1、1,1.Sampling that means when the width increases, precision decreases. Therefore, the increase of confidence level is at the expense of the decrease of precision. When 1- remains unchanged, if n increases，then the width decreases and the precision increases. However, if n is too large, wasting will

2、be caused and sampling becomes meaningless. Therefore, the precision should be selected carefully.,Parameter Estimation,29,Determining the sample size = Z/2 is called permissible error is called standard error 標(biāo)準(zhǔn)誤 we can get the confidence interval with Excel (see a example on p154),Parameter Estima

3、tion,30,Interval estimation for population proportion (p141) If np5, nq5, then p follows N(p, p(1-p)/n) normal distribution confidence interval estimate of a population proportion is (6.30) p142,Parameter Estimation,31,This is another issue of statistical inference, focusing on getting the conclusio

4、n of “Yes” or “No”. Background： After improving technology, does the average product size change significantly? After improving technology, whether the production is stable or not? Is the qualified rate up to the standard? Does the life of the product follow the normal distribution? Etc.,Chapter 7 H

5、ypothesis Tests (p156),32,When considering the above questions, we can assume that the hypothesis is tenable, then according to sample, judge whether the hypothesis is right or not. If it is right, then accept the hypothesis; if not, then reject the hypothesis. These are the content Hypothesis Tests

6、 covers. Generally, the hypothesis to be tested is called the Null Hypothesis (H0), the opposite hypothesis is called the Alternative Hypothesis (H1).,Hypothesis Tests,33,Hypothesis Tests Process,Population,Suppose,The average age of population is 50.(H0),Reject hypothesis,Sample mean is 20,Sample,3

7、4,Its theoretical base is the principle of small probability: In one experiment, the event with small probability hardly happens. Example：H0：= 0=200mm, H1: 0=200mm It is known that the population X follows N（, 2)， if H0 is tenable，then we get i.e. it appears with a large probability. The opposite ev

8、ent appears with a small probability. After sampling, compute:,The Idea of Hypothesis Tests,35,If ，then there is no contradiction. If the opposite appears，then it is proved that the event with small probability happens in one experiment, which contradicts with the principle of small probability and

9、proves that H0 is wrong.,Hypothesis Tests,36,Steps for Hypothesis Tests,37,If the statistic is larger than the critical value, reject H0. If the statistic is smaller than the critical value, accept H0. If the statistic equals to the critical value, then enlarge the sample size, and make a retesting.

10、,Hypothesis Tests,38,Testing hypotheses concerning the population mean When is known, use Z statistic, whenis unknown, use t statistic. As for a large sample, whatever distribution it is, Z statistic can be used as approximation. Testing a hypothesized value of the population proportion Testing the

11、difference between two means The hypothesis test about the population variance.,Content of The Hypothesis Tests,39,Is there a significant difference between the net asset income rate of 1993、1994 of commercial corporations listed in Shanghai Stock Exchange ? Solution: Suppose the income rate of the

12、two years is X1,X2, and follows the normal distribution N(1,12), N(2,22) respectively. Method 1: = 1-2 Method 2: Use the formula at page 170。,An Example About Hypothesis Tests,40,When H0 is true, H0 may be rejected (caused by stochastic factors),we call this kind of error Rejecting Truth Error. From

13、 the prior formula we can know, this kind of probability is , it is also called Type Error or Supplier Risk. When H0is false, H0 may be accepted (caused by stochastic factors),we call this kind of error Accepting Falseness Error. Its probability is , it is also called TypeError or User Risk. In gene

14、ral, we controlin most time，our lecture does not cover the computing aspect of . (If the sample size isnt enlarged, the two types of risks can not be reduced simultaneously ),Two Types of Errors,41, if not ,accept H0,46,p Value Test,The p-value, the observed level of significance, is a measure of th

15、e likelihood of the sample results when the null hypothesis is assumed to be true. The smaller the p-value, the less likely it is that the sample results came from a situation where the null hypothesis is true. If p ,do not reject H0 If p , reject H0.,47,One-Tail Z Test about Mean ( Known),We assume

