3.4 Statistical inference for 2 populations based on two samples


 Rudolf Hood
 2 years ago
 Views:
Transcription
1 3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted as Y 1, Y 2,..., Y n. The population means of the populations from which these samples are taken are denoted by µ X and µ Y, respectively. 1 / 63
2 Statistical inference for 2 populations based on two samples We consider 2 cases 1. Dependent samples  in this case we have n pairs of observations (X 1, Y 1 ),..., (X n, Y n ). 2. Two unrelated samples (X 1,..., X m ) and (Y 1,..., Y n ). 2 / 63
3 3.4.1 Tests for the difference between two population means  dependent samples Each pair of observations comes from either one individual under two different conditions e.g. the weights of a group before (X ) and after (Y ) a diet. or from two related sources e.g. the height of a father (X ) and his son (Y ). 3 / 63
4 Tests for the difference between two population means  dependent samples We wish to test the hypothesis H 0 : µ X = µ Y In the first example, this hypothesis states that the diet has no effect on weight. In the second example, this hypothesis states that on average fathers are as tall as their sons. 4 / 63
5 One and twosided tests As before, we consider both one and twosided tests. In a twosided test the alternative is H A : µ X µ Y In the first example this states that the diet has some effect on weight. In the second example this states that the average height of fathers differs from the average height of their sons. 5 / 63
6 One and twosided tests The alternative in a onesided test may be of the form H A : µ X > µ Y. In the first example this states that the diet on average causes weight loss. In the second example this states that the average height of fathers is greater than the average height of their sons. 6 / 63
7 One and twosided tests The alternative may be of the form H A : µ X < µ Y. In the first example this states that the diet on average causes a gain in weight. In the second example, this states that the average height of sons is greater than the average height of their fathers. 7 / 63
8 Testing procedure for 2 dependent samples When the two samples are dependent, we calculate the differences D i = X i Y i. We treat these differences as one sample and carry out the appropriate one sample test. Let µ D be the population mean of this difference. We have µ D = µ X µ Y. 8 / 63
9 Testing procedure for 2 dependent samples The null hypothesis corresponds to The alternatives H 0 : µ X = µ Y H 0 : µ D = 0. H A : µ X µ Y ; H A : µ X > µ Y ; H A : µ X < µ Y correspond to H A : µ D 0; H A : µ D > 0; H A : µ D < 0, respectively. 9 / 63
10 Testing procedure for 2 dependent samples Suppose a company promoting a diet stated that on average a person loses 4kg on this diet, this hypothesis is H 0 : µ X µ Y = 4 H 0 : µ D = 4. The alternative in this case would be that the diet is not as effective as the company states i.e. H A : µ X µ Y < 4 H A : µ D < / 63
11 Testing procedure for 2 dependent samples The test statistic for the null hypothesis H 0 : µ D = µ 0 is T = D µ 0 S.E.(D), where D is the mean of the sample of differences. This is simply the statistic used for a onesample test for a population mean. If the sample size is large (n > 30), then this statistic has approximately a standard normal distribution. It should be noted that if the sample size is small, then this test assumes that the differences come from a normal distribution. If this condition is satisfied, then the test statistic has a Student tdistribution with n 1 degrees of freedom. 11 / 63
12 Example athletes run 400m both at sea level and at altitude. Their times are given below. Test the hypothesis that altitude does not affect the average times of runners at a significance level of 5%. Runner Sea Level Altitude / 63
13 Example Since two times are given for each athlete, these samples are dependent. We calculate the differences D i = X i Y i, where X i and Y i are the times of the ith athlete at sea level and altitude, respectively. The sample of differences is given by Runner Sea Level Altitude Difference / 63
14 Example i) The hypotheses are ii) The test statistic is H 0 : µ D = 0 against H A : µ D 0. T = D µ 0 S.E.(D), where µ 0 is the mean difference according to the null hypothesis, here µ 0 = / 63
15 Example iii) We calculate the realisation of the test statistic D= 8 s 2 = 1 n (D i D) 2 n 1 i=1 = 0.2 = ( )2 + ( ) ( ) 2 7 s= Hence, S.E.(D) s n = t= / 63
16 Example iv) Since the sample size is small, if these differences come from a normal distribution, this statistic has a Student tdistribution with 7 degrees of freedom. The test is two sided, thus the critical value is t n 1,α/2 = t 7,0.025 = v) Since t = 2.49 > t n 1,α/2 = 2.365, we reject H 0. We conclude that altitude affects the runners times. 16 / 63
