PS211 Exam 2
How is the calculation of standard error different for a t test than for a z test?
you calculate both by doing standard deviation/sqrt(N) in z test, you only use N in t test, you do N-1
Why cant correlation be greater or less than +/- 1
1 = perfect correlation
What is power
A measure of the probability that we will reject the null hypothesis when we should reject the null hypothesis Another way of saying this: probability that we will not make a Type II error
Why do we need comparison distributions
A problem with making meaningful comparisons is that variables are measured on different scales, so we need to put different variables on the same standardized scale.... Using a comparison distribution is part of this.
Explain what a correlation coefficient is.
A statistic that quantifies a relation between two variables
What magnitude of a correlation coefficient is large enough to be considered important, or worth talking about?
According to Cohen, a correlation coefficient of 0.50 is a large correlation and 0.30 is a medium one. However, it is unusual in social science research to have a correlation as high as 0.50. The decision of whether a correlation is worth talking about is sometimes based on whether it is statistically significant, as well as what practical effect a correlation of a certain size indicates.
How are the critical t values affected by sample size and degrees of freedom?
As degrees of freedom go up, the critical values go down. Same thing with sample size - as sample size increases, the critical values decrease. (Ties into the idea that larger sample size is associated with greater power!)
What effect does increasing sample size have on standard error and the test statistic?
As sample size increases, so does the test statistic (if everything else stays the same) and it becomes easier to reject the null hypothesis. Because of this, a small difference might not be statistically significant with a small sample, but might be statistically significant with a larger sample. As sample size increases, standard error decreases.
3 Assumptions of Hypothesis Testing
Assumption 1: the data must be randomly selected, or external validity will be limited Assumption 2: the underlying population distributions for the two variables must be approximately normal Assumption 3: each variable should vary equally, no matter the magnitude of the other variable
How are statistical power and effect size different but related?
Both give us info about the likelihood that we mistakenly failed to reject our null hypothesis (type II error) effect size is also independent of sample size
Why do we calculate confidence intervals
Confidence intervals confirm the results of the hypothesis test while also adding more detail. Specifically, they tell us a range within which the population mean would fall 95% of the time if we were to conduct repeated hypothesis tests using samples of the same size from the same population.
Using everyday language rather than statistical language, explain why the words critical region might have been chosen to define the area in which a z statistic must fall in order for a researcher to reject the null hypothesis.
Critical region is where you will have a statistically significant result
When is pooled variance and pooled standard deviation used?
For independent samples t-test - allows us to find one standard error of the mean which is weighted for our difference groups based on degrees of freedom Pooled standard deviation used to calculate cohen's d for independent samples t-test
Relate effect size to the concept of overlap between distributions.
If two distributions overlap a lot, then we would probably find a small effect size and not be willing to conclude that the distributions are necessarily different. If the distributions (distributions of research and null hypothesis) do not overlap much, this would be evidence for a larger effect or a meaningful difference between them.
List factors that affect statistical power. For each explain how and why each factor affects power. Use drawings of distributions to highlight your answer.
Increase alpha (aka p level from 5% to 10%)... this increases the probability of a Type I error from 5% to 10% Turn a 2-tailed hypothesis into a 1-tailed hypothesis Increase sample size Exaggerate the mean difference between levels of the independent variable. E.g., increase length of intervention time... lessens overlap between curves Decrease standard deviation
What are independent events or observations?
Independent events are things that do not affect each other. For example, the lunch you buy today does not impact the hours of sleep that I will get tonight
What does regression add above and beyond what we learn from correlation?
It enables us not only to quantify the relation between 2 variables, but also to predict a score on a dependent variable from a score on an independent variable.
Explain what a wider confidence interval means as it pertains to being able to describe the true mean of the distribution based on the research hypothesis?
Larger ranges mean less precision in making predictions, just as widening the goal posts in football means that you can be less precise when trying to kick the ball between the posts. Smaller ranges indicate we are doing a better job of predicting the phenomenon within the population. For example, a 95% confidence interval that spans a range from 2 to 12 is larger than a 95% confidence interval from 5 to 6. Although the percentage stayed the same, the width of the distribution has changed.
What characteristics of the comparison distribution are needed to conduct an inferential statistical test? What do these characteristics allow us to do?
Mean and standard error When data are normally distributed, we can compare one particular score/mean to an entire distribution of scores/means Allows us to understand how many standard errors above the mean our value of interest is (e.g., above the mean)
Nonparametric Test
Nonparametric test = an inferential statistical analysis that is not based on a set of assumptions about the population... you use these when assumptions for parametric tests are not met (e.g., when the dependent variable is NOT scale) ex. chi squared
Write the symbols for the null hypothesis and research hypothesis for a one-tailed test.
