PYC3704 ASSIGNMENT PACK 2024

PYC3704-exam-prep - Assignments questions and answers Psychological Research (University of South Africa) lOMoARcPSD| PYC3704 – Psychological Research Topic 1 Inferential Statistics – concerned with inferring numerical properties of statistical populations from sample data Theory:  Accounts for facts and suggest how they are related to each other. A framework of ideas that provides an explanation of something  The theory – if true – implies what we should observe under certain specific circumstances Psychological research: Testing theories of human behaviour / testing of theories against observations  Purpose is to gather and organise data o  Build theories and test them empirically, through observation Theory: A network of postulated relations between constructs / concepts / variables Constructs:  building blocks of theory  theoretical in nature  deliberately created / proposed to explain certain observations  often refer to a hypothetical aspect of a person / situation  cannot be observed, has to be inferred from behaviour  i.e. anxiety = sweat, empty sensation in stomach, heart palpitations are all symptoms caused by unseen underlying process  Anxiety = LATENT variable - hidden / intervening  Behaviour instances of sweating / palpitations = MANIFEST variables  Variable aka indicators / referents / observable consequences / implications  Based on the observable behaviour of an abstract construct / concept – we can derive an anxiety score  To arrive at a score – we must provide a measured operational definition of the construct; a measurement model for each construct  By identifying observable behavioural instances of a construct – one can measure it  DEPENDENT VARIABLE - Variable that is influenced – the EFFECT  INDEPENDENT VARIABLE – Variable that influences – the CAUSE OPERATIONAL DEFINITION : Specifying observable instances of a construct 2 KINDS 1. THEORETICAL definition of a construct – where a construct is defined in terms of other constructs 2. OPERATIONAL definition of psychological constructs – constructs defined in terms of observable instances (necessary for measurement) - allows us to bridge the gap between theoretical constructs and observations by spelling out what must be done to measure the construct  OPERATIONALISING OF CONSTRUCTS – process followed to make abstract concepts empirically observable – making them measurable – transforming theoretical constructs into empirical constructs  INDICATOR – an observable measure of a construct lOMoARcPSD|  Intelligence = construct  Vocab and numerical competence = indicators of the construct intelligence  Based on indicators one can specify behaviours associated with the constructs NOMINAL LEVELS OF MEASUREMENT  Labels – just a name  Can’t measure the difference  Discrete – distinct & separate  Mutually exclusive - choosing one automatically excludes the other  Exhaustive – makes provision for all possible variables ORDINAL LEVELS  When data needs to be ranked in an order of importance  Least to most / always to never  The degree of difference cannot be measured INTERVAL  Same characteristics of nominal and ordinal BUT – can measure the difference or intervals between 2 points  Numbers have a value but no absolute zero point RATIO  Highest level of measurement  Length / weight / time The HYPOTHESIS A statement of relation (or absence thereof) between 2 / more constructs or variables Y = exam performance – dependent variable X = Independent variables (cause for performance) U = unknown causes There are multiple influences on Y The hypothesis - will give a rule that associates values of one construct with those of another construct - will suggest the population for which the relation holds Operational forms of the hypothesis Operational hypothesis: using observations of measurable forms of the construct A = General hypothesis: “Anxiety influences performance negatively” B = “test anxiety influences exam performance negatively” C = Operational hypothesis: “the higher i-pat anxiety scores, the poorer PYC3704 students’ performance” In C, construct anxiety has been operationally defined as i-pat and performance Downloaded by Thomas Mboya () lOMoARcPSD| STATISTICAL HYPOTHESES Comparison of GROUPS DESIGN with CORRELATIONAL designs GROUPS design: research can define a population of subjects for each of the values of the independent variable  “Employees subjected to training based on experiential learning process (independent variable) will, as a result, tend to display a more positive attitude (dependent variable) compared to employees that did not receive training (2nd population group)  The hypothesis will clearly state that there are two population groups, that these groups will be tested and compared with regard to the dependent variable  Sample of population selected randomly  Researcher controls size of each sample of each of 2 distinct populations  Comparison of Pop A with Pop B will allow us to conclude if a relation exists (or not) between the independent and dependent variable  Key word = COMPARE CORRELATIONAL design  “Male students perform better in PYC3704 exams than do female students”  Select single random sample of 200 students  