17 Use of duality for twosided tests We can also use the duality between confidence intervals and two sided tests. In this case since the significance level is 5%, we calculate a 95% confidence interval for the mean difference. Since the samples are dependent, we treat the differences as one sample and use the appropriate formula to calculate a confidence interval for the true mean difference (in the appropriate population, e.g. all those on the given diet). If this interval contains the value from the null hypothesis, then we do not reject H / 63
18 Use of duality for twosided tests Since the sample is small, this formula is This gives D ± t n 1,α/2 S.E.(D) = D ± st n 1,α/2 n 0.2 ± = 0.2 ± 0.19 = [0.01, 0.39] We are testing the hypothesis H 0 : µ D = 0. Since 0 does not belong to this confidence interval, we reject H 0 at a significance level of 5%. We have evidence that altitude affects runners times. 18 / 63
19 Assumptions of this test It should be noted that this test assumes that the differences come from a normal distribution. It is not clear whether this is satisfied. Since, the realisation of the test statistic is close to the critical value, we should be somewhat sceptical of our conclusion (more data should be collected). 19 / 63
20 3.4.2 Tests for the difference between two population means: independent samples In this case the two samples come from two unrelated populations. e.g. the height of Americans and Irish, the times of two different groups of runners. We consider two cases 1. Large sample tests (both samples have at least 30 observations). 2. Small sample tests. 20 / 63
21 Large Samples Assume that we have samples (X 1,..., X m ) and (Y 1,..., Y n ) from populations with population means µ X and µ Y (where m and n are at least 30). We use the difference between the two sample means, X Y, to estimate the difference between the two population means, µ X µ Y. The standard error of this estimate is σx 2 S.E.(X Y ) = m + σ2 Y n. This is approximated using S.E.(X Y ) s 2 X m + s2 Y n. 21 / 63
22 Large Samples Suppose we wish to test the hypothesis that H 0 : µ X µ Y = d i.e. the difference between the two population means is d. In twotailed tests the alternative is H A : µ X µ Y d. If the test is twotailed, we can always label the two samples in such a way that the alternative is H A : µ X µ Y > d 22 / 63
23 Large Samples When both samples are large (m, n > 30), the test statistic is Z = (X Y ) d S.E.(X Y ) This statistic has approximately a standard normal distribution. Critical values and pvalues are calculated in the same way as in one sample tests. i.e. The pvalue for a two sided test is p = P( Z > t ) = 2P(Z > t ). The pvalue for a one sided test is p = P(Z > t). 23 / 63
24 Large samples The critical value for a two sided test is Z α/2 = t,α/2. H 0 is rejected if and only if t > t,α/2. The critical value for a one sided test is Z α = t,α. H 0 is rejected if and only if t > t,α. It should be noted that, as before, the realisation of the test statistic is a measure of the distance between the data and H 0. e.g. when the difference between the sample means is much greater than d, then the realisation of the test statistic will be much greater than / 63
25 Example The average height of 100 Dutch men is 176cm and their standard deviation 12cm. The average height of 50 Japanese men is 169cm and their standard deviation is 10cm. Test at a significance level of 1% the hypothesis that the average heights of Dutch men and Japanese men are equal. 25 / 63
26 Example i) We have H 0 : µ X µ Y = 0; H A : µ X µ Y 0, where µ X is the mean height of all Dutch men and µ Y the mean height of all Japanese men. 26 / 63
27 Example ii) We use the test statistic Z= X Y S.E.(X Y ) s 2 X S.E.(X Y ) m + s2 Y n 12 2 = = The realisation of this test statistic is t = / 63
28 Example iv) From the table for the standard normal distribution, the pvalue for this test is p = 2P(Z > 3.77) = v) Since p < α = 0.01, we reject H 0 at a significance level of 1%. Also, since p < 0.001, we have very strong evidence that the mean heights of Dutchmen and Japanese differ. From the data we may state that Dutchmen are taller on average than Japanese men. 28 / 63
29 Example iv) We can also use the appropriate critical value. Since both samples are large and this is a two sided test, this value is given by Z α/2 = t,α/2 = v) Since t = 3.77 > t,α/2 = 2.576, we reject H 0 at a significance level of 1%. We have strong evidence that the mean heights of Dutchmen and Japanese differ. 29 / 63
30 Duality for two independent samples When both samples are large, the 100(1 α)% confidence interval for the difference between two population means is (X Y ) ± t,α/2 S.E.(X Y ) We can use the duality between confidence intervals and two sided tests. 30 / 63
