Class Note for STAT 528 at OSU 36
Class Note for STAT 528 at OSU 36
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Date Created: 02/06/15
Stat 528 Autumn 2008 Elly Kaizar Inference for the mean of a population One sample t procedures Reading Section 71 o Inference for the mean of a population 0 The t distribution for a normal population 0 Small sample Cl for u in a normal population 0 Robustness of the t procedures 0 Testing hypotheses about a single mean the one sample t test 0 Methods for matched pairs The paired t test The sign test for matched pairs 0 The power of the one sample t test Inference for the mean of a population 0 So far we have based inference for population mean on the Z statistic Y Z a For large n and independent observations Z is approximately NO1 0 Problem in practice we do not know the population stan dard deviation 0 Instead we use the sample standard deviation 5 as an estimate for a Why is this a good idea The distribution of t for a normal population 0 Let X1 X2 Xn be a SR8 from a normal population with population mean M Then the standardized variable X u 7 sW has a t distribution with n 1 degrees of freedom df t 0 We also say t has a tml distribution 0 The quantity s is the estimated standard error for the sample mean It is denoted SE mean in MlNlTAB Properties of the t distribution 04 standard normal xx r twith5df la r I tWIth2df I a twith1df 39 0 III g I w 5 39 39D I t r ll 39 39 E II 39 E 39 39 g 39 01 00 value c The density curve is symmetric with mean zero and is bell shaped like the normal distribution 0 The t distribution has heavier tails than the normal dis tribution more spread out about zero 0 As the degrees of freedom increase the tails become thinner and more of the density is concentrated in the center of the distribution too standard normal distribution A small sample twosided CI for u The normal population case o For one random sample of normal data a C 1001 00 level two sided con dence interval for u is given by 7 5 37 3 tn717a2 x 7 Where tmmg is the critical value of the t distribution with n 1 degrees of freedom 0 The twLag value is tabulated in Table D 1 Look at the bottom of the table for the con dence level C of the two sided interval OR 2 Look up a2 as the upper tail probability p Robustness of the tdistribution o What about if the population is not normal 7 can we still use the t distribution 0 Practical guidelines from the textbook 1 n lt 15 Use t procedures if data are close to normal lf data are clearly non normal or if outliers are present do not use the t procedure 2 n 2 15 Use t procedures except in presence of strong skewness or outliers 3 Roughly 71 Z 40 t The t procedures are valid even for clearly skewed distributions 0 These guidelines relate to the CLT 0 Use plots of the data to help you decide Polymerization example The article Measuring and understanding the aging of craft in sulating paper in power transformers77 contained the following observations on the degree of polymerization for paper specimens for which viscosity times concentration fell in a certain middle range 418 421 421 422 425 427 431 434 437 439 446 447 448 453 454 463 465 Plots of the data showed that a normality approximation for the data is reasonable Note that in 43829 5 1514 71 17 a Calculate a 95 con dence interval for the true average de gree of polymerization as did the authors of the article b Does the interval suggest that 440 is a plausible value for the true average degree of polymerization What about 450 Testing hypotheses about a single mean The one sample t test 0 Data We assume 131332 33 is a random sample from a normal population with mean u 0 We state our hypotheses H0 1 0 for some constant value 0 Ha ultuouuoORugtuo remember to de ne what u is in words for your problem 0 We calculate the test statistic t i 5C H0 i Sx 39 0 Under H 0 the test statistic follows a tml distribution o Pvalue Using Table D calculate the approximate area under the t distribution curve The area is based on the form of Ha as shown on the next page Calculating the Pvalue of the one sample ttest 1 For Ha I lt 0 the P value is PltT g t 01 02 03 01 02 03 a a N O p r N I I I I I I I I I 3 2 t 1 0 1 2 2 For Ha I gt 0 the P value is PltT Z t 01 02 03 01 02 03 I a I N p I O N a 3 For Ha I y 0 the P value is 2PltT Z 01 02 03 I I I I I I I l I It2 3 t1 o 3 2 I I Making conclusions in the onesample ttest o For a test of signi cance level oz If the P Value 3 oz we reject H0 Otherwise if the P Value gt oz we do not reject H 0 0 Now write down your conclusionltsgt in words 0 It is important to think about the assumptions that you made to carry out the t test Remember some assumptions can be validated using plots of the data 10 Example Similar to Exercise 721 The onesample t statistic from a sample of n 5O observations for the two sided test of H0 21 50 versus Ha 21 y 50 has the value t 165 a What are the degrees of freedom for the test statistic t b Locate the two critical values t from Table D that bracket t What are the right tail probabilities for these two values c How would you report the P value for this test d ls the value t 165 statistically signi cant at the 10 level At the 5 level 11 Matched pairs revision and analysis 0 Suppose we have two treatments 0 ln the matched pairs design we try to gain precision in the response by matching pairs of similar individuals 0 Either we assign each treatment randomly to each subject each subject only receives one treatment or both individuals receive both treatments but in a random order 0 Each pair of subjects form their own block 0 To analyze the results of this type of experiment we com pare the responses across the pairs We usually take differences and carry out the statistical inference using the paired ttest 12 Football example Two identical footballs one air lled and one helium lled were used outdoors on a windless day at The Ohio State University7s athletic