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Statistics for Education Research Lecture 2 Normal Distributions & Sampling Distribution of Means Instructor: Dr. Tung-hsien He [email protected] Normal Distributions 1. Theoretically, any variable being measured by infinite times would tend to display a normal distribution. 2. Normal Distributions are described by a mathematical equation (4.1, p. 86) 3. Not every distribution of a variable will match the normal distribution. 4. Distributions of variables, however, will approximate the normal distributions. 5. Normal distributions have various shapes and curves (p. 88. Noted: these distributions are normal distributions] 6. Shapes of normal distributions will be determined by means and standard deviation Features of Normal Distributions 1. Unimodal: Only with one mode 2. Symmetrical 3. Bell-Shaped 4. Maximum Height as Mean 5. Values on X axis are continuous 6. Asymptotic (漸進) to X axis: the curses never touch the X axis 7. Shapes are determined by mean and SD (See p. 88: Figure B/C: Which distributions have larger values of SD? Why they are normal distributions?). 8. The number of normal distributions is infinite (since the requirements for a normal distribution can be easily met). 9. After a variable has been tested for an infinite number of times, the scores of all the tests will approximate the normal distribution. Standard Normal Distribution: 1. The distribution of normally distributed standard scores: z score 2. Formula: See 4.2, p. 89 3. Mean = 0 4. SD = 1 5. The shape (proportion) of the distribution: See figure 4.3, p. 90 6. Determining Proportions: See Table c.1, p. 618 7. Determining Percentiles by Using NSD: See Table c.2, p. 621 E.g.: 70th PR with a mean = 85, SD = 20: (Think about how z-score is computed?) Step 1: Check Table C.2 on p. 621 Step 2: Find B (The Larger Area) = .70 Step 3: Find z that corresponds to B = .70 -> z = .5244 Step 4: Use z score formula: ?-85/20 = .5244 -> ? = 95.488 8. Determining Percentile Ranks: E.g.: PR102 with mean = 85, SD=20: (i.e., PR of raw score 102 is = ?) Step 1: Use z score formula: z = 102-85/20 -> z = 0.85 Step 2: Check C.2 Table on p. 618 Step 3: Find z = .85, Area between mean and z = .3023 Step 4: 0.5+0.3023 = 0.8023 -> PR 102 = 80.23 Normal Distribution the fundamental assumption for inferential statistics: a. a variable will be normally distributed if it is tested for an infinite number of times. b. The selected sample(s) must represent the characteristics of the population from which the sample(s) is/are selected (i.e., sample(s) must match the population) c. Random selections (accompanied by random assignments) are the only way to guarantee the representation of the sample(s). Chain of Reasoning in Inferential Statistics a. Parameters: Populations b. Statistics: Samples c. Test hypothesis about parameters: There will be two types of hypotheses that will be tested: the null hypothesis and alternative hypothesis. d. Estimate parameters based on statistics e. Inferential statistics is all about “sampling” and “hypothesis testing” Two types of Samples: Probability Samples vs. Nonprobability Samples Probability Samples (隨機非隨便樣本) 1. Simple Random Sampling: Every member in the target population has identical chances to be selected a. Sampling with replacement: Selected subjects are included in the next selection procedures b. Sampling without replacement: Selected subjects are excluded from next selection procedures c. The two methods will yield different probabilities. d. Using SPSS to random select cases (i.e., the random seeds) 2. Systematic Sampling: Choosing every kth member of a list that contains all members of a population. 3. Cluster Sampling: Clusters (naturally formed groups) are randomly selected from the population of clusters. 4. Stratified Random Sampling: a. To select samples from a heterogeneous population that contains several subpopulations (strata); b. Strata have to be defined well; c. Random samples of members of each stratum are selected. Nonprobability Samples (No random selection is involved) : 1. Purposive Samples: a. Starting off with a large group of potential subjects; b. Using screening criteria to select those subject who meet these criteria. 2. Convenience Samples (立意樣本): Selecting whoever you want to be subjects. 3. Quota Samples: a. Deciding X percent of a certain kind of subjects and Y percent of another kind of subjects; b. Going out to select whoever these kinds of subjects to be subjects. 4. Snowball Samples: a. Starting off with a convenience sample; b. Recruiting more others related to this sample like family members, friends, . . . 