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STATS 305 CALCULATIONS QUESTIONS AND ANSWERS SOLUTIONS
How do you describe a sample? - By describing the Mean, Medium, and how each unit deviates from the mean (via standard deviation) How do you find the standard deviation - 1. subtract each unit by the mean to "center the data" ( 𝑥m−𝑥̅ )
- take the sum of all the squared centered data to get the "sum of squares" Σ ( 𝑥m−𝑥̅ ) ^
- divide that by df to get the "variance"
- square root the whole equation to get "the standard deviation" what does "centering the data" do when you calculate standard deviation - it converts all the data to the positive or negative deviations from the median. On a chart all the units would fall on a line on the x axis, centered around the median. Why do you square the deviations when you calculate sum of squares - if we added all the deviations together without squaring them we would get 0. We need a positive number. why do you divide by n-1 when you calculate variance - n-1 is used for any sample under 30. It accounts for the total variance in the population what determines how large or small the sum of squares will be in a given data set - sample size how does the variance equation change if you were calculating the variance within your sample alone (and not for the population?) - you could divide by n instead of n- why do you need to square root the variance to get the standard deviation - the value lies beyond the Y scale
How do we find if there is a relationship or correlation between certain variables - 1. plot the variable
- calculate correlation coefficient How do you calculate the correlation coefficient - 1. center the data for both variables (xm-xbar) x (ym- ybar)
- calculate the "sum of products", add all units deviated sum scores
- calculate the covariance by dividing this by n-
- calculate the correlation coefficient by dividing by the product of the two standard deviations what is the range of values that the correlation coefficient includes? - -1 to + How do we interpret correlation coefficients? - 1. strength, The closer to 1 the stronger
- directionality, - is negatively correlated, + is positively correlated What are the benchmarks for weak, moderate, and strong relationships? (according to Cohen) - r>.1 = weak relationship r>.3 = moderate relationship r>.5= strong relationship what strength? r>.1 - weak relationship what strength? r>.3 - moderate relationship what strength? r>.5 - strong relationship
is 𝑥̅ a descriptive or inferential statistic - descriptive if we compare 𝑥̅ to μ is it a descriptive or inferential statistic - inferential How do you know which statistical analysis is important - Depends on
- Study design/research question
- Types of variables (level of measurements, distribution) 3, Whether assumptions of the analyses are met What are variables - a characteristic that varies across observations ex. age are variables columns or rows in spreadsheets - columns what is the independent variable - the variable the research is manipulating ex. treatments what is dependent variable - the variable we are measuring in response to the treatment ex. speed when do we choose correlational treatments - 1. when we want to generalize our findings to the real world
what is importnat to consider about correlational experimenst - it is not good at inferring causality Why is correlation not causation? - the IV(treatment) and DV(response) may have a relationship due to a third (confounding) variable or common cause why woudl you choose experimental experimenst - it is good at inferring causality what is bad about experimental experiments - Manipulating IV in lab setting may sometimes feel detached from the real world how does the IV differ between correlation and experimental research desigms - in correlational, the IV is measured by the researcher in Experimental the IV is manipulated by the reserahcer is this research design better for correlational or experimental statistic design? Gather a sample from population (N=50) ask participants if they have seen an ad or not - correlational is this research design better for correlational or experimental statistic design? Gather a sample from the population (N=50) randomly assign 25 to see a new ad, 25 to not see a new ad
- experimental What is a Within-subjects design - Each participant does more than one experimental conditions (e.g. participant does a baseline treatment and then another treatment 3 weeks later)