complex The kicker was a novice punter and was not informed which football contained the helium Each football was kicked 39 times The kicker changed footballs after each kick so that his leg would play no favorites if he tired or improved with practice Source Lafferty M B 1993 7 OSU scientists get a kick out of sports controversy 7 The Columbus Dispatch 21 Nov 1993 B7 13 The data all distances are in yards Trial Air Helium Trial Air Helium Trial Air Helium 1 25 25 14 25 31 27 22 30 2 23 16 15 34 22 28 31 27 3 18 25 16 26 29 29 25 33 4 16 14 17 20 23 30 20 11 5 35 23 18 22 26 31 27 26 6 15 29 19 33 35 32 26 32 7 26 25 20 29 24 33 28 30 8 24 26 21 31 31 34 32 29 9 24 22 22 27 34 35 28 30 10 28 26 23 22 39 36 25 29 11 25 12 24 29 32 37 31 29 12 19 28 25 28 14 38 28 30 13 27 28 26 29 28 39 28 26 14 A scatterplot Scallerplot of Air vs Helium Air Helium 15 The paired t procedure the setup 0 Suppose we have pairs of data values 3317111 27112 39 39 39 717 eg ln our example the pairs of values are the heliumfilled airfilled distances for each kick 0 Clearly the 13 and y values are not independent 0 lnstead we calculate the differences dz39 yz39 55239 foreachz391n 0 We assume d1 d2 dn is a random sample from a normal population with mean Md and stdev 0d Md is the population mean of the differences between the 13 and y values ad is the population stdev of the differences 16 The paired t procedure c We want to test H0 id 0 for some constant value no Ha Md lt 0 Md 0 OR Md gt 0 0 We calculate the test statistic t Sdx where cl is the sample average of the differences and 5d is the sample stdev of the differences 0 Under H 0 the test statistic follows a tml distribution 0 We calculate the Pvalue in the same way as the one sample t test For a test of signi cance level oz if the P Value 3 oz we reject H0 otherwise if the P Value gt oz we do not reject H 0 17 Example Question There is a belief that on average a helium lled ball travels further than the air lled ball Does the OSU data support this belief Computational Hint We use the MlNlTAB command Calc gt Calculator to calcu late the difference in the distances Air Helium 18 Inference for nonnormal populations o If the data do not seem to be drawn from a normal popula tion then the t procedures may not be valid 0 Three possible strategies 1 Learn about other probability distributions For exam ple there plenty of skewed distributions eg exponential gamma Weibull Use methods for these distributions in stead of the methods for the normal distribution 2 Transform your data to make it look as normal as pos sible77 common transformations include log and square root Can be hard although not impossible to inter pret the results When using a transformation 3 Use distributionfree tests These tests do not assume a particular distribution for the population Often these test are based on other parameters of the distribution such as the median rather than the mean These tests can be less powerful in practice 19 The sign test for matched pairs 0 Example of a distributionfree test 0 As before consider pairs of data values 5517111 3327112 557171171 0 We Will test H0 population median of differences 0 versus Ha population median of differences y O 0 Let dZ39 yZ39 33 1 n be the differences 0 Excluding the differences that are zero let X denote the count out of the remaining m differences that are positive 0 Then under H0 X is Binomialm05 lf the median is zero then half the nonzero differences are above zero and the other half are below zero 0 lf 1 is the observed X value then the P value is 2 X PX g as long as this is g 1 20 The sign test for matched pairs cont o For the football example Out of n 39 differences m 37 differences are nonzero Thus under H0 X is Binomiallt37 05 Out of the 37 we observe 17 that are above zero P value 2 X PX g 17 2 X 03714 07428 No evidence to reject H0 0 See the textbook for the one sided test 0 Note lf the population is normally distributed then this test will be less powerful at detecting differences as compared with the paired t test 21 The power of the one sample ttest o The power calculation for the one sample t test is simi lar to the power calculation for the Z test 0 But the math is much harder lnstead we use MlNlTAB o Stat gt Power and Sample Size gt 1Sample t 0 Under Options select the Alternative Hypothesis and Signi cance Level 0 Then enter any two of the following three items 1 Sample sizes 2 Differences 3 Power values 0 Enter the Standard deviation the sample stdev in this case and click OK 22 An agricultural eld trial example An agricultural eld trial compares the yield of two varieties of tomatoes for commercial use The researchers divide in half each of 10 small plots of land and plants each tomato variety on one half of each plot After harvest they compare the yields in pounds per plant at each location The ten differences Variety A Variety B give the following statistics 3 046 and 5 092 ls there convincing evidence that Variety A has the higher mean yield Let Md denote the population mean of the difference in the yields We test H0 Md 0 versus Ha id gt O The MlNlTAB output for the paired t test is One Sample T Test of mu 0 vs gt 0 95 Lower N Mean StDev SE Mean Bound T P 10 0460000 0920000 0290930 0073307 158 0074 Conclusion 23 Agricultural trial cont The tomato experts who carried out the eld trial suspect that the relative lack of signi cance is due to low power They would like to detect a mean difference in yields of 06 pounds per plant at the 005 signi cance level Based on the previous study use 092 as an estimate of both the population a and the value of 5 in future samples 0 What is the power of the test with n 12 against the alter native of u 06 o If the sample size is increased to n 30 plots of land what will be the power against the same alternative 0 What sample size is needed to achieve 80 power 24
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