Statistics

صفحه 1:
Statistics From BSCS: Interaction of experiments and ideas, 2°¢ Edition. Prentice Hall, 1970 and Statistics for the Utterly Confused by Lloyd Jaisingh, McGraw-Hill, 2000 

صفحه 2:
وگ is the fraction of the variation in the values of y that is explained by the least-squares regression line of y on x. Exanple: IP? = 0.000 ta the gropk to the kA, his wee ‏تاه با‎ 00% oP pae's qrade is acerunited Por by the foe Cites ‏مق‎ wih otieadsare. Che other 99% could be due ‏و( اه له و و‎ Chose Pteuddace 

صفحه 3:
What is statistics? a branch of mathematics that provides techniques to analyze whether or not your data is significant (meaningful) Statistical applications are based on probability statements Nothing is “proved” with statistics Statistics are reported Statistics report the probability that similar results would occur if you repeated the experiment 

صفحه 4:
Statistics deals with numbers * Need to know nature of numbers collected — Continuous variables: type of numbers associated with measuring or weighing; any value in a continuous interval of measurement. * Examples: — Weight of students, height of plants, time to flowering — Discrete variables: type of numbers that are counted or categorical + Examples: — Numbers of boys, girls, insects, plants 

صفحه 5:
Can you figure out... Which type of numbers (discrete or continuous?) — Numbers of persons preferring Brand X in 5 different towns —The weights of high school seniors —The lengths of oak leaves —The number of seeds germinating — 35 tall and 12 dwarf pea plants — Answers: all are discrete except the 2"? and 3" examples are continuous. 

صفحه 6:
Populations and Samples ٠ Population includes all members of a group — Example: all 98 grade students in America — Number of 9* grade students at SR - No absolute number ٠ Sample — Used to make inferences about large populations — Samples are a selection of the population — Example: 6'* period Accelerated Biology + Why the need for statistics? — Statistics are used to describe sample populations as estimators of the corresponding population — Many times, finding complete information about a population is costly and time consuming. We can use samples to represent a population. 

صفحه 7:
Sample Populations avoiding Bias Individuals in a sample population — Must be a fair representation of the entire pop. —Therefore sample members must be randomly selected (to avoid bias) — Example: if you were looking at strength in students: picking students from the football team would NOT be random 

صفحه 8:
Is there bias? A cage has 1000 rats, you pick the first 20 you can catch for your experiment A public opinion poll is conducted using the telephone directory You are conducting a study of a new diabetes drug; you advertise for participants in the newspaper and TV All are biased: Rats-you grab the slower rats. Telephone-you call only people with a phone (wealth?) and people who are listed (responsible?). Newspaper/TV-you reach only people with newspaper (wealth/educated?) and TV( wealth?). 

صفحه 9:
Statistical Computations (the Math) * If you are using a sample population — Arithmetic Mean (average) z. =x N «= {1,2,3,4,5};2=3 The sum of all the score. divided by the total number of scores. — The mean shows that % the members of the pop fall on either side of an estimated value: mean | 

صفحه 10:
“Looking at profile of data: Distribution * What is the frequency of distribution, where are the data points? Distribution Chart of Heights of 100 Control Plants er of plants in Class (height of plants: Nu cm) eat 3 0.0.0.9 10 هر 21 2.0.2.9 30 3.03.9 20 40.4.9 14 5.05.9 6.06.9 2 

صفحه 11:
Histogram-Frequency Distribution Charts Number of Plants in each Class sa Number of plants ineach This is called a “normal” curve or a bell curve This is an “idealized” curve and is theoretical based on an infinite number derived from a sample 

صفحه 12:
Mode and Median * Mode: most frequently seen value (if no numbers repeat then the mode = 0) * Median: the middle number —If you have an odd number of data then the median is the value in the middle of the set —If you have an even number of data then the median is the average between the two middle values in the set. 