31 Example Calculate a 95% confidence interval for the difference between the mean height of Dutch and Japanese men. Test the hypothesis that on average Dutch men are 10cm taller than Japanese men (data from previous example). We are testing H 0 : µ X µ Y = 10 against H A : µ X µ Y / 63
32 Example Since the samples are large, the confidence interval for the difference between the population means is (X Y )±t,α/2 S.E.(X Y ) t,α/2 =t,0.025 = 1.96 Hence, the confidence interval is ( ) ± = 7 ± 3.64 = [3.36, 10.64] Since 10 [3.36, 10.64], we do not reject H 0 at a significance level of 5%. The is no evidence against the hypothesis that on average Dutchmen are 10cm taller than Japanese men. 32 / 63
33 Small sample tests In the case where at least one of the samples is small, the test for the difference between two population means assumes that the observations come from normal distributions with equal variances (i.e. σ 2 X = σ2 Y = σ2 ). 33 / 63
34 Test for equality of variances Before we carry out the test for a difference between two population means, we should carry out an F test for the equality of two variances. We test H 0 : σ 2 X = σ2 Y against H A : σ 2 X σ2 Y The test statistic, F, is the ratio between the two sample variances. F = max{s2 X, s2 Y } min{s 2 X, s2 Y }. When this ratio is close to one we do not reject the null hypothesis that the population variances are equal. Ratios much greater than 1 indicate that the null hypothesis is not true. 34 / 63
35 Test for equality of variances Suppose the observations in both samples come from a normal distribution. F has an F distribution with j 1 and k 1 degrees of freedom, where j and k are the number of observations in the sample with the largest and smallest variance, respectively. We reject H 0 if and only if the realisation of the test statistic, f, satisfies f > F j 1,k 1,α/2, where P(F > F j 1,k 1,α/2 = α/2). 35 / 63
36 Test for equality of variances Critical values of the F j 1,k 1,p are given in Table 9. j 1 and k 1 correspond to the column and row number, respectively. This test is normally carried out at a significance level of 5%. Each cell contains 4 critical values. The first is for p = 0.05, the second (in brackets) for p = (this is the appropriate value), the third for p = 0.01 and the fourth for p = Note that if the two variances are not equal, the assumptions of the test for a difference between two means, presented below, do not hold. The appropriate procedure in this case is not covered in the course. 36 / 63
37 Test for difference between two means (small samples) Given the hypothesis regarding the equality of variances was not rejected, we use a pooled estimate of the variances, s 2 p, where s 2 p = (m 1)s2 X + (n 1)s2 Y m + n 2 This is a weighted average of the sample variances, in which the sample with the largest number of observations has the largest weight. The standard error of the difference between the sample means is. S.E.(X Y ) = s p 1 m + 1 n 37 / 63
38 Test for difference between two means (small samples) Suppose we wish to test the null hypothesis The test statistic used is H 0 : µ X µ Y = d. T = (X Y ) d S.E.(X Y ). Given the assumptions of the test are satisfied (the observations come from normal distributions with a common variance), then this statistic has a student t distribution with m + n 2 degrees of freedom. A 100(1 α)% confidence interval for the difference between the two population means, µ X µ Y, is given by (X Y ) ± t m+n 2,α/2 S.E.(X Y ). 38 / 63
39 Test for difference between two means (small samples) The critical value for the two sided test with is t m+n 2,α/2. H A : µ X µ Y d. We reject H 0 iff t > t m+n 2,α/2. If the test is two sided, we can always label the two samples such that the alternative is of the form H A : µ X µ Y > d. The critical value for such a test is t m+n 2,α. We reject H 0 iff t > t m+n 2,α. 39 / 63
40 Example The average height of 13 Dutch men is 176cm and their standard deviation 12cm. The average height of 11 Japanese men is 169cm and their standard deviation is 10cm. Test at a significance level of 5% the hypothesis that the average heights of Dutch men and Japanese men are equal. 40 / 63
41 Example Since the sample sizes are small, we first test the assumption that the population variances are equal. i) We have ii) The test statistic is H 0 : σ 2 X = σ2 Y against H A : σ 2 X σ2 Y F = max{s2 X, s2 Y } min{s 2 X, s2 Y }. iii) The realisation of this test statistic is f = max{122, 10 2 } min{12 2, 10 2 } = / 63
42 Example iv) We read the appropriate critical value. Since there are 13 observations in the sample with the greatest variance, j 1 = 12. Similarly, k 1 = 10. Since α = 0.05, the critical value is F 12,10,0.025 = v) Since f < F 12,10,0.025 = 3.62, we do not reject H 0. Hence, we may assume that the two population variances are equal. 42 / 63
