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  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 1 INFERENTIAL STATISTICS
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 2 Inferential Statistics The problem with descriptive statistics is that they may be too simplistic. Suppose you are interested in finding out whether there is a difference in memory between 16 year olds and 25 year olds. You make the following hypotheses: - Null Hypothesis: There will be no significant difference between the number of words correctly recalled on a memory test between 16 year olds and 25 year olds. - Alternate hypothesis: There will be a significant difference between the number of words correctly recalled on a memory test between 16 year olds and 25 year olds. You carry out an experiment to test your hypotheses, and summarise the data by finding the mean: - Mean score for 16 year olds: 49.1 - Mean score for 25 year olds: 55 This is a difference of 5.9 points. BUT…is this enough of a difference to be able to conclude that the 25 year olds’ scores were genuinely and significantly worse? Moreover, is this enough of a difference to reject the null hypothesis and accept the alternate hypothesis? Or could this result have occurred simply due to chance factors? An inferential statistics test helps us decide, by working out the probability of the difference between the two sets of scores occurring just due to chance (rather than because of any real effect). This probability (p) is expressed as a number between 0.00 and 1.00 which can be converted to a percentage. E.g. 0.00 = 0% chance, 1.00 = 100% chance. 0.50 = ……..…. 0.10 = …..……. 0.05 = ……..…. 0.01 = ………... Whenever we conduct an inferential stats test, we must select a probability value. So if I find a difference between the memory scores for 25 years and 16 year olds and I have used the p<0.5 level of significance. Then this means there is a 50% probability that the difference I found was due to chance (fluke). However, if I find a difference using the p<0.05 level of significance, then there is only a 5% probability that the difference I found was due to chance. Levels of significance In general, psychologists use the 0.05 level of significance; meaning that there is a 5% chance that the results would have occurred this way even if there was no real difference between the groups. This probability of 5% is recorded as p=0.05
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 3 Type 1 and type 2 errors In most circumstances we can accept the alternate hypothesis when there is a 5% probability that results occurred due to chance factors (and therefore 95% probability that results occurred due to real differences between the two conditions). However, we are dealing with probabilities and not absolute certainties. This means we can never be absolutely sure that our results mean what we think they mean. Even with a 5% probability that the results occurred sue to chance factors, we are only 95% certain. This means that sometimes we might accept the alternate hypothesis when actually the null hypothesis was true – a false positive. Conversely, there may be times when we reject the alternate hypothesis and accept the null hypothesis when actually the alternate hypothesis was true – a false negative. They are called type 1 and type 2 errors respectively. Type 1 error False positive Accept alternate hypothesis although really the null hypothesis was true. (Reject null hypothesis when actually it was true). Type 2 error False negative Accept null hypothesis although really the alternate hypothesis was true. (Accept null hypothesis was actually it was false). Circle the level of significance is more likely to result in a type 1 error? 0.01 0.05 0.10 0.5 Explain your answer. ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………..…………………………………………………………………… Circle the level of significance is more likely to result in a type 2 error? 0.01 0.05 0.10 0.5 Explain your answer. ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………..…………………………………………………………………… You are conducting a study on the effect a new drug has on health… - What level of probability do you think you should use before the drug is made available on prescription? You think your best friend’s partner might be cheating on them… - What level of probability that you are right do you need before you tell them? View slide
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 4 View slide
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 5 Types of data (level of measurement) Previously we have simply considered data as with being qualitative (words) or quantitative (numerical). However now we must consider quantitative data more closely as the type of quantitative data we produce will govern which inferential test for statistical analysis we choose to use. When you are operationalizing the DV, there are several further issues to consider. What sort of data will it produce (level of measurement)? Will you be able to display the data on a bar chart? Would the data fit the requirements of any inferential statistical test? Example of data collected Level of measurement The score on an IQ test. Interval Participants are in one of three conditions: A 1 month programme of cardiovascular exercise, 1 month of weight training, and one month of no exercise. At the end of which they are asked “Are you happy in general with your life?” to which participants give a yes or no response. Nominal The top ten British soaps are given as a list in a questionnaire; participants must put them in an order of most preferred to least preferred. Ordinal The time taken to complete a cognitive task. The number of words remembered on a memory test. A rating scale asks you to rate how much you like chocolate on a scale of 1-10. A short questionnaire consisting of 20 questions on self-esteem is question is presented as a statement to which the participant has to respond ‘agree’ or ‘disagree’. Each ‘agree’ statement is worth one point and they are added together so that the participants receive a total score out of 20. Investing the number of doctors who are left-handed compared with the number of right-handed. An observation of whether males or females students are more likely to be late for lessons. The teacher marks each student who turns up before the bell as ‘on-time’ and any student who turns up after the bell as ‘late’. The number of micrograms of adrenaline in a urine test. An observation of a person’s body language, a count of how many times someone touches their ears when in conversation.