صفحه 13:
Variance (s2) * Mathematically expressing the degree of variation of scores (data) from the mean + A large variance means that the individual scores (data) of the sample deviate a lot from the mean. « Asmall variance indicates the scores (data) deviate little from the mean 

صفحه 14:
مس سار و و سس ولج( بصخم ‎eu of X = sore,‏ = 2 ‎ween, = td of scores or usher‏ =[ لل 22 _ و ‎N‏ OR use the OBR Pucrtivs is Bare Worksheet for Calculating the Variance for 7 scores For this problem the population variance is 0.57 XX (X- Wf 5 ۲ 1 3 1 1 4 ‏م6‎ 0 4 0 0 3 1 1 4 0 0 S| 1 bg 4 hip Iunnw xecokte eduluxesrled DC rdovardevs. hice 

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منم ‎Por a Bised GBOPLE‏ جومم سا مطولطله) ‎anv of; X = sore, uch,‏ = ‎totd oF scores or uches-(1‏ = 4 و vy Sa L(x 0 xX) 7۱-1 (often.read as “x bar”) is the mean (average value of Worksheet for Calculating the Qote the ‏نون موی‎ ts haryer...why? ‘Variance for 7 scores لي 1 وو كر ‎ge EON)‏ قر ع عر م اع 7-1 7-1 د 3 1 1 ۱ For this problem the population variance is 0.57 3 1 1 4 6 S| 1 28 4 جما .وجل صا ©( ©لج لمموجسب ادلب جاداصمج. بصنم | ‎hip‏ 

صفحه 16:
ights in Centimeters of Five Randomly Selected Pea Plants Grown at 8-1 Pla Height Deviations Squares of nt (cm) from mean deviation from mean (x) (x; x) (x,- x)? A 10 2 4 B 7 1 1 © 6 2 4 D 8 0 0 E 9g 1 1 2 ۲ < = (x; x) =0 = (x- x)? = 10 40 X, = score of value; X (bar) = mean; = = sum of 

صفحه 17:
عمنو() سا مطلطل() ا۳) “Dhere were Pave phate; 09; therePore o- 42 2 ‎ge LX)‏ 99 10/6 :6 7-1 ‎der‏ با بط ام و همطل سا سس واه وا ‎helps‏ ه00 مت روم عامومجو با مات ای امه ات با 

صفحه 18:
Standard Deviation An important statistic that is also used to measure variation in biased samples. S is the symbol for standard deviation Calculated by taking the square root of the variance So from the previous example of pea plants: The square root of 2.5 ; s=1.6 Which means the measurements vary plus or minus +/- 1.6 cm from the mean 

صفحه 19:
What does “S” mean? * We can predict the probability of finding a pea plant at a predicted height... the probability of finding a pea plant above 12.8 cm or below 3.2 cm is less than 1% * Sis a valuable tool because it reveals predicted limits of finding a particular value 

صفحه 20:
Pra Plact Oerwal Disttbuticd Curve uth Grt Dev: 

صفحه 21:
The Normal Curve and Standard Deviationer ian: Cock veriod ke io ‏اه اه سس‎ cae 00% of uches Pal wets +1 or Cl of the sees 98% oF uches Pal wikis +O 8, © ‏صب‎ ‏لس اه رام‎ (299%) Pal whic © ‏لد‎ dev write Same as others Probably less than others Definitely less than others wae) ome trae tom Probably more than others Definitely more than others 2 ۲ 4 ۲ رس امسر راو حلله مد اهنا 

صفحه 22:
Standard Error of the Sample Means KA Standard Error A The mean, the variance, and the std dev help estimate characteristics of the population from a single sample So if many samples were taken then the means of the samples would also form a normal distribution curve that would be close to the whole population. The larger the samples the closer the means would be to the actual value But that would most likely be impossible to obtain so use a simple method to compute the means of all the samples 

صفحه 23:
A Simple Method for estimating standard error 5 سب < بر ‎fn‏ (Gtoodard error te the colculited stocdard deviatiza divided by the square root oF the Or ‏اه ام‎ te popubatios (Gtoodard error ‏و عم بل چاه‎ used to test the retobiiiy oP the cot ‏.سم‎ 1 here ore (0 core phan ‏ولج و لس‎ deviation ‏خا‎ 0.2 Ge, = 0.8] syrovick IO = 0.8/9.09 = 0.009 0.009 represeds vor std dev too scp oF 40 ‏ام‎ ‎AP there were (OO phrase the standard error would drop te D.DDE ky? erase whee we tohe horwer suxopkes, our ‏واه بو سوه وتو‎ to the fru weoo ude of the popukiica. Dhus, the dstrbuica of the ‏مروت‎ wero would be tess spread cut ced woud hove a luwer stoodacd deviation. 