43 Example We now proceed to test the hypothesis regarding the equality of the two means. i) We have H 0 : µ X µ Y = 0 against H A : µ X µ Y 0. ii) The test statistic for this test is where T = X Y S.E.(X Y ), S.E.(X Y )=s p 1 m + 1 n s 2 p= (m 1)s2 X + (n 1)s2 Y m + n / 63
44 Example iii) We calculate the realisation of the test statistic. The pooled variance is s 2 p = (13 1) (11 1) = 124. The standard error of the difference between the two sample means is 1 S.E.(X Y )=s p m + 1 n 1 = The realisation of the test statistic is t = / 63
45 Example iv) The critical value for the test is t m+n 2,α/2 = t 22,0.025 = v) Since t = 1.53 < t m+n 2,α/2 = 2.074, we do not reject H 0. There is no evidence that the mean height of Dutchmen differs from the mean height of Japanese. 45 / 63
46 Use of duality It should be noted that we can carry out this test using the duality between two sided tests and confidence intervals. The formula for a 100(1 α)% confidence interval for the difference between two population means when the sample sizes are small is (X Y ) ± t m+n 2,α/2 S.E.(X Y ). Since we are carrying out a test at a significance level of 5%, we calculate a 95% confidence interval. This is given by ( ) ± t 22,0.025 S.E.(X Y )=7 ± =7 ± 9.46 = [ 2.46, 16.46] 46 / 63
47 Use of duality Since we are testing H 0 : µ X µ Y = 0, We reject the null hypothesis if and only if 0 does not belong to this confidence interval. Since 0 belongs to this confidence interval, we do not reject H 0 at a significance level of 5%. There is no evidence that the mean height of Dutchmen differs from the mean height of Japanese. 47 / 63
48 Confidence intervals for the difference between two population proportions Suppose we have two independent, large samples from distinct populations. Suppose the ith sample has n i observations and the number of individuals showing the trait of interest in the ith sample is x i. Let the proportion of individuals exhibiting these traits in the ith population be p i and the proportion of individuals exhibiting these traits in the ith sample be ˆp i, where ˆp i = x i n i. The difference between the two sample proportions is used to estimate the difference between the two population proportions. 48 / 63
49 Confidence intervals for the difference between two population proportions The standard error of the difference between the two sample proportions is p 1 (1 p 1 ) S.E.(ˆp 1 ˆp 2 ) = + p 2(1 p 2 ) n 1 n 2 As before, the standard error of this difference depends on the (unknown) population proportions. When we calculate a confidence interval for the difference between population proportions, this standard error can be approximated using ˆp 1 (1 ˆp 1 ) S.E.(ˆp 1 ˆp 2 ) = + ˆp 2(1 ˆp 2 ). n 1 n 2 49 / 63
50 Confidence intervals for the difference between two population proportions An approximate 100(1 α)% confidence interval for the difference between two population proportions is given by (ˆp 1 ˆp 2 ) ± t m+n 2,α/2 S.E.(ˆp 1 ˆp 2 ) Note that this procedure should not be used e.g. to calculate a confidence interval for the difference between the level of support of two political parties in a single population. 50 / 63
51 Example of 300 male applicants for an engineering course were accepted and 40 of 80 female applicants. Calculate a 95% confidence interval for the difference in the proportion of males and females accepted for the course. 51 / 63
52 Example The sample proportions are ˆp 1 = = 0.4; ˆp 2 = = 0.5 The estimate of the standard error of the difference between the two proportions is given by ˆp 1 (1 ˆp 1 ) S.E.(ˆp 1 ˆp 2 )= + ˆp 2(1 ˆp 2 ) n 1 n = / 63
53 Example The confidence interval for the difference between the two proportions is (ˆp 1 ˆp 2 ) ± t,α/2 S.E.(ˆp 1 ˆp 2 )=0.1 ± =0.1 ± = [ 0.023, 0.223]. 53 / 63
54 Hypothesis testing Suppose we want to test the hypothesis The test statistic for this test is H 0 : p 1 = p 2. Z = (ˆp 1 ˆp 2 ) S.E.(ˆp 1 ˆp 2 ). This statistic has approximately a standard normal distribution. Let t be the realisation of this statistic. 54 / 63
55 Hypothesis testing In order to estimate the standard error of the difference between the two sample proportions under the null hypothesis, we calculate the pooled proportion p. This is the total number of individuals with the trait in both samples divided by the total number of individuals in both samples p = x 1 + x 2 n 1 + n 2. We have [ 1 S.E.(ˆp 1 ˆp 2 ) p(1 p) + 1 ]. n 1 n 2 55 / 63
56 Hypothesis testing For two tailed tests with H A : p 1 p 2 the critical value is Z α/2 = t,α/2. We reject H 0 if and only if t > t,α/2. The pvalue is 2P(Z > t ). 56 / 63
57 Hypothesis testing In the case of one tailed tests, we can always number the samples such that the alternative is The critical value is Z α = t,α. H A : p 1 > p 2. We reject H 0 if and only if t > t,α. The pvalue is P(Z > t). 57 / 63