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 6
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 7
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 8 Types of Statistics tests for inferential analysis There are four main types of inferential tests that you need to know about. 1. Spearman’s Rho 2. Mann Whitney U 3. Wilcoxon matched pairs 4. Chi-Squared In order to know when to use which test you must know 1. The type of data used in the study 2. The design used in the study 3. Whether the study is looking for a relationship (correlation) or a difference between variables.
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 9 Choosing Inferential Statistical Tests of Analysis 1. A psychologist at the local university carries out a study to investigate the claim that eating a healthy breakfast improves reading skills. He has access to 400 five-year-old children from 10 local schools. In total 175 children attend a morning breakfast club in which a healthy breakfast (porridge and fruit) is given to the children. The remaining children do not attend the breakfast club. Each child is given the same reading comprehension test. Which inferential test should be used to analyse the results and why? ......................................................................................................................................... ......................................................................................................................................... ......................................................................................................................................... ......................................................................................................................................... 2. A psychologist was interested in the effects of a calorie restricted diet on memory functioning. The psychologist’s hypothesis was that participants’ scores on a memory test are lower after a calorie restricted diet (800 calories per day) than before a restricted diet. He gave the volunteers a memory test when they first arrived in the research unit and a similar test at the end of the four-week period during which participants had been on a calorie restricted diet. He recorded the memory scores on both tests. Which inferential test should be used to analyse the results and why? ......................................................................................................................................... ......................................................................................................................................... ......................................................................................................................................... ......................................................................................................................................... 3. A psychologist was interested in finding out about local people’s opinions on the building of the London 2012 Olympic Games development in Stratford, east London. She predicted that there may be an association between occupation and negative opinions of the Olympics development. A sample of 50 participants from 5 different occupations (teachers, students, business owners, administrative workers and factory workers) took part in the study. Each participant was given a questionnaire with 10 different Likert scale questions. Such as: ‘The 2012 Olympic games development will be of benefit to the whole community of Stratford, east London.’ - Strongly agree----------Agree----------Indifferent----------Disagree----------Strongly disagree
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 10 The psychologist was particularly interested in the strong negative opinions and so she only recorded the number of ‘strongly disagree’ responses given by each of the participants. Which inferential test should be used to analyse the results and why? ......................................................................................................................................... ......................................................................................................................................... ......................................................................................................................................... 4. A researcher had recruited 100 students by putting up posters around college. Half the students were given fish oil capsules (Omega-3) for a month. The other half, were given capsules that looked the same but contained only sugar. All students were asked to take an IQ (intelligence) test at the start of the experiment, and another one at the end of a month. Which inferential test should be used to analyse the results and why? ......................................................................................................................................... ......................................................................................................................................... ......................................................................................................................................... 5. Bickman (1974) tested the effects of perceived authority on obedience. Confederates dressed in a sports jacket and tie, a milkman’s uniform or as a guard, and made requests to passers-by, for example, asking them to pick up some litter. Bickman recorded whether or not the participant obeyed in each of the conditions. Which inferential test should be used to analyse the results and why? ......................................................................................................................................... ......................................................................................................................................... ......................................................................................................................................... 6. A psychologist was interested in the whether or not natural light affected happiness. He gave participants questionnaires to assess their levels of happiness. The higher the score on the questionnaire, the more happy the participant felt. Each participant was also asked to estimate the number of hours spent outside during day time. For each participant, the psychologist recorded a score for happiness as well as their number of hours spent outside during day time. Which inferential test should be used to analyse the results and why? ......................................................................................................................................... ......................................................................................................................................... .........................................................................................................................................