16、 the population follows a normal distribution, when n 30, a non-normal distribution can be approximated by a normal distribution. 2.Null hypothesis only uses or Test Statistic Z,48,Rejection Region,Z,0,reject H,0,Z,0,reject H,0,H0:0 H1: 0,H0:0 H1: 0,Only when statistic is significantly less than tha

17、t it will be rejected.,Smaller value does not contradict with H0, therefore, H0 will not be rejected.,49,One-Tail Z Test: Finding Critical Z Values,Z,.05,.07,1.6,.4505,.4515,.4525,1.7,.4599,.4608,.4616,1.8,.4678,.4686,.4693,.4744,.4756,Z,0,Z,= 1,1.96,.500 -.025.475,.06,1.9,.4750,Table of Standard No

18、rmal Distribution (Part),When = 0.025，compute Z?, = .025,50,Test about p-value,P(Z 1.50) = 0.0668,Z,0,1.50,p-value=,.0668,Z value of sample statistic,In Z table, find: 1.50,.4332,.5000-.4332 .0668,Determine the direction of test by using the alternative hypothesis.,51,Test about p-value,0,1.50,Z,rej

19、ect H0,(p = 0.0668) ( = 0.05). Do not reject H0, = 0.05,Test statistic is not within the rejection region.,52,Two-Tailed Z Test for Mean ( Known),Assume a population follows the normal distribution, when n 30，the population of a non-normal distribution can be approximated by a normal distribution. T

20、hy null hypothesis is an equality. Test Statistic Z,53,Rejection Regions,H0,critical value,critical value,1/2,1/2,sample statistic,rejection region,non-rejection region,sampling distribution,1 - ,confidence level,rejection region,54,Two-Tailed Test: Finding Critical Z values,Z,.05,.07,1.6,.4505,.451

21、5,.4525,1.7,.4599,.4608,.4616,1.8,.4678,.4686,.4693,.4744,.4756,Z,0,Z,= 1,1.96,-1.96,.500 -.025.475,.06,1.9,.4750,When = 0.05，compute Z?, /2 = .025, /2 = .025,55,Test about p-value,P(Z -1.50 or Z 1.50) = 0.1336,Find the probability of Z1.50,.5000-.4332 .0668,multiply 2,56,Test about p-value,(p = .13

22、36) ( = .05) do not reject the null hypothesis,57,A manager of a hotel claims that the mean of the bills at the end of every week is smaller or equal to 400 yuan. However, an accountant of this hotel finds that the total income is increasing in the recent month. This accountant will use a sample con

23、sisted of the bills of recent weekends to testify the managers claim. a.What kind of hypothesis should be used? H0: 400, H0: 400, H0:=400 b.In this example, whats the meaning of rejecting H0?,Exercises for Hypothesis tests,58,A new weight-reducing method claims that the participant will averagely lo

24、se at least 8 kg in the first week. 40 participants consist of a random sample, the sample mean of the reduced weight is 7 kg, the standard deviation is 3.2 kg. Compute: a. When a=0.05, what is the rejection criterion? b. What is your conclusion about this weight-reducing method？,Exercises for Hypot

25、hesis tests,FITTNESS CLUB,Welcome,59,Relationship among variables Function Correlation (statistical relationship) Y depends on X, but isnt merely determined by X. Example: pricedemand for product temperaturedemand for air-conditioning RegressionAccording to observant data,establish regression model

26、and make statistical reference on variables having statistical relationship.,Chapter 10 Regression,60,What does regression do?,Solve the following problems: Determine whether there has statistical relationship among variables, if has, show the formula. Forecast the value of another variable accordin

27、g to one variable or a group of variables.,61,Linear Regression Assumptions,Normality Every value of X , Y follows the normal distribution The error probability follows the normal distribution Homoscedasticity (Constant Variance) Independence of Errors Linearity,62,Example: X-price，Y-demand for the

28、product We have data: 1. Scatter plot 2. Regression equation(Ordinary Least Square Estimation) 3. Correlation coefficient r Testing the regression model 4.Forecasting 5.Regression can be linearitied,Simple Linear Regression,63,Linear Regression Model,Variables consist of a linear function.,Y,X,i,i,i