58 Example of 300 male applicants for an engineering course were accepted and 40 of 80 female applicants. Test at a significance level of 5% the hypothesis that the proportion of males accepted equals the proportion of females accepted. 58 / 63
59 Example The hypotheses are ii) The statistic used is H 0 : p 1 = p 2 against H A : p 1 p 2 Z = ˆp 1 ˆp 2 S.E.(ˆp 1 ˆp 2 ). 59 / 63
60 Example iii) We calculate the realisation of the test statistic. The pooled proportion is p = x 1 + x = n 1 + n The estimate of the standard error of the difference between the two sample proportions under H 0 is [ 1 S.E.(ˆp 1 ˆp 2 ) p(1 p) + 1 ] n 1 n 2 [ 1 = ] / 63
61 Example The sample proportions are ˆp 1 = = 0.4; ˆp 2 = = 0.5. The realisation of the test statistic is t = iv) The critical value for this test is t,0.025 = v) Since t = 1.61 < t,0.025 = 1.96, we do not reject H 0 at a significance level of 5%. There is no evidence that the admission rates vary according to sex. 61 / 63
62 Example In iv) we could calculate the pvalue p = 2P(Z > t ) = 2P(Z > 1.61) = = Since p > 0.05, we do not reject H 0. There is no evidence that the admission rates vary. 62 / 63
63 Example As in the one sample case, we can use the duality between twosided hypothesis tests and confidence intervals. Since different approximations are used to estimate the standard error of the difference between the sample proportions, this duality is only approximate. Suppose we wish to test H 0 : p 1 = p 2 (i.e. p1 p2 = 0). In this example the 95% confidence interval for the difference between the two proportions was [0.023, 0.223]. Since 0 belongs to this confidence interval, we do not reject H 0 at a significance level of approximately 5%. There is no evidence that the admission rates vary. 63 / 63
6. Duality between confidence intervals and statistical tests
6. Duality between confidence intervals and statistical tests Suppose we carry out the following test at a significance level of 100α%. H 0 :µ = µ 0 H A :µ µ 0 Then we reject H 0 if and only if µ 0 does
More informationChapter 8 Introduction to Hypothesis Testing
Chapter 8 Student Lecture Notes 81 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate
More informationHypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...
Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More information93.4 Likelihood ratio test. NeymanPearson lemma
93.4 Likelihood ratio test NeymanPearson lemma 91 Hypothesis Testing 91.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationHYPOTHESIS TESTING: POWER OF THE TEST
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
More informationModule 5 Hypotheses Tests: Comparing Two Groups
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
More informationIntroduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.
Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.
More informationTHE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.
THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM
More informationAn Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10 TWOSAMPLE TESTS
The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10 TWOSAMPLE TESTS Practice
More information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More informationMATH 214 (NOTES) Math 214 Al Nosedal. Department of Mathematics Indiana University of Pennsylvania. MATH 214 (NOTES) p. 1/6
MATH 214 (NOTES) Math 214 Al Nosedal Department of Mathematics Indiana University of Pennsylvania MATH 214 (NOTES) p. 1/6 "Pepsi" problem A market research consultant hired by the PepsiCola Co. is interested
More informationBA 275 Review Problems  Week 6 (10/30/0611/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394398, 404408, 410420
BA 275 Review Problems  Week 6 (10/30/0611/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394398, 404408, 410420 1. Which of the following will increase the value of the power in a statistical test
More informationHYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationHypothesis Testing or How to Decide to Decide Edpsy 580
Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at UrbanaChampaign Hypothesis Testing or How to Decide to Decide
More informationEstimation of σ 2, the variance of ɛ
Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated
More informationTwo Related Samples t Test
Two Related Samples t Test In this example 1 students saw five pictures of attractive people and five pictures of unattractive people. For each picture, the students rated the friendliness of the person
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationChapter 9, Part A Hypothesis Tests. Learning objectives
Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population
More informationGeneral Method: Difference of Means. 3. Calculate df: either WelchSatterthwaite formula or simpler df = min(n 1, n 2 ) 1.