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 11
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 12
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 13
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 14
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 15
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 16 Additional Notes ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ….…………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …….………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……….……………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………….…………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …………….………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………….……………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………….…………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …………………….………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………….……………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 17 Chi-square (X2) When do I use Chi-square? - The Hypothesis states a difference between two variables or an association - The sets of data must be independent - The data are in frequencies – nominal data Example: Psychological research suggests an association between gender and choice of subjects for A Level. For example, maths teachers often report that they have more male students for maths than female students. A psychologist wonders whether there will be an association between gender and choice of subject for design technology. She decides to investigate this by asking the textiles and woodwork GCSE teachers at a local school, how many of their students are male and female. The psychologist found the following results: - 5 out of 17 textiles students were male - 10 out of 19 woodwork students were male 1. Write a non-directional hypothesis for this study ……………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………….………………… 2. Write a null hypothesis for this study ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… 3. Draw up a contingency table
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 18 4. Find observed value by comparing observed and expected frequencies for each cell in your contingency table. Row X column/total = expected frequency (e) Subtract expected value from observed value, ignoring signs (0- E) Square previous value (O-E)2 Divide previous value by expected value (O-E)2/E Cell A 17X15/36 = 7.08 5 - 7.08 = 2.08 4.3264 0.6110 Cell B 17X21/36 = 9.92 12 – 9.92 = 2.08 4.3264 0.4361 Cell C 19X15/36 = 7.92 10 – 7.92 = 2.08 4.3264 0.5463 Cell D 19X21/36 =11.08 9 – 11.08 = 2.08 4.3264 0.3905 X2 = sum of (O-E)2/E 5. Calculate degrees of freedom Calculate the degrees of freedom (df) by multiplying (rows – 1) X (columns – 1) 6. Find the critical value of chi square - Look up the value in the critical values table. Level of significance for a one-tailed test 0.10 0.05 0.25 0.01 Level of significance for a two-tailed test 0.20 0.10 0.05 0.02 1 (df) 1.64 2.71 3.84 5.41 2 3.22 4.60 5.99 7.82 3 4.64 6.25 7.82 9.84 4 5.99 7.78 9.49 11.67 Observed value of X2 must be EQUAL TO or GREATER THAN the critical value in this table for significance to be shown. - State the critical value For a ……….tailed test, df = ……., the critical value of chi square (p…………….) = 7. State the conclusion As the observed value (…….…) is …………………..…….. the critical value (……), we must ……………………... the null hypothesis (at p<0.05) and conclude that there is …………………………. ………………………………………………….……………………………………………………………………………….………… ……………………………………………………………………………………………………………………………………………… ……………………………….…………………………………………………………………………………...…………………………
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 19 Spearman’s Rho (p) Is there a correlation between the number of vitamin treatments a person has, and their score on a memory test? Null hypothesis: …………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………… Number of participants: N = __________ Pp. Variable One (number of vitamin treatments) Variable Two (score on memory test) Difference between Ranks (d) d2 Raw Score Rank Raw Score Rank 1 2 22 2 1 34 3 3 36 4 4 49 5 3 42 6 6 57 7 5 82 8 8 82 d2 = Note: There are two formulae that could be used to work out rho Step 1: Assign ranks to the raw data, for each variable separately. Rules for ranking: A. Give the lowest score a rank of 1; next lowest a rank of 2; etc. B. If two or more scores are identical, this is a "tie": give them the average of the ranks they would have obtained had they been different. The next score that is different gets the rank it would have had if the tied scores had not occurred. Step 2: Subtract one set of ranks from the other, to get a set of differences, D. Step 3: Square each of these differences, to get D2 . Step 4: Add up the values of D2 , to get ΣD2 =
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 20 Step 5: Multiple ΣD2 by 6 to get 6* ΣD2 = Step 6: N*(N2 -1) or N3 – 1 = Step 7: Table of critical values of rho State the conclusion with regards to the hypothesis As the observed value for rho (…………) is ………………………. than the critical value (……………) we could ……………………… the null hypothesis (at p…………….) and therefore conclude that………………………………..……………………………………………………………………………………………………… ……..……………………………………………………………………………………………………………………………………… …………………………………………………………………….………………………………………………………………………… Level of significance for a one- tailed test 0.05 0.01 Level of significance for a two- tailed test 0.10 0.02 N = 4 1.000 5 .900 1.000 6 .829 .886 7 .714 .786 8 .643 .738 9 .600 .700 10 .564 .648 11 .536 .618 12 .503 .587 13 .484 .560 14 .464 .538 15 .443 .521 16 .429 .503 17 .414 .485 18 .401 .472 19 .391 .460 20 .380 447 Observed value of rho must be EQUAL TO or GREATER THAN the critical value in this table for significance to be shown. State the observed and critical values The observed value for rho = ______ ; critical value for p  0.05; one tailed with N = ______ is ________.