Null hypothesis: H0: μ1 ≥ μ2 Research Hypothesis: HR: μ1 < μ2 Null hypothesis: H0: μ1 ≤ μ2 Research Hypothesis: HR: μ1 > μ2
How is a confidence interval calculated? What does the range of numbers mean?
Point estimate (e.g., our d-bar, mean of our sample for a z-test, etc) ± (critical value * standard error of the comparison distribution) (critical value * standard error of the comparison distribution) = margin of error Provides a range of plausible values for the population parameter, expands beyond the point estimate to an interval estimate So if the mean of the comparison distribution (what we're comparing our findings - d-bar, mean difference, etc - to) is within this range of numbers, it suggests that a statistically significant result (according to a t-test, z-test, etc) should potentially not be trusted.
Explain what pooling the variance means and why it is important?
Pooled variance is a weighted combination of the variability in both groups in an independent samples t-test. We assume that larger samples do a better job of estimating the population than smaller samples do, so we would want the variability measure based on the larger sample to count more.
What is the difference between standard deviation and standard error?
Standard error is smaller than the standard deviation, adjusted for sample size
How to create a distribution of mean differences
Step 1: randomly choose 3 pairs of scores (e.g., pre and post-intervention scores), replacing each pair before randomly selecting the next Step 2: for each pair, calculate a difference score by subtracting the first weight from the second weight Step 3: Calculate the mean of the differences in weights for these three subjects. Then complete these 3 steps again. Randomly choose another 3 people from the population, calculate their difference scores, and then calculate the mean of the 3 difference scores. And then complete these 3 steps again, and again, and again.
What is included in APA format
Test statistic, df, p value, cohens d, and confidence interval
What is the central limit theorem? Think about all statements of the CLT.
The CLT refers to how a distribution of sample means is a more normal distribution than a distribution of scores, even when the population distribution is not normal. Statement 1: Repeated sampling approximates a normal curve, even when the original population is not normally distributed. Statement 2: A distribution of means is less variable (smaller st. dev) than a distribution of individual scores
As they relate to comparison distributions, explain the difference between mean differences and differences between means?
The comparison distribution for the paired-samples t-test is made of mean differences - the average of many difference scores. The comparison distribution for the independent samples t-test is made up of differences between means, or the differences we can expect to see between group means if the null hypothesis is true.
Why does the standard error become smaller simply by increasing the sample size? What does this effect have on our inferences?
The larger the sample size, the narrower the distribution of means and the smaller the standard error is. When there's a larger N, it means that there are many more scores that are close to the mean that balance out any extreme scores (outliers) that might have affected standard error (aka spread) It becomes easier to reject the null hypothesis
What is the standard size of the critical region used by most psychologists?
The most extreme 5% of the comparison distribution curve (2.5% on either end). The area in the tails of the comparison distribution in which the null hypothesis can be rejected.
Using everyday language rather than statistical language, explain why the word cutoff might have been chosen to define the point beyond which we reject the null hypothesis.
These are the extreme values at which we reject the null hypothesis (the cut-off point at which we reject or fail to reject the null hypothesis)
What does the phrase "free to vary," referring to a number of scores in a given sample, mean for statisticians
This refers to the number of scores that can take on different values when a given parameter is known/estimated from the sample if we know that the mean of 4 scores is 6, and we know that three of those scores are 2, 4, and 8, the last score can only be 10 (in order to get a mean of 4). So the df is the number of scores in the sample minus 1
When should you use an independent samples t test
Used when we do not know the population parameters (i.e., the population mean and standard deviation) We are comparing two groups that are composed of unrelated participants (or "observations"), i.e., no participant is in more than one condition/group (between subjects design)
Explain what s2 difference is and why is it important?
Variation of the distribution of differences between means This is an estimate of spread, which is adjusted for sample size. Then gets adjusted to standard deviation form by taking the square root!
How do we calculate the percentage of means below a particular positive z statistic
We add the percentage between the mean and the positive z score to 50%, which is the percentage of scores below the mean (50% of scores are on each side of the mean)
interval estimate
based on a sample statistic and provides a range of plausible values - such as a range of means - for the population parameter.
What does it mean to have a distribution of mean differences? How are they theoretically constructed? Why are they theoretically constructed?
We can understand the meaning of a distribution of mean differences by reviewing how the distribution is created in the first place. A distribution of mean differences is constructed by measuring the difference scores for a sample of individuals and then averaging those differences. This process is performed repeatedly, using the same population and samples of the same size. Once the collection of mean differences is gathered, they can be displayed on a graph (in most cases, they form a bell-shaped curve).