General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either WelchSatterthwaite formula or simpler df = min(n
More informationChapter 8. Hypothesis Testing
Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing
More informationPractice problems for Homework 12  confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems.
Practice problems for Homework 1  confidence intervals and hypothesis testing. Read sections 10..3 and 10.3 of the text. Solve the practice problems below. Open the Homework Assignment 1 and solve the
More informationSTA218 Introduction to Hypothesis Testing
STA218 Introduction to Hypothesis Testing Al Nosedal. University of Toronto. Fall 2015 October 29, 2015 Who wants to be a millionaire? Let s say that one of you is invited to this popular show. As you
More informationChapter 2. Hypothesis testing in one population
Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance
More informationCHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING
CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING MULTIPLE CHOICE 56. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ 50, the following information is known: n = 64, = 53.5, and σ = 10. The standardized
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationName: Date: Use the following to answer questions 34:
Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin
More informationHypothesis testing  Steps
Hypothesis testing  Steps Steps to do a twotailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
More informationChapter 9: Hypothesis Testing GBS221, Class April 15, 2013 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College
Chapter Objectives 1. Learn how to formulate and test hypotheses about a population mean and a population proportion. 2. Be able to use an Excel worksheet to conduct hypothesis tests about population means
More informationConfidence Intervals for the Difference Between Two Means
Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means
More informationHypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the
More informationConfidence Intervals (Review)
Intro to Hypothesis Tests Solutions STATUB.0103 Statistics for Business Control and Regression Models Confidence Intervals (Review) 1. Each year, construction contractors and equipment distributors from
More informationHYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE)  CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More informationThe NeymanPearson lemma. The NeymanPearson lemma
The NeymanPearson lemma In practical hypothesis testing situations, there are typically many tests possible with significance level α for a null hypothesis versus alternative hypothesis. This leads to
More informationMind on Statistics. Chapter 13
Mind on Statistics Chapter 13 Sections 13.113.2 1. Which statement is not true about hypothesis tests? A. Hypothesis tests are only valid when the sample is representative of the population for the question
More informationDifference of Means and ANOVA Problems
Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationMeasuring the Power of a Test
Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection
More informationNull Hypothesis H 0. The null hypothesis (denoted by H 0
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
More informationChapter Five: Paired Samples Methods 1/38
Chapter Five: Paired Samples Methods 1/38 5.1 Introduction 2/38 Introduction Paired data arise with some frequency in a variety of research contexts. Patients might have a particular type of laser surgery
More informationAMS7: WEEK 8. CLASS 1. Correlation Monday May 18th, 2015
AMS7: WEEK 8. CLASS 1 Correlation Monday May 18th, 2015 Type of Data and objectives of the analysis Paired sample data (Bivariate data) Determine whether there is an association between two variables This
More informationLecture 13 More on hypothesis testing
Lecture 13 More on hypothesis testing Thais Paiva STA 111  Summer 2013 Term II July 22, 2013 1 / 27 Thais Paiva STA 111  Summer 2013 Term II Lecture 13, 07/22/2013 Lecture Plan 1 Type I and type II error
More informationClass 19: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.1)
Spring 204 Class 9: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the
More informationBasic concepts and introduction to statistical inference
Basic concepts and introduction to statistical inference Anna Helga Jonsdottir Gunnar Stefansson Sigrun Helga Lund University of Iceland (UI) Basic concepts 1 / 19 A review of concepts Basic concepts Confidence
More informationBA 275 Review Problems  Week 5 (10/23/0610/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380394
BA 275 Review Problems  Week 5 (10/23/0610/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete
More informationC. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.
Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample
More information[Chapter 10. Hypothesis Testing]
[Chapter 10. Hypothesis Testing] 10.1 Introduction 10.2 Elements of a Statistical Test 10.3 Common LargeSample Tests 10.4 Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests
More informationChapter 1112 1 Review
Chapter 1112 Review Name 1. In formulating hypotheses for a statistical test of significance, the null hypothesis is often a statement of no effect or no difference. the probability of observing the data
More informationHypothesis testing. Power of a test. Alternative is greater than Null. Probability
Probability February 14, 2013 Debdeep Pati Hypothesis testing Power of a test 1. Assuming standard deviation is known. Calculate power based on onesample z test. A new drug is proposed for people with
More informationHypothesis testing for µ:
University of California, Los Angeles Department of Statistics Statistics 13 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative
More informationTwosample hypothesis testing, II 9.07 3/16/2004
Twosample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For twosample tests of the difference in mean, things get a little confusing, here,
More informationSPSS on two independent samples. Two sample test with proportions. Paired ttest (with more SPSS)
SPSS on two independent samples. Two sample test with proportions. Paired ttest (with more SPSS) State of the course address: The Final exam is Aug 9, 3:30pm 6:30pm in B9201 in the Burnaby Campus. (One
More informationHow to Conduct a Hypothesis Test
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
More informationHypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam
Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests
More informationE205 Final: Version B
Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random
More informationBasic Statistics Self Assessment Test
Basic Statistics Self Assessment Test Professor Douglas H. Jones PAGE 1 A sodadispensing machine fills 12ounce cans of soda using a normal distribution with a mean of 12.1 ounces and a standard deviation
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationMAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters
MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for
More informationCHAPTER 11 CHISQUARE: NONPARAMETRIC COMPARISONS OF FREQUENCY
CHAPTER 11 CHISQUARE: NONPARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationHypothesis Testing hypothesis testing approach formulation of the test statistic
Hypothesis Testing For the next few lectures, we re going to look at various test statistics that are formulated to allow us to test hypotheses in a variety of contexts: In all cases, the hypothesis testing
More informationMath 251, Review Questions for Test 3 Rough Answers
Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections  we are still here Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 TwoSided Alternatives 9.5 The t Test 9.6 Comparing the
More informationLesson 1: Comparison of Population Means Part c: Comparison of Two Means
Lesson : Comparison of Population Means Part c: Comparison of Two Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis
More informationChapter 7 Section 1 Homework Set A
Chapter 7 Section 1 Homework Set A 7.15 Finding the critical value t *. What critical value t * from Table D (use software, go to the web and type t distribution applet) should be used to calculate the
More informationSampling and Hypothesis Testing
Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus
More informationSections 4.54.7: TwoSample Problems. Paired ttest (Section 4.6)
Sections 4.54.7: TwoSample Problems Paired ttest (Section 4.6) Examples of Paired Differences studies: Similar subjects are paired off and one of two treatments is given to each subject in the pair.
More informationTesting: is my coin fair?
Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible
More information5/31/2013. Chapter 8 Hypothesis Testing. Hypothesis Testing. Hypothesis Testing. Outline. Objectives. Objectives
C H 8A P T E R Outline 8 1 Steps in Traditional Method 8 2 z Test for a Mean 8 3 t Test for a Mean 8 4 z Test for a Proportion 8 6 Confidence Intervals and Copyright 2013 The McGraw Hill Companies, Inc.
More informationExample for testing one population mean:
Today: Sections 13.1 to 13.3 ANNOUNCEMENTS: We will finish hypothesis testing for the 5 situations today. See pages 586587 (end of Chapter 13) for a summary table. Quiz for week 8 starts Wed, ends Monday
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More informationHypothesis Testing  One Mean
Hypothesis Testing  One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis
More informationSolutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG820). December 15, 2012.
Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG8). December 15, 12. 1. (3p) The joint distribution of the discrete random variables X and
More information7. Tests of association and Linear Regression
7. Tests of association and Linear Regression In this chapter we consider 1. Tests of Association for 2 qualitative variables. 2. Measures of the strength of linear association between 2 quantitative variables.