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 21 Wilcoxon T Test How to do the Wilcoxon T Test 1. Work out the difference between the scores (subtract score two from score one) 2. Once you have worked out the difference, rank from low to high, ignoring the signs (i.e. the lowest number receives the rank of 1). Rules for ranking: A. If there are two or more of the same number calculate the rank by working out the mean of the ranks that would have been given. B. If the difference is zero, omit this from the ranking and reduce N accordingly. 3. Calculate T – work out the sum of the ranks of the less frequent sign. This is your observed T value. 4. Compare your observed T value with the critical value in the table. Participant Likability for more frequently seen face Likability for less frequently seen face Difference Rank 1 5 2 2 4 3 3 3 3 4 6 4 5 2 3 6 4 5 7 5 2 8 3 4 9 6 3 10 4 6 11 5 2 12 3 4 T = sum of the ranks of the less frequent sign When do I use a Wilcoxon T Test?  The hypothesis states a difference between two sets of data  The sets of data are pairs of scores from one person or a matched pair o Repeated measures design o Matched pairs design  Data are ordinal or interval The Study Alternative Hypothesis: Participants will rate the more frequently seen face as more likeable than the less frequently seen face. Null Hypothesis: ………………………………………………………………………………. …………….…………………………………………………….……………………………………… …………….…………………………………………………….……………………………………… …………….………….…………………………………………………….…………………………
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 22 Table of critical values of T Observed value of T must be EQUAL TO or LESS THAN the critical value in this table for the significance to be shown. State your critical value For a ……………..-tailed test, N =……………….., the critical value of T (p…………….)= ………………. State your conclusion As the observed value (……) is …………………..…….. the critical value (……), we must …………………….. the null hypothesis (at p<0.05) and conclude that there is ……………………………… in the likeability score for faces seen more or less frequently.
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 23 Mann Whitney U Test How to do the Mann Whitney U Test 1. Consider each score one at a time, compare this score – ‘the target’ with all the scores in the other group. - Give 1 point for every score that is higher than the target score - Give ½ point for every equal score 2. To get the observed value of u – this is the lower total number of allocated points 3. Finding the critical value of u - N1 = Number of participants in group 1 - N2 = Number of participants in group 2 When do I use a Wilcoxon T Test? • The Hypothesis states a difference between two sets of data • The two sets of data are from separate groups of participants – E.g. Independent groups design • The data are ordinal or interval The Study Alternative Hypothesis: Male participants interviewed on a high bridge will give higher ratings on the attractiveness of a female interviewer than those interviewed on a low bridge. Null Hypothesis: ………………………………………………………………………………. …………….…………………………………………………….……………………………………… …………….…………………………………………………….……………………………………… …………….………….…………………………………………………….…………………………
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 24 Critical Values tables State the critical value: For a ……….…….-tailed test, N1 = ……….... And N2 = ……..……, the critical value of U (p<…………….…) = ………………………… State your conclusion As the observed value (……..……) is ……………..……….than the critical value (……………), we can …………………….… the null hypothesis (at p…………..) and therefore conclude that participants interviewed on a high bridge ………………………………………….………………………………………… ….…………………………………………………………………………….…………………………………………….……………… ………………………….………………………………………….………………………………………….………………………… ….…………………………………………………………………………….…………………………………………….……………… ………………………….………………………………………….………………………………………….…………………………
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 25 Additional Notes ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ….…………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …….………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……….……………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………….…………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …………….………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………….……………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………….…………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …………………….………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………….……………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………
  • A2 Psychology [UNIT 4: SECTION C PSYCHOLOGICAL RESEARCH AND THE SCIENTIFIC METHOD ] 26 ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ….…………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …….………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……….……………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………….…………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …………….………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………….……………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ………………….…………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… …………………….………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………… ……………………….……………………………………………………………………………………………………………………… ………………………………………………………………………………………………………………………………………………