5. Returning Questionnaires in Surveys (Why?) 6. Volunteers Sampling Distributions of Mean: Key Concept & Assumption for Inferential Statistics a. Meaning: the distributions of all possible means of a certain size of samples that have been selected and tested for an infinite number of times (theoretically possible only) b. Interpretation: Theoretically, a certain side of samples can be chosen for infinite times from a population, and each time when a sample is drawn, there will be a corresponding mean for this particular sample. Because samples can be infinitely chosen (only theoretically possible), there will be an infinite number of sample means. Sampling distribution of mean represents the distribution of these means. c. Properties: 1. Shape: (a) As sample size increases, the sampling distribution of the mean for simple random samples of n cases will approximates a normal distribution (b) if the sample size is no smaller than 30, the sampling distribution of the mean (n=30) will approximate normal distributions. 2. A normal Distribution; 3. Variance & Standard Deviation of Sampling Distribution of Mean: (a) Variance: 6.8, p. 150 (b) Standard Deviation (Standard Error): 6.9, p. 150 4. Mean of the Distribution of Sampling Means equals 5. As sample size (n) increases, variability of the sampling distribution of the mean decreases (Why?) 6. See figure 6.7, 6.8 on p. 152 & p. 153 for standard sampling distribution of the mean (z score) Hypothesis Testing: (A must-known concept for you to understand inferential statistics): 1. Meaning: Making inferences about the nature of the population on the basis of observations of a sample drawn from the population 2. Logics: as the differences between hypothesized value for the population mean and the sample mean are computed and found to be very large, the hypothesis is rejected. 3. Figure 7.1 on p. 166 for detailed explanations: Types of Hypothesis 1. A conjecture about a or more population parameters 2. Testing a specific hypothesis does not mean prove or disprove the conjecture; it only tells how likely (i.e., the probability) this hypothesis may be true. a. Null Hypothesis [虛無假設]: (1) Symbol: Ho (2) It means “no relation between variables”: the relation index is equal to 0 (3) It can also mean: “no difference in means”: mean1 is equal to mean2 (3) E.g. 1: Ho: = 455 (4) E.g. 2: the mean of the experimental group is equal to the mean of the control group. (5) If Ho is retained, it means: (a) the null hypothesis is very likely to be true (or happen) at certain level of confidence; (b) the probability for the null hypothesis to be true is very high; (c) there is no relation between two variables; (e) differences in two means are so small and nonsignificant that the differences can be discarded (i.e., the differences may be caused by sampling error) b. Alternative Hypothesis [對立假設] : (1) Symbol: Ha (2) Relations between variables exist (i.e., the relation index is not equal to 0); two or more means are different from each other or one another (i.e., mean1 is not equal to mean2) (3) Against the Null Hypothesis (4) The researchers’ expected outcome (researchers’ hypothesis) (5) Example: Ha: 455 (6) If Ha is retained, it means: (a) The null hypothesis (Ho) is rejected. (b) A significant relation or significant differences are detected. (c) Ha is very likely to be true (or to happen) at certain level of confidence. (d) The probability for the alternative hypothesis to be true is very high. (e) Differences in two means are so significant that the two means are very likely to be different. Meaning of Accepting Ho: = 455, but Rejecting Ha: 455 when the sample mean was found to be 454: 1. Condition: We assume (hypothesize) that of a population is 455. We formulate the following hypotheses: Ho: = 455, Ha: 455 Then we select and test a sample, and find its sample mean = 454. Based on the sample mean, it is very likely that the population mean will be 455. So, Ho is retained but Ha is rejected. 2. Interpretation a. Because the difference ( - X bar = 455-454 = 1) between the expected mean and the observed mean is very small, i.e., nonsignificant, Ho is retained and Ha is rejected . b. Since we formulate two hypothesis, namely, Ho: = 455, Ha: 455, and Ho is retained, we can draw a conclusion: It is very likely that the population mean is 455. c. Where does the difference, that is, 1, come from? It comes from sampling error! Meaning of Rejecting Ho: = 455, but Accepting Ha: 455 when the sample mean was found to be 80,000: 1. Condition: We assume (hypothesize) that of a population is 455. We formulate the following hypotheses: Ho: = 455, Ha: 455 Then we select and test a sample, and find its sample mean = 80,000 Based on the sample mean, it is very unlikely that the population mean will be 455. So, Ho is rejected but Ha is retained. 