what is interval data - a type of continuous variables where the data is measured along a scale what is ratio data - a type of continuous variable where the data is measure by intervals, with an absolute 0 point, or meaningful origin ex. Height Assumption of normalitiy means - that mean = median = mode the interval mean +/- 1 standard deviation contains 68% of the values in the population mean +/- 2 standard deviations contains 95% of the values in the population mean +/- 3 standard deviations contains 99.7% of the values in the population What is the relation between the mean of a sampling distribution and population mean? - its identitical Standard error - standard deviation of the sampling distribution sigma/ square root n What happens to the standard error as N increases? - it gets smaller and the sample distribution gets narrower What sampling distributions do we get for small and large N, when population distribution is skewed? - if we sample with a large enough n we will see a normal distribution
what is the central limit theorem - For any population with mean mu and standard deviation (small sigma), the distribution of sample means for sample size n will have a mean of mu and a standard deviation of sigma/(square root of n), and will approach a normal distribution as n approaches infinity. what is a hypothesis - Assumption/prediction made about a population parameter (NOT about a sample estimate) what is null hypothesis significance testing - when you test you statistic to prove whether a null hypothesis is true or not traditional statistical experiments what is the Null Hypothesis (H0) - a hypothesis that the treatment will have no effect. This is what we test statistically, what we want to prove/disprove what is Alternative Hypothesis (H1) - a hypothesis that there is a significant enough effect the research or experimental hypothesis how do we choose an alpha significance level - by deciding how much risk do we want to take. Are we are willing to accidently reject H0 even if H0 is true 5% of the time? or would we rather do 1% and use alpha=. when do we use the statistic test X^2 (chi-square) - to test frequency distributions (usually when our data is nominal)
ex. a 95% confidence interval (CI) may indicate that true weight gain in the population is between 2g and 18g per month if alpha is .01 how sure is our confidence interval - 99% if alpha is .05 how sure is our confidence interval - 95% how do you form a confidence interval - (1-alpha) x 100 what confidence interval do you expect for a larger sample size? - as samlpe size increases our estimate becomes more precise so our confidence intervals become more narrow what confidence interval do you expect for a smaller alpha - as alpha decreases our estimate becomes less precise our confidence intervals become wider what is an effect size - a quantitative measure of magnitude of the experimental effect the larger the effect size the stronger relationship between two variables name four effect sizes - 1. pearons correlation coefficent (r) or correlation ratio squares (R^2)
- cohen's d
- omega or omega square
- Eta squared
what are cohen (1988) effect size scale - r= .10 (small effect) • r= .30 (medium effect) r= .50 (large effect) what is a type 1 error - false positive, when we reject the H0 even though it is true what is a type II error - false negative, when we retain the H0 even though it is false how do we calculate power - 1 - β what is power in statistics - Probably of correctly rejecting the null hypothesis i.e. probability of correctly concluding that two treatment groups differ...when they actually do β in statistics is the probability - of committing type II error alpha in statistics is the probability - of committing type I error what is the NULL hypothesis for a single-mean Z-test for the question Do PSYC 305 students at McGill differ on IQ from the general population (mu = 100)? - H0: mu = 100 (McGill students ahve same mean as general population) what is the ALTERNATIVE hypothesis for a single-mean Z-test for the question
they transform all scores to a standard normal distribution does the shape of the distribution change when you plot the z-score units? - no, only your units change If alpha = .05 (two-tailed), then our critical z-score value is about - 1. If H0 of our z-score is true - than 2.5% of observed z-test will be less than -1. 2.5% will be greater than +1. (2.5+2.5=5%, the desired level for alpha) what are the limitations of the z-test - knowing the true value of the population standard deviation is unrealistic how is a t-statistic obtained (in comparison to a z-statistic) - by replacing the population standard deviation with the sample standard deviation does the t-statistic follow the standard normal distribution - no, since it does not use standard deviation. Instead it follows the t-distribution What is the t distribution? - The t-distribution is a distribution very similar to the standard normal distribution. It is bell-shaped and has a mean of zero, but has a larger standard deviation than the standard normal distribution, and therefore, has thicker tails than the standard normal distribution. A "flattened" z distribution
