ࡱ> tvsc[ $bjbj #ΐΐ=JJmmmAt!Rsssf$???????$OCE?m&f"&&?sA(((&H sm?(&?((=-@B?b?n'=?A0A=\Fn'F0B?FmB?T&&(&&&&&??(&&&A&&&&F&&&&&&&&&J S: Review of Basic Statistical Concepts The purpose of this review is to summarize the basic statistical concepts. Introductory statistics dealt with three main areas: descriptive statistics, probability, and inference. Descriptive StatisticsSample data may be summarized graphically or with summary statistics. Sample statistics include the mean, variance, standard deviation, and median. For the following definitions let x1, x2, , xn represent the values obtaining from a random sample of size n drawn from a population of interest. Sample Mean EMBED Equation.3 The mean is just the average of the n values observed. Sample Variance EMBED Equation.3 The sample variance equals the mean squared deviation from  EMBED Equation.3 . A small  EMBED Equation.3  means that the observed values cluster around the average, while a large variance means that they are more spread out. Thus, the variance is a measure of the spread in the sampled values. Sample Standard Deviation EMBED Equation.3 The sample standard deviation, s, is often a more useful measure of spread than the sample variance, s2, because s has the same units (inches, pounds, etc.) as the sampled values and  EMBED Equation.3 .  StatGraphicsCommon descriptive statistics can be obtained by following: Describe > Numeric Data > One-Variable Analysis > Tabular Options > Summary Statistics ExampleThe file LMF contains the three-year return for a random sample of 26 mutual funds. All of these funds involve a load (a type of sales charge). StatGraphics output is to the right. EMBED SGStatFolio  Random Variables and their Probability Distributions Random VariableA variable whose numerical value is determined by chance. The key elements here are that the variable assumes a number (sales volume, rate of return, test score, etc.) and that the sample selection process generates the numbers randomly, i.e., by a random selection. (In these notes, a random variable will be designated by a capital letter, such as X, to differentiate it from observed values x. For instance, X might represent the height of a man to be selected randomly. Once the man has been selected, his height is given by the value x, say x = 68 inches.) Probability DistributionAlthough the values of a random variable are subject to chance, some values are more likely to occur than others. For instance, the height of a randomly selected man is more likely to measure 6 than 7. It is the random variables probability distribution that determines the relative likelihood of possible values.Standardized Values For the value x drawn from a population with mean m and standard deviation s, the standardized value  EMBED Equation.3 . For example, if incomes have a mean and standard deviation of $48,000 and $16,000, respectively, then someone making $56,000 has a standardized income of  EMBED Equation.3  because their income is one-half standard deviation above the mean income. The advantage of standardizing is that it facilitates the comparison of values drawn from different populations. Standardized Random Variables For the random variable X with mean m and standard deviation s,  EMBED Equation.3  is the Standardized random variable. (Note: The Standardized Variable always has mean 0 and standard deviation 1.) The Normal Distribution In this course we will make use of (at least) four distributions designed to model continuous data: the Normal, t, F, and Chi-Square. Of these, the normal distribution is by far the most important because of its role in statistical inference. Much of the logic behind what we do and