2. Interpretation: a. Because it is very unlikely that the population mean will be 455. So, Ho is rejected in favor of Ha. b. Because differences between the expected population mean and the observed mean are very huge, i.e., significant, Ho is rejected (i.e., Ho: = 455, Ha: 455). Thus, we can reach the following conclusion: it is very unlikely that mean of population is 455. Rule of Thumb: 1. If differences between the expected mean and the observed mean are very small, i.e., nonsignificant, you should retain Ho but reject Ha. 2. If differences between the expected mean and the observed mean are very huge, i.e., significant, your should reject Ho but retain Ha. To retain or to reject Ho , that is a question (see p. 166): a. Since it is impossible to know the true of a population, we can only hypothesize its value. If we hypothesize to be 455, and its standard deviation of the population, , is hypothesized to be 100. Then we select a sample of 144 subjects and find its sample mean = 535. We formulate the following hypotheses: Ho: = 455, Ha: 455 Q: Based on the sample mean, should we reject or retain Ho? (a) At first sight, we should reject Ho because the difference between the observed 535 and hypothesized 455 is 80, and it seems very huge. But, that is our feeling only. How will statistics tell us? There are a few key points that need to be taken into account before we can answer this question: (b) The sampling distribution of means is our solution to this question because: (1) the mean of the sampling distribution of means is the population mean; (2) the sampling distribution of means is a normal distribution, and the standard deviation of the sampling distribution of means (standard error, 標 準誤) = /√n, when (population SD) = 100 (hypothesized); n (number of subjects) =144 (3) When we select a sample and get its mean, we don’t expect this mean to be perfectly equal to , particularly when we do not know what the is. But we are sure if we sample the population for infinite times, one of the sample means will be equal to the population mean. And this mean is exactly the mean of the sampling distribution of means. (4) Why may or may not the observed mean be equal to the population mean? It is because when we draw a sample, we will make errors called “sampling errors” (any sampling procedure will yield sampling errors, including random selections). Thus, the observed sample mean stems from two resources: “sampling error + true population mean” (c) In our example, the observed sample mean is 535 but we hypothesize the population mean to be 455. The difference between the two numbers is 80. Since we know observed sample mean = sampling error + true population mean, there are at least two possible reasons to account for why the observed mean is 535: (1) Possible Explanation 1: true population mean = 455, sampling errors = 80 Thus, Ho: = 455 So, we retain Ho: = 455, but reject Ha: 455 (2) Possible Explanation 2: true population mean ≠455, sample errors ≠ 80 (i.e., either > or < 80), but sampling error + true population mean = 535 Thus, Ha: 455 So, we reject Ho: = 455, but retain Ha: 455 (d) Now we can test the possibility (i.e., probability) of the two possible explanations by estimating whether the probability of the sample error is 80 or not. (e) Remember: [1) the mean of the sampling distribution of means is the population mean and a normal distribution; (2) the standard deviation (standard error) of the sampling distribution of means is /√n . (f) Now, we can compute the probability of the occurrence of 535 when it is put into the sampling distribution of means whose mean is hypothesized as 455. Z = x1 –x2/SD -> Z score of 535: x1 = 535 x 2= = 455 SD = Standard Error = /√n = 100 / √144 = 100/12 = 8.33 Z = x- /( /√n) = 535-455/8.33= 80/8.33 = 9.6 (g) What does Z = 9.6 tell us? (1) Check the standard normal distribution graph. Z = 9.6 will fall extremely farther on the right end of the distribution. In other words, the probability for Z = 9.6 to take place in this standard normal distribution is very extremely low. (2) Check Z score Table to find the large portion of Z = 9.6. 1- the large portion = probability for Z = 9.6 to take place when = 455. The value of probability means the chance for Ho to be retained. The higher the value is, the higher probability is and the more likely Ho is retained. (3) z = 3.2905 (see Table C.2, p. 621), the larger area = .9995, probability = 1-.99995 = .0005. Probability for z = 9.6 will be much smaller than .0005. (4) In other words, for a population whose mean is hypothesized as 455 (in the z score distribution, the mean will be 0), the probability for this sample mean, 535, (in the z score distribution is 9.6) is less than 0.0005. Thus, based on our sample mean = 535, it is extremely impossible that the population mean will be 455. (h) Results: Look at the two hypotheses: Ho: = 455 Ha: 455 Probability for Ho to be retained is less than 0.0005. (i) Conclusion: (1) Based on the statistics of the sample mean, that is, 535, the probability for the