29、,0,1,Slope,Y-Intercept,Independent (Explanatory) Variable,Dependent (Response) Variable,Random Error,64,Population Linear Regression Model,i,= random error,X,YX,i,X,0,1,Y,X,i,i,i,0,1,Observed Value,Observed Value,Y,65,Sample Linear Regression Model,e,i,= random error,Y,X,Y,b,b,X,e,i,i,i,0,1,Y,b,b,X,

30、i,i,0,1,Unsampled Observed Value,Sampled Observed Value,66,Ordinary Least Squares,The least squares method provides an estimated regression equation that minimizes the sum of squared deviations between the observed values of the dependent variable yi and the estimated values of the dependent variabl

31、e .,e,2,Y,X,e,1,e,3,e,4,Y,b,b,X,e,i,i,i,0,1,Y,b,b,X,i,i,0,1,OLS Min,e,e,e,e,e,i,i,2,1,1,2,2,2,3,2,4,2,Predicted Value,67,Coefficient & Equations,Y,b,X,b,X,Y,n,X,Y,X,n,X,b,Y,b,X,i,i,i,i,i,n,i,i,n,0,1,1,1,2,2,1,0,1,Sample regression equation,Slope for the estimated regression equation,Intercept for th

32、e estimated regression equation,b,68,Evaluating the Model,Test Coefficient of Determination and Standard Deviation of Estimation Residual Analysis Test Coefficients of Significance,69,Measures of Variation in Regression,1. Total Sum of Squares (SST) Measure the variation between the observed value Y

33、i and the mean Y. 2. Explained Variation (SSR) Variation caused by the relationship between X and Y. 3. Unexplained Variation (SSE) Variation caused by other factors.,70,Variation Measures,Y,X,Y,X,i,SST (Yi - Y)2,SSE (Yi -Yi)2,SSR (Yi - Y)2,Yi,Y,b,b,X,i,i,0,1,71,Coefficient of Determination,0 r2 1,r

34、,b,Y,b,X,Y,n,Y,Y,n,Y,i,i,i,i,n,i,n,i,i,n,2,0,1,2,1,1,2,1,2,Explained variation,Total variation,SSR,SST,A measure of the goodness of fit of the estimated regression equation. It can be interpreted as the proportion of the variation in the dependent variable y that is explained by the estimated regres

35、sion equation.,72,Correlation Coefficient,A numerical measure of linear association between two variables that takes values between 1 and +1. Values near +1 indicate a strong positive linear relationship, values near 1 indicate a strong negative linear relationship, and values near zero indicate lac

36、k of a linear relationship.,73,Coefficients of Determination (r2) and Correlation (r),74,Test of Slope Coefficient for Significance,1. Tests a Linear Relationship Between X & Y 2.Hypotheses H0: 1 = 0 (No Linear Relationship) H1: 1 0 (Linear Relationship) 3.Test Statistic,75,Example Test of Slope Coe

37、fficient,H0: 1 = 0 H1: 1 0 .05 df 5 - 2 = 3 Critical value:,Statistic: Determine: Conclusion:,Reject at = 0.05,There is evidence of a relationship.,76,Multiple Regression Model,There exists linear relationship among an dependent variable and two or more than two independent variables.,Y,X,X,X,i,i,i,

38、P,Pi,i,0,1,1,2,2,slope of population,intercept of population Y,random error,Dependent Variable,Independent Variables,77,Example: New York Times,You work in the advertisement department of New York Times(NYT). You will find to what extent do ads size(square inch ) and publishing volume (thousand) inf

39、luence the response to ads(hundred).,You have collected the following data: response size volume 1124881313572644106,78,Example (NYT) Computer Output,Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Param=0 Prob|T| INTERCEP 1 0.0640 0.2599 0.246 0.8214 ADSIZE 1 0.2049 0.0588 3.656 0.0399 CIRC 1 0.2805 0.0686 4.089 0.0264,79,Interpretation of Coefficients,1.Slope (b1) If the publishing volume remains unchanged,when ad