a specified percetage of the distributio? This iterval ca be expressed as: µ ± Z*. Remember that the Z-value represets the umber of stadard deviatios from a mea of zero. 2

3 So, what is the value of X that ecompasses the middle 50% (Z = ±0.67 look up a Z-value for 25%)? X u = *10 = 76.7 kg X l = *10 = 63.5 kg. What is the value of X that cuts off the upper 5% (Z = 1.645)? X u = *10 = 86.5 kg What is the value of X that cuts off the lower 10% (Z = -1.28)? X u = *10 = 57.2 kg Cetral Limit Theorem The CLT is cocered with a sample mea, x i i= 1 X =. For ay, draw from a ormal populatio, X has exactly a ormal distributio with a mea = µ ad a stadard deviatio = S x = >>> X ~ N µ,. If the paret populatio is ot ormally distributed ad is large, the X ~ N µ,. Thus, for ay X: E ( x) = µ 3

4 Var( X) 2 2 = = = = stadard error of the sample mea = SE. If X is ormally distributed (X ~ N(µ,)), the X µ Z = ~N(0,1), for ay. If X is some other distributio, the X µ Z = teds towards a N(0,1) as gets large. Note the differece betwee the idividual X ad the sample mea X: 1. X~N(µ,) implies: Z = X µ ~ N(0,1). 2. X >>> X µ Z = is ~N(0,1) as gets large, ad is exactly ~N(0,1) if X~N(µ,). Itroducig Statistical Hypothesis Testig Statistics is largely cocered with makig ifereces about a populatio from a sample collected from that populatio. Most ofte, the ifereces are made o oe or more populatio meas. Statistical ifereces are costructed o a framework that icludes a ull hypothesis ad a alterative or research hypothesis. The ull hypothesis represets a coditio of o chage or o differece or equality while the alterative hypothesis represets the coditio that is true if the ull hypothesis is false. The ull ad alterative hypotheses should be stated a priori. Whe testig hypotheses cocerig populatio meas, sample meas are calculated from radomly sampled data collected withi the populatio. The, the probability of that sample mea occurrig give that the ull hypothesis is true is calculated. This probability ca be graphically represeted by the proportio of the area uder the curve, like we calculated earlier. This calculated probability is the compared to a objective criterio for drawig a statistical 4

5 coclusio about our sample mea. This compariso is stated i a decisio rule. Thus, our criterio is used to reject or ot reject the ull hypothesis. This is a importat cocept, i that the calculated probability represets the sample data give that the ull hypothesis is true, ad NOT that the ull hypothesis is true give the data. Thus, we ever accept the ull hypothesis as beig true, we oly fail to reject or do ot reject the ull hypothesis. This is a philosophy of falsificatio, largely codified by Karl Popper. The reaso we ca ever accept a ull hypothesis is because i statistics, we ever actually have a complete accoutig of a populatio. I some istaces, our sample mea may actually be a extreme occurrece that creates a error i our coclusio of the statistical test. There are two types of errors that occur i statistical hypothesis testig: 1) Type I Error ad 2) Type II Error. Type I Error occurs whe we reject a ull hypothesis that is i fact true, ad a Type II Error occurs whe we do ot reject a ull hypothesis that is i fact false. As the researcher, you set the acceptable levels of Type I ad II Errors that you are willig to accept. The Type I Error rate or sigificace level or alpha level is ofte set at 0.05 or 95%, while the Type II Error rate or 1-power or beta level is ofte set at These values are arbitrary ad ca be chaged. The idea of statistical power will be discussed i more detail later, but i summary it tells you about the reliability of your statistical test for makig a coclusio about the sample mea. You typically icrease power of a test by icreasig the sample size. Aother poit about hypotheses is that they ca be directioal; you ca specify which tail of the distributio (or both) for which you are testig. This is called oe tail versus two tail testig, ad it is importat that you correctly state your hypotheses to reflect which or both tails you are cosiderig. Zar provides a excellet treatmet of this topic i sectio 6.3; it may very well be the best sectio i the book, i my opiio! 5

6 Testig for Departures from Normality The Chi-square ad KS GOF tests ca be used to assess whether or ot a sample came from a ormal populatio, though Zar does ot recommed these methods because they have low power. Graphical methods ca be used to visually assess whether the sample appears to be from a ormal populatio. Shapiro ad Wilk developed a test for ormality by calculatig a W statistic. This is oe of the ormality tests provided i SAS. Let s do a example i SAS to show how you ca go about testig for ormality. Example: We wat to test if total height (i feet) i two samples of Appalachia oaks are ormally distributed. We will use the PROC UNIVARIATE routie i SAS to calculate the Shapiro-Wilk W statistic ad test the hypotheses: Ho: The white oak sample comes from a ormal distributio. Ho: The red oak sample comes from a ormal distributio. Ha s: ot Ho. α = 0.05 From SAS, we get: W (white oaks) = 0.96 (p = 0.24) >>> do ot reject Ho. W (red oaks) = 0.97(p = 0.44) >>> do ot reject Ho. 6