why we do it is based upon an understanding of the properties of the normal distribution, and of the theorems involving it, particularly the Central Limit Theorem. PropertiesNormal distributions are bell-shaped. (In fact, it is sometimes called the Bell Curve.) Normal distributions are symmetric about their mean. Normal distributions follow the 68-95-99.7 rule: (Approximately) 68% of the area under the curve is within one standard deviation s of the mean m (Approximately) 95% of the area under the curve is within two standard deviations s of the mean m (Approximately) 99.7% of the area under the curve is within three standard deviations s of the mean m If the random variable X is normal with mean m and standard deviation s, then the random variable  EMBED Equation.3  is standard normal, i.e., is normal with mean equal 0 and standard deviation equal 1.  EMBED SGStatFolio  Finding probabilities in ExcelCumulative Probabilities for any normal random variable X, i.e., P(X  EMBED Equation.DSMT4  x), are easy to find in Excel. Follow: fx > Statistical > NORMDIST and enter TRUE in the Cumulative field. Probabilities of the form P(X > x) or P(a < X < b) can be obtained by subtraction. ExampleTo find P(-1.2 < Z < 2), note that P(-1.2 < Z < 2) = P(Z < 2) P(Z  EMBED Equation.DSMT4  -1.2) and use the Excel output to the right. Answer = 0.9772  0.1151 = 0.8621  Critical Valuesz a is defined by P(Z > za) = a. Critical values are used in the construction of confidence intervals and (optionally) in hypotheses testing. To find the critical value associated with the significance level a, follow: fx > Statistical > NORMINV and enter 1 - a in the Probability field. ExampleFrom the Excel output to the right we see that z0.05 = 1.645 The Distribution of the Sample Mean Because, when we take a random sample, the values of a random variable are determined by chance, statistics such as the sample mean that are calculated from the values are themselves random variables. Thus the random variable  EMBED Equation.3  has a probability distribution of its own. If we intend to use the sample mean  EMBED Equation.3  to estimate the mean m of the population from which the sample was drawn, then we need to know what values the random variable  EMBED Equation.3  can assume and with what probability, i.e., we need to know the probability distribution of  EMBED Equation.3 . It can be shown (using advanced calculus) that  EMBED Equation.3  has the following properties: The mean of  EMBED Equation.3  equals the mean of X, i.e.,  EMBED Equation.3 . This just says that the sample mean  EMBED Equation.3  is an unbiased estimator of the population mean m. The variance of  EMBED Equation.3  is less than that of X. In fact  EMBED Equation.3 . This states that there is less variability in averaged values (and the variability decreases as the size of the sample increases) than there is in individual values. Hence, you might not be surprised if a randomly selected man measured 7, but you would be suspicious if someone claimed that 100 randomly chosen men averaged 7! If the variable X is normally distributed, then  EMBED Equation.3  will also be normal. The properties above, however, dont describe the shape of the distribution of  EMBED Equation.3  (needed for making inferences about m) except in the special case where X is normal! They only contribute information about the mean and spread of the distribution. In general, the shape of the distribution of  EMBED Equation.3  may be difficult to determine for non-normal populations and small samples. However: For large samples the Central Limit Theorem states that  EMBED Equation.3  will be at least approximately normal. (Most introductory statistics texts consider a sample large whenever n > 30.) ExampleThe dean of a business school claims that the average weekly income of graduates of his school 1 year after graduation is $600, with a standard deviation of $100. Find the probability that a random sample of 36 graduates averages less than $570. Solution: Let X = weekly income of a sampled graduate 1 year after graduation. We are asked to find  EMBED Equation.3  for 36 graduates.  