Does the t-distribution shape vary - yes, according to degrees of freedom the t-distribution approaches a normal distribution as df becomes larger or smaller - larger what is the NULL hypothesis for a single-sample t-test? Do PSYC 305 students at McGill differ on IQ from the general population (mu = 100)? - H0: mu = 100 what is the ALTERNATIVE hypothesis for a single-sample t-test? Do PSYC 305 students at McGill differ on IQ from the general population (mu = 100)? - H1: mu does not equal to 100 what is an independent samples t -test - It is used to test whether the null mean and the observed mean are equal when separate and independent samples are used (between subject design - participants are randomly assigned to one of two conditions) what is a paired-samples t-test - a test Used to compare the means on a scale or test between two dependent groups (pre/post scores) Ex: Is there a difference between the test average before and after you learn the material what is the NULL hypothesis for an independent samples t-test - H0: mu #1 = mu # what is the ALTERNATIVE hypothesis for an independent samples t-test - H1: mu#1 does not equal mu#
what are treatment levels? - different values or categories of the independent variable/factor what is a single-factor (one-way) design - Involve a single IV with two or more levels can be independent or repeated-measures what is a factorial design - Experimental studies with two or more IV and two or more Levels for each ex. 2x2 (2 levesl and 2 levels) why would you not just test all possible differences with a t-test? - the alpha level would become inflated This would lead to an inflated experiment-wise Type I error rate what are the two tests during a one way anova - 1. F test
- Post Hoc when do you calculate a post-hoc test - if an F test is significant What does ANOVA mean - analysis of variance what are the steps of the ANOVA - 1. Divides the variance observed in data into different parts resulting from different sources
- Assesses the relative magnitude of the different parts of variance
- Examines whether a particular part of the variance is greater than expectation under the null hypothesis what are the two sources of variance - 1. MSM
- MSR what is model variance - variance explained by our model how the groups vary that we can explain with our manipulation what is residual variance - variance unexplained by our model is the within group variabiltiy that we cannot explain with our manipulation - usually just random what does the F test do - it assessed the magnitude of the model difference/residual difference Which F distribution is commonly used for ANOVA - the right skewed distribution the theoretical sampling distribution of F values; it gives the probability of various outcomes when all samples come from identical populations when do you reject your the null hypothesis for an F test - If your F value is greater than or equal to the critical value, you may reject the null hypothesis or if p-value is less than.
what is the strength of .14 for w^2 - large what is the strength of .01 for w^2 - small what is the multiple comparisons problem - Increasing the number of t-tests you perform increases the possibility of a false positive when is the multiple comparisons problem an issue in one-way anova? - when you run post-hoc tests in order to distinguish which group is significantly different from the other how do you solve the multiple-comparisons problem? - by decreasing the alpha level what is the bonferroni correction - adjustment made to p-value, accounts for problems of multiple data comparisons .05/ number of H when do we use the bonferroni correction - when we calculate multiple t-tests during the post-hoc tests of one-way ANOVA what is an omnibus - the overall test which tell us if there is a significance but does not tell us how the specfic groups distinguish between each other. what are the two post-hoc test we can use for one-way anova - 1. scheffe
- Tukey's Honestly Significance Difference (HSD)
distinct of Scheffe test - 1. can be used if groups have different sample sizes
- less sensitive to departures from normalityy assumption/ homogenous variance
- conservative test why is the scheffe test so conservative - because it uses a larger critical value. So the F ratio it calculates has to be very large What are the distinctions of the Ukeys HSD test - 1. typically used if groups have equal sample sizes
- uses Q which test statistic does Tukeys HSD use - Q: the studentized range statistic Which test statistic does scheffe use - F test when do we reject the null hypothesis in the Tukey HSD test (when sample sizes are equal) - if the observed Q values is greater or equal than Qrit. or if the absolute difference between (group mean and total mean) is greater or equal to HSD value. what does the HSD value - it is the absolute difference between two means required for a statistically significant differences when do you reject the null hypothesis in a tukey HSD test (when sample sizes are unequal) - if the absolute difference (between group mean and overall mean) is greater than Qcrit/square root of two x the square root of MSr (1/ni + 1/nj) Why do we need assumptions - it effects the shape of our distribution