hypothesized population mean = 455 to take place is less than 0.00003. In other words, the probability for Ho: = 455 to be retained is less than 0.0005. Because this probability is too low, Ho: = 455 should NOT be retained. Ho: = 455 should be rejected. Instead, Ha: 455 should be retained. Thus, based on our sample mean, 535, it is extremely impossible that the population mean will be 455. (2) Hence, the difference between the hypothesized population mean and the observed sample mean is so huge and so significant that the sampling errors can not be used to explain this difference (in this case, the sampling error should be less than 80). (3) The probability for = 455 is smaller than .05 [p <.05]; the difference between the sample mean and assumed population mean is statistically significant. The population mean should not be 455. Criterion for Rejecting Ho: 1. Researchers can set up the region of rejection (in the normal distribution) to reject a Ho. 2. This proportion of area is referred to “level of significance” and notated as , i.e., is a probability) 3. It equals the maximum probability of rejecting Ho. 4. In the field of language education, is usually set at 0.05. can also be set at 0.1, 0.01, 0.001, or even smaller. 5. Depending on the type of distribution used, a cutting value, i.e., critical values in the statistic term, will be computed. 6. For the z distribution, if = .05, then the critical value of z score for rejecting or retaining Ho will be 1.6449 (check Table C.2 on p. 621.) Step 1: 1-.05 = .95 Step 2: Find B (large area) = .950 from Table C.2 Step 3: Find z score corresponding to B = .950 Step 4: z = 1.6449 If = 0.025, the critical z score will be 1.96 7. The smaller value of an : (1) the smaller rejecting area (2) the more difficult to reject the Ho (3) the less likely to reject the Ho (4) the more likely to accept Ho But (5) the easier to reject Ha (b) the more likely to reject Ha (7) the more difficult to accept Ha (8) the less likely to accept Ha (9) a larger critical value. 8. The smaller an , the more conservative it is. In the field of medicine, is conventionally set a very conservative level such as 0.01 or 0.001. It is because an extremely conservative will make it very unlikely to reject Ho. Thus, in order to reject Ho in favor of Ha, the differences between two means must be very, very large. In medicinal experiments, taking new drugs must make huge differences in patients in order to reject the Ho and accept the Ha. So, a very conservative is used. 9. If the p values are lower than the level of significance, i.e., , it means the probability to retain Ho is too small. Thus, Ho should be rejected. E.g., p = 0.00034 < = 0.05, i.e., p < 0.05, it means the probability to accept Ho is too small. So, rejecting Ho and accepting Ha. Results can be written as: The difference is statistically significant at the level of 0.05. 10. If the p value is larger than the level of significance, i.e., , it means the probability to retain Ho is high. E.g., p = 0.45 < = 0.05, , i.e., p > 0.05 it means the probability to accept Ho is huge. So, retaining Ho and rejecting Ha. Results can be written as: The difference is not significant at the level of 0.05. Important Note: The p-value indicates the probability to accept Ho Errors in Hypothesis Testing: 1. Meaning: No matter Ho is rejected or retained, researchers will take the risk of making errors in their decisions. 2. Two Types of Errors: (1) Type I Error: a. Meaning: Rejecting a true null hypothesis b. Explanation: Researchers decide to reject a null hypothesis, but this null hypothesis is actually true. c. Reasons: Researchers reject the Ho because the sample mean falls in the rejecting area. That is, the probability for the Ho to be retained is very low (but not 0). However, Ho may still stand true, although it is very unlikely. Because the probability for the Ho to be retained is not 0 (e.g., p-value = 0.0001), there should be a very slightest chance (i.e., 0.0001) that the Ho may be true and should be retained. Researchers decide to reject the Ho because of its low probability, not because the probability is zero. Thus, when the Ho is rejected, researchers may make a Type I error since the Ho can be true but it has been rejected. (2) Type II Error: a. Meaning: Retaining a false null hypothesis b. Explanation: Researchers decide to retain a null hypothesis, but this null hypothesis is actually false. c. Reasons: Researchers retain the Ho because the sample mean does not fall in the rejecting area. That is, the probability for the Ho to be retained is very high (but not 1). However, Ho may still stand false, although it is very unlikely. Because the probability for the Ho to be retained is not 1 (e.g., p = 0.99), there should be a very slightest chance (i.e., 0.01) that the Ho should be rejected because it’s