EMBED Equation.3  Note: Without the Central Limit Theorem we could not have approximated the probability that a sample of graduates average less than $570 because the distribution of incomes is not usually normal.  Statistical Inference: Estimation Point EstimateA single number used to estimate a parameter. For example, the sample mean  EMBED Equation.3  is typically used to estimate the population mean m.  Interval EstimateA range of values used as an estimate of a population parameter. The width of the interval provides a sense of the accuracy of the point estimate.  Confidence Interval Estimates for m Confidence intervals for m have a characteristic format:  EMBED Equation.3 *standard error, where CV stands for Critical Value and the standard error is the (usually estimated) standard deviation of  EMBED Equation.3 . Case I: X normal or n >30, and s is known A (1 - a)*100% confidence interval estimate for m is given by  EMBED Equation.3 Case II: n  EMBED Equation.DSMT4  30 and s is unknownA (1 - a)*100% confidence interval estimate for m is given by  EMBED Equation.3 , with n-1 degrees of freedomCase III: X is normal and s is unknownA (1 - a)*100% confidence interval estimate for m is given by  EMBED Equation.3 , with n-1 degrees of freedom Case III requires some explanation. When X is normal, and we must use the sample standard deviation s to estimate the unknown population standard deviation s, the studentized statistic  EMBED Equation.3  has a t distribution with n-1 degrees of freedom. Hence, we must use the critical value t a/2 from the t distribution with n-1 degrees of freedom. The properties of the t distributions are similar to those for the standard normal distribution Z, except that the t has a larger spread to reflect the added uncertainty involved in estimating s by s. Note: For large samples, where n  EMBED Equation.DSMT4  30, there is very little difference between the t distribution with n-1 degrees of freedom and the standard normal distribution Z. Therefore, for large samples (Case II in the table above) some texts replace t a/2 with z a/2 even when X is normal and s is unknown!  ExampleA manufacturer wants to estimate the average life of an expensive component. Because the components are destroyed in the process, only 5 components are tested. The lifetimes (in hours) of the 5 randomly selected components are 92, 110, 115, 103, and 98. Assuming that component lifetimes are normal, construct a 95% confidence interval estimate of the components life expectancy.Solution: Using Excel,  EMBED Equation.3  hours, and  EMBED Equation.3  hours. From the discussion above, the critical value is t 0.025%V Z \ x  ( ) < = > ? d e   ~j> hUVjhCJEHUj> hUVjhCJEHUj> hUVh6CJ]jhCJEHUj> hUVjhCJU hCJH* h6CJ h5CJ hCJh,%&   ( @ w nkd$$Ifl0|n(+&0+64 la$If$If w x y ~|vvv$IfkdT$$IflF|n(F 0+6    4 la       & ' l m n x y # + . ; = S U d y | 9:NOPQTù㮢Õ㕉ぞvnjhUj> hUVjhUh56CJ\]h6CJ]hj hCJEHUj> hUVh5CJH*\h5CJ\j hCJEHUjh> hUV hCJjhCJUjvhCJEHU+  ~|vvv$IfkdR $$IflF|@ n(F~ .0+6    4 la z ~|vp$If$Ifkd$$IflF|F n(F (0+6    4 laz { | 9R$If$IfnkdJ$$Ifl0|bn(B&0+64 laRSTU~||zuooi` $Ifgdt$If$If$a$kd$$IflF|n(FVV0+6    4 la   '()9;Rz 8FHnprtVWjklm+JҳҳҩҞҩ҇{jhCJEHUj_ @ hUVjhCJEHUj @ hUVjhCJUh5CJOJQJ\h6CJ]h5CJ\ hCJ h6]htht6hth56\] h5\h0P$Ifnkd$$Ifl0|n(F&0+64 laPRz|*+IJNP$If$a$nkdO$$Ifl0|n(a&0+64 laJbc248:`bdfvLNj<>XZFHxzjohCJEHUjh> hUVh6CJ]h h+ CJh5CJ\jWhCJEHUj @ hUVjhCJUh5CJOJQJ\h56CJ\] hCJ78\ F_`awul $Ifgd hCJUVjhCJU hCJh5CJ\(p r t v $If$IfnkdU-$$Ifl0|n(R&0+64 la     !!!