We have no control over number of males vs. females  Number of males vs. females is outcome of random process  Key word = RELATIONSHIP  You do NOT select populations associated with two levels of the independent variable as you would in the groups design. Topic 2: Probability Probability = a measure of uncertainty  Research findings have many potential sources of error and bias, therefore not accepted as conclusive  Therefore evaluated in terms of likelihood or probability Probability Theory Estimate of likelihood of something happening / an event – where various outcomes (results) are possible PROBABILITY OF EVENT {P (E) = Number of favourable events (specific events) _______________________________________ No of possible outcomes Probability that toss of fair coin will land on heads = ½ Probability of drawing an ACE from a pack of cards = 4 (aces) / 52 cards = 1/13 All outcomes are equally probable and we know the number of possible outcomes The theoretical probability of an event occurring can be approximated by the RELATIVE FREQUENCY = the proportion of times that an event occurs Downloaded by Thomas Mboya () lOMoARcPSD| P (E) = Number of observations of E ------------------------------------------------------ = f(E) / N Number of times experiment performed OBSERVED RELATIVE FREQUENCY = an approximation of the true probability of an event occurring if experiment is repeated an infinite number of times If we repeat it more and more – the relative frequency would approach the theoretical probability  Two events are INDEPENDENT if the occurrence of one has no effect on the probability of the other occurring. The composition of the one sample in no way affects the composition of the other  An example of two independent events is as follows; say you rolled a die and flipped a coin. The probability of getting any number face on the die in no way influences the probability of getting a head or a tail on the coin.  Events are MUTUALLY EXCLUSIVE if occurrence of one prevents the occurrence of the other - they cannot occur at the same time Events are DEPENDENT if the fact that probability A occurring DOES affect the probability of B occurring. The likelihood of the second event depends on what happens in the first event. SAMPLE SPACE / POPULATION of an experiment = all possible outcomes = “S” I flip of coin = S: {heads; tails} – two possible outcomes LAW OF LARGE NUMBERS  If an experiment is done repeatedly  Outcomes are independent of one another  The observed proportion of favourable occurrences of the event will eventually approach its theoretical probability In coin flip – chances of heads or tails is not influenced by previous flip All flips are independent of each other P (E) = favourable outcomes / total outcomes = ½ = 0.5 If we repeatedly flip coin, the proportion of heads will eventually get close to 0.5 Relative frequency = theoretical probability  According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. For example, a single roll of a six-sided die produces one of the numbers 1, 2, 3, 4, 5, 6, each with equal probability. Therefore, the expected value of a single die roll is According to the law of large numbers, if a large number of dice are rolled, the average of their values (the sample mean) is likely to be close to 3.5, with the accuracy increasing as more dice are rolled. Downloaded by Thomas Mboya () lOMoARcPSD| Characteristics and Rules of probability Probability value = p-value of the event occurring / a measure of likelihood  Always lies between 1 and 0  Sum of all probabilities of all events in a sample space (S) = 1  0 ≥ P ≤ 1  0 = complete certainty event WON’T happen  1 = complete certainty event WILL happen If we get a p-value of 0.20 of an event happening we can infer that there is a 0.80 probability of an event NOT happening  When 2 / more events are mutually exclusive (they cannot occur at the same time) – we use ADDITIVE RULE: P: (A or B) = P(A) + P(B) Sum of probabilities Look for word OR – heads or tails = 0.5 + 0.5 = 1 Independent events allows for overlap between probabilities Probability of drawing ace or heart from pack of cards P(ACE) = 4/52 = 1/13 P (HEART) =13/52 = ¼ P(ACE or HEART) = 1/13 + ¼ - 1/52 (as one of the hearts is an ace and will be counted twice) MULTIPLICATIVE RULE = P(A & B) = P(A) X P(B) Look for word AND Probability of next two cards being a circle and a star P(Circle and Star) = P(02. X 02.) = 0.04 CONDITIONAL PROBABILITY - a particular probability is conditional on something else happening Event A conditional on Event B happening = P (A B) THE PROBABILITY MODEL  Sampling space of a coin = 2 possible outcomes of heads or tails – both equally likely = ½  Sampling space of a dice = 6 possible outcomes = 1/6 - each outcome equally likely - each # has a 1 in 6 chance of being rolled FREQUENCY DISTRIBUTION How observations are distributed / the frequency of a score THE NORMAL CURVE  Cumulative probability equated with the area under the curve  Discrete Variables = variables that take whole numbers / integers as values (1,2,3)  Continuous Variables = age, weight, length – can take on any value i.e. age = 38yrs, 2 months, 6 days, 3 hrs and 30 min.  