false. Thus, when the Ho is retained, researchers may make a Type II error since the Ho can be false but it has been retained. Level of Confidence [信心水準]: In a survey study, the opposite of value (1- ] . (a) Meaning: the degree of confidence that researchers have in not making the error when they decide to reject Ho. (b) Interpretation: if the = .05, researchers realize when a null hypothesis is rejected, they may make a mistake 5 times out of 100 times. That is, they are 95% confident in their results of rejecting Ho. Levels of Significance vs. Types of Errors: (1) if the value of is reduced from 0.05 to 0.01, the probability of making Type I error decreases, whereas the probability of making Type II error increases. (2) if the value of is raised from 0.01 to 0.05, the probability of making Type I error increases, whereas the probability of making Type II error decreases. (3) = 0.05 is more likely to reject Ho than = 0.01 (Why?) 5. One-Tailed [單尾] vs. Two-Tailed [雙尾] Hypothesis Testing: (1) One-Tailed (Directional): Ho: 455; Ha: 455 (2) Two-Tailed (Non-Directional): Ho: = 455; Ha: 455 (3) Critical values [決斷值] for rejecting Ho is different: Figures 7.6 (p. 178), Figure 7.7 & 7.8 (p. 180/181) one-tailed critical value of z = 1.645 (critical value = 1.645) as = .05, whereas two-tailed z = 1.96 (4) One-tailed is more likely to reject Ho (Why?) Standard Sampling Distribution of the Mean: Using Student’s t distribution if is unknown: (1) Standard error: s/√n as s: standard deviation of the sample Student’s t distributions: a. For small samples, sampling distribution of the mean departs considerably away from normal distributions; b. a family of distributions; c. as sample sizes increase (n=30), distributions of sampling distribution of the mean approximates normal distribution; d. t distributions with a mean equal to 0 and SD = 1 (Exactly like z- score distribution) e. All testing procedures are identical to z-score distribution f. Each t distribution is related to degree of freedom (df] [自由度): n-1 (1) df: the number of elements of data that are free to vary in calculating a statistic. [2] why df: each t distribution responds to a df. [3] if one restriction is added, the number of freedom will be one less. [4] x = Σ/n, but s = √Σ（x-x）2/n-1 -> x is added as a restriction; thus, df = n-1 [5] when df increases, t distribution approximates normal distribution; when df=29, t distribution becomes a normal distribution. Thus, the statistic techniques that use t distributions must have a df over 29, that is, n at least is 30. g. Critical values for t distributions can be found in Table c.3, p. 622. Statistical Precision: (1) the inverse of a standard error; (2) the smaller a standard error is, the greater the statistical precision; (3) as the sample size is increased, the precision is increased accordingly. Example: Scenario: A researcher hypothesizes that GPA of student athletes is less than 2.5. To test this hypothesis, the researcher selects 20 subjects and find the GPA mean of this sample is 2.45, s = 0.54, s2 = 0.29, is set as 0.05 level. What inferences can the researcher make? Computation: 1. t distributions is used since population variance is not known. 2. Hypotheses: Ho: = 2.5; Ha: ≠ 2.5 3. t = x - /standard error 4. standard error = s /√n -> 0.54/ √20 = 0.12 5. t = 2.45-2.50/0.12 = -0.42 6. = 0.05, df=19, one-tailed t critical value = -1.729 7. t = -0.42 > -1.729, outside the Ho rejection area; thus, Ho should not be rejected but retained -> = 2.5 8. = 2.5, p > .05 means: a. The difference in the observed mean and hypothesized mean is very small (i.e., nonsignificant); b. The probability for sampling error to account for this difference is higher than 5%; c. The probability for Ho to be true is larger than .05 Confidence Interval (CI: 信賴區間): The estimation of (a) CI: A range of values that we are confident contains the population parameter (i.e., ]. [b] CI = X± (tcv)*standard error [c] E.g.: A researcher hypothesizes that GPA of student athletes is less than 2.5. To test this hypothesis, the researcher selects 20 subjects and find the GPA mean of this sample is 2.45, s = 0.54, s2 = 0.29, is set as 0.05 level. What is the CI? Computation: 1. standard error = s /√n -> 0.54/ √20 = 0.12 2. Critical Values of t when n = 20, df= 19, = 0.05, two-tailed -> tcv= 2.093 3. CI95 = 2.45 ±[2.093]*0.12 = 2.45 ± 0.25 = (2.20, 2.70) Interpretation: We are 95% confident that the will fall between 2.20 and 2.70. Statistics’ A, B, & C A: In theory, an infinite number of samples can be selected from a population. All the possible sample means will be normally distributed. B: The mean of all the possible sample means is the hypothesized population mean. C: For a particular sample mean, there is a corresponding probability. This probability indicates the chance for the null hypothesis to be true.