*!,!2!4!6!>!@!!!!""""""" ###4#H#͎ͥͅwmmͅcmͅh6CJH*]hCJOJQJh6CJH*OJQJ]h6CJ]h56CJ\]hCJH*OJQJ hCJH*jMhCJUj0hCJUhh5CJ\ hCJj-hQhQCJEHUjyV hQCJUV hQCJjhQCJU& !D#~|vv$Ifkd%h$$IflF| n(e z0+6    4 laD#F#H#X####$If$Ifnkdh$$Ifl0|n(8'0+64 laH#X#######$$$$%%%%a%b%u%v%w%x%&&&&&&''''''''((&('(vjd hCJjhCJEHUjhCJEHUj> hUVh5CJOJQJ\jHhhCJEHUjJzV hUVjhCJEHUjl@ hUVjhCJU h>*CJjoihCJU hCJH*h6CJ] hCJh'########$$\(](*~||||||zu||p & F$a$kd0$$IflF| n(8@ z0+6    4 la '(:(;(<(=(i(j(}(~(((((((((((((((((( )**&*(*N*P*R*T*******}qfjQ> hUVj`hCJEHUh5CJOJQJ\h5CJ\jhCJEHUj> hUVj~hCJEHUje> hUVh56CJ\]jhCJEHUjhCJUj΍hCJEHUj> hUV hCJ(***n++++,,,,,,,,,,,,,,=-B-Z-[-n-o-p-q-...H.J.^.\/^///// 0%020G0T0U0h0i0j0÷ëÒ}jhCJEHUh5CJ\j%hCJEHUh5CJOJQJ\jMhCJEHUjuhCJEHUj> hUVh56CJ\]h6CJ] hCJjhCJUj8hCJEHU1*, - -0000012222q3$If$If & F & Fj0k000011h2i2|2}2~2222222222t3333333 4 444v4x4z4|4556︬xrk h>*CJ hCJh5CJOJQJ\j<hCJEHUj> hUV h>*CJh5CJ\jhCJEHUj > hUVjՠhCJEHUj > hUVh56CJ\] h>*h hCJjhCJU&q3r3s3t3333~4~||zuoo$If$a$kd$$IflF| ( 0+6    4 la~44445$Ifnkd$$Ifl0|('0+64 la5556677<8>8829$If$a$nkd$$Ifl0|('0+64 la 666N6P6666666667777777788&8(8L8N88899*9,9.909F9H9J9L9Ź祙xlbjhknCJUjhknCJEHUjB|V hknUVhCJOJQJh56CJ\]jFhCJEHUj> hUVh6CJ]jEhCJEHUj> hUVjhCJUh5CJOJQJ\ hCJ h>*CJh>*CJOJQJ&29499&::$If $Ifgdknnkdm$$Ifl0| (:^0+64 laL9z9|9~999999 : :l:n:::::::: ;;4;6;;;;;;;<<<<X<h<<< ="=≠̸̭̊̕~t̕̕hknh5CJj̺hknCJEHUjZ|V hknUVh56CJ\]jhknCJEHUj)|V hknUVjhCJUhCJOJQJ hCJjhknCJUjhknhknCJEHUj8}V hknCJUV* hknCJ(::&;;R<$Ifnkd2$$Ifl0| (:^0+64 laR<T<V<@@NCPCTCVCXChC1Eƍȍ$If$Ifnkd$$Ifl0| (:^0+64 la==========>>>>>>>>>R?T????? @@@@@@@@@@AA0A2A4A6AAA8Bퟕ틟~ojhknhknCJEHUjL}V hknCJUVjhknCJUhknhkn5CJ hknCJhCJH*OJQJ hCJH*h56CJ\]jhCJEHUj> hUVjhCJUh5CJ\ hCJh5CJOJQJ\+8B:BvBzBBBBBBBBBBBBBBBBBBBBBCCC2C6CPCRCXCfChCHEIE\E]EǼǩ|wmbj> hUVjhCJU h>*hjhCJUh5CJOJQJ\h6CJ]hkn5CJ\h5CJ\h5CJH*\hCJH*OJQJ hCJH*hknhknCJ hkn5CJhknhkn5CJ hknCJ hCJh56CJ\]%]E^E_EkElEEEEEEEE֌،"$:<>h~čfgz{|}ÎĎ%&c㤚yuuoo h)CJhjhCJEHUj%> hUVj2hCJUh6CJH*]h6CJ]hCJOJQJU hCJH*h56CJ\]j5hCJEHUj> hUV hCJjhCJUjhCJEHU+ = 2.776. (Note: In Excel, shown below, to find the critical value associated with the t distribution and significance level a, follow: fx > Statistical > TINV and enter a in the Probability field.)  Thus a 95% CIE for the mean lifetime of the components is given by  EMBED Equation.3  or (92.2, 115.0) hours  Statistical Inference: Decision Making In hypothesis testing we are asked to evaluate a claim about something, such as a claim about a population mean. For instance, in a previous example a Business dean claimed that the average weekly income of graduates of his school one year after graduation is $600. Suppose that you suspect the deans claim may be exaggerated. Hypothesis testing provides a systematic framework, grounded in probability, for evaluating the deans claim against your suspicions. Although hypothesis testing uses probability distributions to arrive at a reasonable (and defensible) decision either to reject or "fail to reject" the claim associated with the null hypothesis of the test, H0, it does not guarantee that the decision is correct! The table below outlines the possible outcomes of a hypothesis test. (Note: We avoid "accepting" the null hypothesis for the same reason juries return verdicts of "not guilty" rather than of "innocent") Decision: TRUTHAccept H0 Reject H0H0 Truecorrect decisionType I errorH0 FalseType II errorcorrect decision Type I errorThe error of incorrectly rejecting H0 