صفحه 24:
Probability Tests What to do when you are comparing two samples to each other and you want to know if there is a significant difference between both sample populations (example the control and the experimental setup) How do you know there is a difference How large is a “difference”? How do you know the “difference” was caused by a treatment and not due to “normal” sampling variation or sampling bias? 

صفحه 25:
Laws of Probability The results of one trial of a chance event do not affect the results of later trials of the same event. p= 0.5 (a coin always has a 50:50 chance of coming up heads) The chance that two or more independent events will occur together is the product of their changes of occurring separately. (one outcome has nothing to do with the other) Example: What’s the likelihood of a 3 coming up ona dice: six sides to a dice: p = 1/6 Roll two dice with 3’s p = 1/6 *1/6= 1/36 which means there’s a 35/36 chance of rolling something else... Note probabilities must equal 1.0 

صفحه 26:
Laws of Probability (continued) The probability that either of two or more mutually exclusive events will occur is the sum of their probabilities (only one can happen at a time). Example: What is the probability of rolling a total of either 2 or 12? Probability of rolling a 2 means a 1 on each of the dice; therefore p = 1/6*1/6 = 1/36 Probability of rolling a 12 means a6 anda 6 on each of the dice; therefore p = 1/36 So the likelihood of rolling either is 1/36+1/36 = 2/36 or 1/18 ۰ ۰ ۰ 

صفحه 27:
The Use of the Null Hypothesis Is the difference in two sample populations due to chance or a real statistical difference? The null hypothesis assumes that there will be no “difference” or no “change” or no “effect” of the experimental treatment. If treatment A is no better than treatment B then the null hypothesis is supported. If there is a significant difference between A and B then the null hypothesis is rejected... ۰ ۰ 

صفحه 28:
T-test or Chi Square? Testing the validity of the null hypothesis Use the T-test (also called Student’s T- test) if using continuous variables from a normally distributed sample populations (ex. Height) Use the Chi Square (X?) if using discrete variables (if you are evaluating the differences between experimental data and expected or hypothetical data)... Example: genetics experiments, expected distribution of organisms. 

صفحه 29:
T-test ٠ T-test determines the probability that the null hypothesis concerning the means of two small samples is correct * The probability that two samples are representative of a single population (supporting null hypothesis) OR two different populations (rejecting null hypothesis) 

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ات و لول با موه وله مه وت با ارت با وا تاه ۳ وا[ لد لم ۹۰( عع سيق ۲۲ 0 1 ‎Mean sample ۹ N= # in sample |‏ < را ‎Xy= mean Sample Q N= in Sample 2‏ ‎Sample |‏ 5 ‎variance of Sample 2‏ = >33 ‎If samples are equal in Size see nak eal ‎t= XX | ‎y= +S 7 

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nt of O, Used by Germinating Seeds of Corn and Pea Plants ml OJhour | at 25°C Reading | Com Pea ALow to de his hin BXOCL Number 1 0.20 0.25 2 0.24 0.23 3 0.22 oat 4 0.21 0.27 5 0.25 0.23 6 0.24 3 7 0.23 0.25 8 0.20 0.28 9 0.21 0.25 10 0.20 0.30 Total 2.20 2.70 Mean 0.22 0.27 Variance | 0028 0106 (Cxxcet Pie tocated tc (Bor Bio Pile Potter 