There is NO GAP in the scale  Therefore sampling space (possible outcomes) between two peoples ages will be infinite  Age of 100 men recorded, how many are 30?  P = # no 30yr olds / possible outcomes = # of 30 yrs olds / infinity = 0 Downloaded by Thomas Mboya () lOMoARcPSD|  Therefore – The probability of any value in a continuous variable = 0 – therefore probability does not work with continuous variables – therefore use normal curve / bell curve that can be plotted by distribution  If the curve is based on sufficiently large data set, it will eventually approach the theoretical probability distribution – i.e. majority of scores clustered around the average and tail off the ends of the distribution  Shape of the curve depends on the mean ( μ ) and the standard deviation ( σ )  Most observations occurring at the mid-point of the curve  Symmetrical - area under the curve to the right of the mean = area to the left of the mean (0 to ∞ = ∞ to 0  Continuous  Curves are asymptotic – the two tails never touch the horizontal axis  Representative curve has μ of 0 and a σ of 1  Area below curve provides probability of an event  Total area under curve = 1  0.5 of the area lies to left of mean and 0.5 of the area lies to right of mean  Any movement away from the mean will divide the distribution into a larger and smaller portion  At σ o= 1, larger portion = 0.8413 and smaller = 0.1587 Measure on x axis = Z-SCORES  z-score = the number of standard deviations that a particular scores lies above or below the mean  Magnitude of z-score tells us how many standard deviations a score is from the mean  The sign (+ or -) tells us if score is greater than or smaller than mean. Student gets 56% in an exam and wants to know where she stands in relation to other students _ Mean mark = 52% (x) Standard deviation = 4 (s) Assume normal distribution Z = x – mean of x / s = 56 – 52 / 4 = 4/4 = 1 Therefore students mark is 1 std. deviation (s) above the average Using z-table – 1 = 0.84 (larger portion as it’s above average) Student did better than 0.84 of the group If mean of x = 65 and s = 6 56 – 65/6 = -9/6 = -1.5 Below average by 1.5, therefore 0.066 (smaller portion) _ Z DISTRIBUTION: z = x – μ / σ (for a population) and z= x – x / s (for a sample)  Variability of scores – how they are spread out / differ from one another  Variability has a variance and a standard deviation _ Mean – add up all scores and ÷ by number of scores = x X = raw scores N = number of scores _ Variability determined by subtracting the x (mean) of the sample from each raw score _ Downloaded by Thomas Mboya () lOMoARcPSD| DEVIATION SCORE = x – x s 2 = VARIANCE s = STANDARD DEVIATION: s = √s 2 : used to indicate average extent to which scores in a distribution differ from one another SAMPLING AND SAMPLING DISTRIBUTION  Purpose of sampling – use small number of cases to draw conclusions about a larger group  Large group = population  Summary value for populations = parameters  Small group taken = sample  Summary value for samples = statistics Apply results of sample to the population LAW OF DIMINISHING RETURNS Once a certain number of cases are studied, each successive case will not add to our understanding of the results 1. RANDOM SAMPLING: drawing a sample from a population in such a way that every possible sample of a particular size has the same probability of being selected (names in a hat) 2. SYSTEMATIC SAMPLING: Selecting individuals at fixed intervals 3. STRATIFIED SAMPLING: dividing population into homogeneous subgroups and drawing random samples from subgroups 4. CLUSTER SAMPLING: sampling individuals from well-delineated areas / clusters – who have characteristics found in the larger population There is NO guarantee that random sample will represent characteristics of population – there will be a MARGIN OF ERROR  Values for population parameters seldom known  Value of sample statistics calculated and can be used as estimates for corresponding population  Samples will be different even if selected from same population, therefore statistics will vary sample to sample SAMPLING DISTRIBUTION OF A STATISTIC (MEAN)  The set of all possible values of the statistic (mean) when all possible samples of a fixed size are taken from the population  i.e. the variation of the statistic (mean) from sample to sample  mean of sample provides accurate estimate of the population mean CENTRAL LIMIT THEOREM  If a simple random sample of size n is selected from a population with mean µ and a standard deviation σ , the sampling distribution of means obtained from all possible samples is approx. normal with mean µ and a standard deviation σ / √n  DISTRIBUTION BECOMES MORE NORMAL AS THE SAMPLE SIZE INCREASES _ STANDARD ERROR OF THE MEAN = STANDARD DEVIATION OF THE SAMPLE MEAN: σ x By what average amount the sample mean deviates from the population mean of the sampling distribution Standard error = σ / √n = population deviation / square root of sample size Downloaded by Thomas Mboya () lOMoARcPSD| It specifies how well a sample mean approximates the population mean - how much error between _ x and µ 1. AMOUNT OF ERROR DECREASES WITH SAMPLE SIZE – BIGGER SAMPLE = SMALLER ERROR 2. THEREFORE X MORE CLOSELY APPROXIMATES µ FOR