More informationOpgaven Onderzoeksmethoden, Onderdeel Statistiek
Opgaven Onderzoeksmethoden, Onderdeel Statistiek 1. What is the measurement scale of the following variables? a Shoe size b Religion c Car brand d Score in a tennis game e Number of work hours per week
More informationUNDERSTANDING THE INDEPENDENTSAMPLES t TEST
UNDERSTANDING The independentsamples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly
More informationMCQ TESTING OF HYPOTHESIS
MCQ TESTING OF HYPOTHESIS MCQ 13.1 A statement about a population developed for the purpose of testing is called: (a) Hypothesis (b) Hypothesis testing (c) Level of significance (d) Teststatistic MCQ
More informationHypothesis Testing: Two Means, Paired Data, Two Proportions
Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this
More informationChapter 8 Hypothesis Tests. Chapter Table of Contents
Chapter 8 Hypothesis Tests Chapter Table of Contents Introduction...157 OneSample ttest...158 Paired ttest...164 TwoSample Test for Proportions...169 TwoSample Test for Variances...172 Discussion
More informationNull Hypothesis Significance Testing Signifcance Level, Power, ttests. 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom
Null Hypothesis Significance Testing Signifcance Level, Power, ttests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is
More informationHypothesis Testing and Confidence Interval Estimation
Biostatistics for Health Care Researchers: A Short Course Hypothesis Testing and Confidence Interval Estimation Presented ed by: Susan M. Perkins, Ph.D. Division of Biostatistics Indiana University School
More informationHypothesis Testing. Hypothesis Testing CS 700
Hypothesis Testing CS 700 1 Hypothesis Testing! Purpose: make inferences about a population parameter by analyzing differences between observed sample statistics and the results one expects to obtain if
More informationBusiness Statistics. Lecture 8: More Hypothesis Testing
Business Statistics Lecture 8: More Hypothesis Testing 1 Goals for this Lecture Review of ttests Additional hypothesis tests Twosample tests Paired tests 2 The Basic Idea of Hypothesis Testing Start
More informationChapter 4 Statistical Inference in Quality Control and Improvement. Statistical Quality Control (D. C. Montgomery)
Chapter 4 Statistical Inference in Quality Control and Improvement 許 湘 伶 Statistical Quality Control (D. C. Montgomery) Sampling distribution I a random sample of size n: if it is selected so that the
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Open book and note Calculator OK Multiple Choice 1 point each MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data.
More informationSolutions to Worksheet on Hypothesis Tests
s to Worksheet on Hypothesis Tests. A production line produces rulers that are supposed to be inches long. A sample of 49 of the rulers had a mean of. and a standard deviation of.5 inches. The quality
More informationUnit 26 Estimation with Confidence Intervals
Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference
More informationSection 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities
Today: Section 2.2, Lesson 3: What can go wrong with hypothesis testing Section 2.4: Hypothesis tests for difference in two proportions ANNOUNCEMENTS: No discussion today. Check your grades on eee and
More information1 Confidence intervals
Math 143 Inference for Means 1 Statistical inference is inferring information about the distribution of a population from information about a sample. We re generally talking about one of two things: 1.
More information13 TwoSample T Tests
www.ck12.org CHAPTER 13 TwoSample T Tests Chapter Outline 13.1 TESTING A HYPOTHESIS FOR DEPENDENT AND INDEPENDENT SAMPLES 270 www.ck12.org Chapter 13. TwoSample T Tests 13.1 Testing a Hypothesis for
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More informationChapter 7 Part 2. Hypothesis testing Power
Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship
More informationCHAPTER 9 HYPOTHESIS TESTING
CHAPTER 9 HYPOTHESIS TESTING The TI83 Plus and TI84 Plus fully support hypothesis testing. Use the key, then highlight TESTS. The options used in Chapter 9 are given on the two screens. TESTING A SINGLE
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More informationGeneral Procedure for Hypothesis Test. Five types of statistical analysis. 1. Formulate H 1 and H 0. General Procedure for Hypothesis Test
Five types of statistical analysis General Procedure for Hypothesis Test Descriptive Inferential Differences Associative Predictive What are the characteristics of the respondents? What are the characteristics
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Sample Practice problems  chapter 121 and 2 proportions for inference  Z Distributions Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide
More informationThe alternative hypothesis,, is the statement that the parameter value somehow differs from that claimed by the null hypothesis. : 0.5 :>0.5 :<0.
Section 8.28.5 Null and Alternative Hypotheses... The null hypothesis,, is a statement that the value of a population parameter is equal to some claimed value. :=0.5 The alternative hypothesis,, is the
More informationSTAT 145 (Notes) Al Nosedal anosedal@unm.edu Department of Mathematics and Statistics University of New Mexico. Fall 2013
STAT 145 (Notes) Al Nosedal anosedal@unm.edu Department of Mathematics and Statistics University of New Mexico Fall 2013 CHAPTER 18 INFERENCE ABOUT A POPULATION MEAN. Conditions for Inference about mean
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More information