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صفحه 40:
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“D table (sucople table with D probubilties 0 20/2[ (لنما مهم) 20 ۰ ‎tails)‏ 0.1 8 1.64 0.05 5 1.96 0.01 33 2.576 Ose ‏امس و‎ ool b show ۳۲2775 1792۳3 ‏عط ج) ما سس‎ tro) sap wen ©. Ose ‏وه وا ات بویا ه‎ ۰ skPirad dPRereure (ether ‏جا لصي‎ Or bee tron) betveen somrpke wrcn ® onl soaps wer © D kble wee! 6 < ‏)لو‎ probubly of) 0096, ©96 ‏لم‎ 0 96 وبصي جما خا يليت جلا جم من وا مه معط لو جا ‎dpk rePers‏ 2 0 41 ‎Blo your nd hypoheots‏ نمی ده تچ و موجه بط سا مت ما و ‎chercraiver kypetbests‏ و سوه ور ما اج یی با نا لو ات علج تم له سر وت موه مه ‎dpa ©: rePers to‏ 2 < ۵/9 ر1 ‎ww dPPereue bet the a OF th ic or the expericectal hypriests (\‏ ات + ‎dP Pereuce expenied). S/our chercaive hypothesis ts lookioy Por‏ ‎dPPereue‏ 

صفحه 43:
Example z-test * You are looking at two methods of learning geometry proofs, one teacher uses method 1, the other teacher uses method 2, they use a test to compare success. * Teacher 1; has 75 students; mean =85; stdev=3 * Teacher 2: has 60 students; mean =83; stdev= 2 - ‏و2‎ = (85-83)/V3*2/75 + 22 < 5 ات = 2/0.4321 = 4.629 

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Example continued 0 را ‎Detad (ts ont beter thon wetud 2‏ با ات جوا با ‎ALO = cherentie Wyzokeoty wink! be teat Drtheal (ts beter ‏دجا‎ sober © ‎Whe bw ove kid 2 tot (ower the «nll hypokeoty doe pride fra here wd be wo WPPereccr) Gp Por the probably oP 0.06 (O% onnPexnee or 08% vexP ecw) tht Deherd ove tort beter tho ‏لس‎ © ... hat ohort ke = Za 1.645 ‎So 4.629 is greater than the 1.645 (the null hypothesis states that method 1 would not be better and the value had to be less than 1.645; it is not less therefore reject the null hypothesis and indeed method 1 is better ‎ ‎table (secple table wits O probabstics) ‎3 Za (one tail) |Za/2 (two tails) ‎0.1 8 1.64 ‎0.05 5 1.96 ‎0.01 233 2.576 ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ 

صفحه 45:
Chi square * Used with discrete values * Phenotypes, choice chambers, etc. * Not used with continuous variables (like height... use t-test for samples less than 30 and z-test for samples greater than 30) 2 * O= observed valu y2 aye + E= expected valu 6 مدا حمسي 5 )داه أعصمصام 5) | مجه ومجسمجوط. نمي ‎hep:‏ 

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http://course1.winona.edu/sberg/Equation/chi- squ2.gif Observed individuals — Expected individuals with a given phenotype with a given phenotype Greek me 2 0-6( 2 ayes e Summation => add together a term for each condition 

صفحه 47:
Interpreting a chi square Calculate degrees of freedom # of events, trials, phenotypes -1 Example 2 phenotypes-1 =1 Generally use the column labeled 0.05 (which means there is a 95% chance that any difference between what you expected and what you observed is within accepted random chance. Any value calculated that is larger means you reject your null hypothesis and there is a difference between observed and expect values. 

صفحه 48:
How to use a chi Square 0.05 3.84 5.99 182 9.49 11.07 12.59 14.07 15.51 16.92 18.81 0.01 0.001 6.64 10.83 921 13.82 11.84 16.27 18.28 18.47 15.09 20.52 16.81 22.46 18.48 24,32 20.09 26.12 21.67 27.88 23.21 29.59 Significant 0.10 2.71 4.60 6.25 18 9.24 10.64 12.02 13.36 14.68 15.99 chart Probability 0.20 1.64 3.22 4.64 5.99 729 8.56 9.80 11.03 12.24 13.44 0.70 0.50 0.30 1.07 241 3.66 4.88 6.06 1,23 8.38 9.52 10,66 11.78 Nonsignificant “080 0.90 0.95 Degrees of Freedom اسان نله اجه زا جه "|| /نجاها. 

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‎Spare ۳‏ دا ‎Me a Expected ۳5‏ ‎Exanple 1 )۱۶-۵۵(< )28< :‏ ‎boo‏ ‎2, Bs BS, 225. = 8+ 4 ‘Goo G00 "40 ‎ ‎a 0.31540. S097 yas feedom = N-| = &suyo~