LARGER AND LARGER SAMPLES 3. Sample size increases – the standard deviation gets smaller Ho = the null hypothesis H1 = the alternative hypothesis  We test Ho directly and try to reject the null hypothesis so that we can accept H1  ∝ = level of significance determined in advance  DECISION RULE: if p-value is smaller than or equal to ≤ the level of significance ∝ , then reject H1  P-value is the probability of the result occurring under Ho  H1 is directional due to smaller than / larger than sign  Two-tailed – non-directional test – p-value is twice that of a directional test Topic 3 – General Principals of statistical hypothesis testing Theory: a network of relations between constructs Research Hypothesis: a relation proposed between 2 / more constructs BASIC PURPOSE OF RESEARCH IS TO TEST THEORY 1. Refine research hypothesis to get OPERATIONAL HYPOTHESIS – which specifies: - How constructs are measured - What is research population - What design used to test relation - Nature / rule of relation 2. Translate operational hypothesis into STATISTICAL HYPOTHESIS to test if possible relations exist. Statistical hypothesis is a statement about the value of a particular population parameter Research hypoth: ‘Unisa students tend to have higher intelligence scores than others’ Two populations – Unisa students and others – therefore groups design The distribution of IQ scores between the 2 populations being compared Ho: µ = 100 Mean IQ of others is 100 The null hypothesis: no relationship between two measured phenomena. It can never be proven, only rejected or fail to reject The assertion that the things you are testing is not related and your results are the product of random chance events The null hypothesis is the hypothesis that would say there is no difference between the specified group (short) and the general population (men in general). Since the mean score of the population is 20, Ho would just be Ho: µ=20 (assuming µ is the mean for short men. So µ (for short men) is compared with 20 (the population mean for all men), and the null hypothesis say they do not differ. Downloaded by Thomas Mboya () lOMoARcPSD| H1: µ ¿ 100 Mean IQ of Unisa students is greater than 100 The alternative hypothesis: is the expected conclusion (why the research was completed in the first place). “In terms of their level of intelligence, UNISA students tend to differ from others” Ho: µ = 100 Mean IQ of others is 100 (try to reject this) H1: µ ≠ 100 Mean IQ of Unisa students differs from others The object is to try and reject Ho in favor of H1 64 Unisa students n = 64 σ = 16 Sample mean of IQ score = 104 Sampling distribution of the mean under Ho: a frequency distribution of all the values of the mean we would get should we repeatedly select a random sample: SAMPLING ERROR If Ho is true – then all samples would have a mean of 100 _ Sampling distribution of the mean = σ x = σ / √n = 16/ √64 = 2 (Std. error of the mean – diff between sample and pop. Mean) Z-score of sample mean: z = mean of x - μx / σ x = 104 – 100/2 = 2 (# of std. dev. Above mean) p-value = 0.022 (one tailed – right hand tail of sampling distribution because of H1 ¿ 100 Ho states that true population mean Ho: µ = 100 and any sample result (104) is due to chance / random sampling error P-value expresses a 0.022 chance of random sampling error A small p-value indicates the probability of the sample result occurring under Ho is small – therefore Reject Ho in favour of H1. If Ho is true, there is a 0.022 chance that 100 is likely Sampling distribution of the mean (Std error) σ x derived under Ho not H1, because H1 does not Have a specific mean of the population distribution (H1: µ 100) Statistical hypothesis testing is about testing Ho The two-tailed / Non-directional test Ho: µ = 100 Mean IQ of others is 100 (try to reject this) H1: µ ≠ 100 True pop mean is either ¿∨¿ 100 but not equal to  Non-directional p-value is twice the size of on-tailed p-value  Therefore 2 tailed p-value of 0.04 must be divided by 2 to get the one-tailed P-value indicates the likelihood of the result under Ho TEST STATISTICS = zx, tx, x 2 : A variable which has a known, theoretical probability distribution Transform sample mean (104) to z-score (refer to z-table) Downloaded by Thomas Mboya () lOMoARcPSD| Value in z-table is the z-statistic – the probability associated with a set of values  Before research is conducted , decide a level of significance ∝  What size p-value is considered small enough to justify rejecting Ho  A cut off value = 0.05 / 0.01  DECISION RULE: If p-value of sample results is SMALLER than ∝ - then reject Ho If P-value is not smaller, then Ho is NOT REJECTED / WE FAIL TO REJECT  If Ho is rejected it implies that research hypothesis is CONFIRMED. P-value gives the probability that Ho is MISTAKENLY REJECTED – TYPE 1 ERROR  We will never know if Ho is true or false as we cannot study an entire population  Level of significance ∝ limits the probability of making a Type 1 error as it sets the maximum  probability  If p-value greater than () level of significance - Ho not rejected = research hypoth. NOT confirmed  Then risk of TYPE 2 ERROR – not rejecting Ho when Ho is actually false and H1 is true .  