when, in fact, it's true. In a hypothesis test conducted at the significance level a, the probability of making a type I error, if H0 is true, is at most a. Type II errorThe error of incorrectly failing to reject H0 when, in fact, it's false. For a fixed sample size n, you cannot simultaneously reduce the probability of making a Type I error and the probability of making a Type II error. (This is the statisticians version of there is no such thing as a free lunch.) However, if you can afford to take a larger sample, it is possible to reduce both probabilities. Decision Making: Hypothesis Testing ExampleSuppose that a sample of 36 graduates of the business school averaged $570 per week one year after graduation. Test the deans claim, against your suspicion, at the 5% level of significance. Solution: 1. H0: m = $600 (the dean s claim) HA: m < $600 (your suspicion) 2. a = 0.05 (the probability of rejecting the dean s claim if she s right) 3. Draw some pictures (see box to the right) 4. Critical Value: -z 0.05 = -1.645 5. From the sample - Standardized Test Statistic:  EMBED Equation.3  6. Conclusion: There is sufficient evidence to reject the deans claim at the 5% level of significance. EMBED Word.Picture.8  the P-value Approach to Hypothesis Testing P-value The smallest significance level at which you would reject H0. The p-value is calculated from the test statistic, and is doubled for two-sided tests. Note: a and the p-value are the  before and  after significance levels for the test. We can reach a decision to accept or reject H0 by comparing the two significance levels. Rule: If the p-value > a, then we "fail to reject" H0 If the p-value  EMBED Equation.DSMT4  a, then we reject H0, i.e., we reject H0 for small p-values ExampleSuppose that a sample of 36 graduates of the business school averaged $570 per week one year after graduation. Use the p-value to test the dean s claim, against yoȍRÎĎefxvvvtomvvv$a$kdL$$IflF"P (.X l0+6    4 la$If cdnqޑdf͒ΒԒܒݒlmnVXbdt BvxDFJLբ h>*hjhCJOJQJhCJOJQJ heZCJ hjhCJ hjhCJH* h5\hh)h)5CJ h)CJh6CJ] hCJ hCJH*?@͚ΚϚКњԚDE$&筣~~t~ej%V hCJUV*jhCJU hCJhCJOJQJ h>*CJjhUmHnHujhCJUjP> hUVj hCJUj9hCJEHUj?> hUVjhCJUh hCJ hCJH*h56CJ\]%ҚӚԚ TVv $Ifgd$a$skd$$Ifl0ZV(0+684 la$If &(*,.RT|~\ּ׼"'½চ}thh56CJ\]h5CJ\ h9~CJjhCJEHUjU> hUVjhCJEHUj?> hUVjhCJUU h>*hh6CJ] hCJH*hCJOJQJ hCJjhCJUjhhCJEHU(/$If$Ifnkd[$$Ifl0"Z(8'0+64 laur suspicion, at the 5% level of significance.Solution: Steps 1-3 are the same as before. 4. Critical Values are not used in this approach. 5. From the sample - Standardized Test Statistic:  EMBED Equation.3  p-value =  EMBED Equation.3  = 0.0359 < 0.05 = a, where we have used the fact that the test is left-tailed! 6. Conclusion: There is sufficient evidence to reject the dean s claim at the 5% level of significance.Notice that we rejected the Deans claim under both the critical value and p-value approaches. This was not a coincidence: the two approaches always lead to the same decision. Since p-values are routinely computed by StatGraphics and Excel, we will usually use p-values to conduct significance tests. Note:/0:\ļb$If$Ifskd$$Ifl0"Z(8'0+64 la !" !"#$`kd+$$IflZ('0+684 la Many of the (hypothesis) tests conducted in this course are two-sided, and assume that we are sampling from a normal population with unknown variance. When this is the case, Statgraphics will automatically return the correct p-value for the two-sided t test.  EMBED SGStatFolio  #$jhUjI> hUVjhUh hCJ (/ =!"8#$% `!:Ku=;Pr>@xڍ hWֶ3nV]J0fڤK]:J)fSz_/IHF>$oj,C1p-"" 0uՔ1f14.a&3`|=矏9 o~<9՘ʌIJ$tKL}@-ݦDKq.Ԭb2z]j>֦a~ASxx2CfBոU]%x3ߏ+#Sk^ɮDAkz)7#^]u*0iI8ɒgeהyNQ}^yy/)/プCes,{K#G$K#T*gi<8KAA^P. 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