β indicates risk of Type 2 error.  Cannot set β in advance because H1 does not specify a mean, therefore can’t get sampling  distribution of the mean under H1  Smaller ∝ = larger β Statistically significant  This refers to the fact that a statistical result can be significant but not very useful from a practical point of view.  Statistical significance simply means that the result (for example, a difference between means or a correlation between variables) is greater than can be expected by chance, but the actual effect can be very small. If you have a very large sample, like 10000 people, even a small difference or relationship may be statistically significant, but it may be unimportant from a practical point of view. To INCREASE the power of a Statistical test we: 1. Increase the sample size 2. Decrease sampling error, measurement error, error due to external variables t-Test (for Independent and Dependent Samples). The t-test is the most commonly used method to evaluate the differences in means between two groups. The groups can be independent (e.g., blood pressure of patients who were given a drug vs. a control group who received a placebo) or dependent (e.g., blood pressure of patients "before" vs. "after" they received a drug, see below). Theoretically, the t-test can be used even if the sample sizes are very small (e.g., as small as 10; some researchers claim that even smaller n's are possible), as long as the variables are approximately normally distributed and the variation of scores in the two groups is not reliably different Dependent samples test. The t-test for dependent samples can be used to analyze designs in which the within-group variation (normally contributing to the error of the measurement) can be easily identified and excluded from the analysis. Specifically, if the two groups of measurements (that are to be compared) are based on the same sample of observation units (e.g., subjects) that were tested twice (e.g., before and after a treatment), then a considerable part of the within-group variation in both groups of scores can be attributed to the initial individual differences between the observations Downloaded by Thomas Mboya () lOMoARcPSD| and thus accounted for (i.e., subtracted from the error). This, in turn, increases the sensitivity of the design. One-sample test. In so-called one-sample t-test, the observed mean (from a single sample) is compared to an expected (or reference) mean of the population (e.g., some theoretical mean), and the variation in the population is estimated based on the variation in the observed sample. Topic 4: Statistical Hypothesis Testing: Single Sample Group Designs (tx) GROUPS DESIGN = a design which compares 2 / more population means or proportions Statistical hypoth. Must be stated in terms of parameters (means)- but actual comparison of parameters is performed on samples selected from population Researcher makes inferences about the population from the observation of the sample.  When comparing two groups on some variable of interest (males and females and aggression): draw two samples from general research population  Statistical procedure will be determining if 2 samples seem to come from different populations  Researching if gender (independent variable) has sufficient effect on the dependent variable (aggression) = TWO GROUP GROUPS DESIGN: When 2 samples selected from a population i.e. males and females - comparing samples from different groups  When properties of one group are known, then researcher would only select one sample from where the properties unknown = SINGLE SAMPLE GROUPS DESIGN – interested in a single variable form a single sample  CORRELATIONAL DESIGN: emphasis is on measure of the relationship / association between two variables from a single sample in order to establish difference between 2 means. Statistical test for means tx A test statistic (such as a t-statistic) measures how far the observed values differ from the expected values “under the null hypothesis.” That is to say, you compare the values (for example difference between means) which you find in your sample with what you would expect these values to be if the null hypothesis were true. The further the observed value differs from the expected value, the larger the test statistic is likely to be. For example, if you are comparing two means with a t-test, you expect the difference to be zero if the null hypothesis were true, so you compare the observed difference between the two means in your two samples with an expected difference of zero. The further from zero this is, the bigger your t-statistic would be. The p-value is the probability of this outcome (getting a test statistic of the particular size which you calculated). A larger value for the test statistic is less probable than a smaller one. The greater the test statistic is, the less likely (or less probable) it is that this is purely a chance event. This implies that the smaller the p-value, the more probable that the null hypothesis is not true. In summary: in general, the bigger the test statistic, the smaller the p-value, and the more likely it becomes that the alternative hypothesis is valid. Effect of anxiety on performance