How is the calculation for an effect size for a t-test different from the effect size for a z test?
We use standard error, not standard deviation. However, when calculating cohen's d for independent samples t-test, we use pooled standard deviation (spooled) rather than standard error, because we want a measure of variability not altered by sample size - pooled variance is affected by sample size!
How does power relate to Type I and Type II errors
When our results were determined to be statistically significant but our effect size is small, it indicates that a) our finding wasn't super meaningful, and b) that we may have committed a Type I error (i.e., If we fail to reject our null hypothesis, but find a large effect size, it suggests that maybe we didn't have enough power and committed a Type II error
When should you use a paired samples t test
When the data we are comparing were collected using the same participants in both conditions, we would use a paired samples t test; each participant contributes 2 values to the analyses. (Within subjects design)
What does statistically significant mean to statisticians?
Where we can reject the null hypothesis, does not mean that it is important or meaninful though pattern differed from what we expected by chance
Describe a perfect correlation, including its possible coefficients.
You could have a perfect negative correlation (-1.00) or a perfect positive correlation (1.00).
The equation for a line is Ŷ = a + b(X). Define the symbols a and b.
a = intercept, b = slope
point estimate
a summary statistic from a sample that is just one number used as an estimate of the population parameter (such as a mean). But point estimates are rarely exactly accurate, so we can increase accuracy by using an interval estimate when possible
What are critical values (and all their names) and the critical region (region of rejection)?
a test statistic value beyond which we reject the null hypothesis; often called a cutoff The critical region is the area in the tails of the comparison distribution (e.g., z-distribution, t-distribution, etc) in which the null hypothesis will be rejected if the test statistic falls there. The probability used to determine the critical values, or cutoffs, in hypothesis testing is a p level (often called alpha)
How does the regression line relate to the correlation of the two variables?
allows us to determine equation for a straight line and then predict score
Negative Correlation
an association between two variables in which participants with high scores on one variable tend to have low scores on the other variable.
Positive Correlation
an association between two variables such that participants with high scores on one variable tend to have high scores on the other variable as well, and those with low scores on one variable tend to have low scores on the other variable
Parametric Test
an inferential statistical analysis based on a set of assumptions about the population ex. pearson correlation coefficient, paired samples t test, independent samples t-test
Why is the population mean of the theoretical comparison distribution equal to 0 in the two-tailed, paired-samples t-test?
because it is based on the null hypothesis, which proposes that the difference is 0
How are the sign of the correlation coefficient and the sign of the slope related
both tell direction of the association
When we have a straight-line relation between two variables, we use a Pearson correlation coefficient. What does this coefficient describe? Explain how the correlation coefficient can be used as a descriptive or an inferential statistic.
coefficient shows a linear relationship between two scale variables, describes direction and strength, it is a descriptive statistic when capturing relationship between two variables is an inferential stat when testing to understand whether the correlation is significantly diff from 0
What are Cohen's guidelines for small, medium, and large effects?
d = .2 indicates a small effect d = .5 indicates a medium effect d = .8 (or above) indicates a large effect
Explain why the numerator of the paired samples t-test is different than the numerator of the independent samples t-test? What source of variation is included in the independent samples equation that is not included in the paired samples t-test?
diff comparison distributions, independent samples t test includes weighting for sample size in the comparison distribution standard deviation
confidence interval
interval estimated based on sample staistitic, includes population mean if population is sampled from repeatedly
What specific danger exists when just reporting a statistically significant difference between groups?
just bc it is significant does not mean its meaninful
What does the symbol μM stand for?
mean of a distribution of means
Null Hypothesis vs Research Hypothesis
null = there is no correlation research = there is a correlation
What affects the size of the area for the critical region?
sample size/degree of freedom
How are deviation scores used in assessing the relation between variables? Explain how a correlation is calculated
steps for calculating pear correlation coefficient: Calculation the deviation of each score from its mean, multiply the 2 deviations for each person, and sum the products of the deviations Calculate a sum of square for each variable, multiply the sums of squares, and take the square root Divide the sum from step 1 by the square root in step 2
Define effect
the outcome of some event, For statisticians, the outcome is any change in a dependent variable, and the event creating the outcome is an independent variable. When statisticians calculate an effect size, they are calculating the size of an outcome.
If we calculate the confidence interval around the sample mean difference used for a t test, and it does not include the value of 0, what can we conclude?
then we say this supports the meaningfulness of a finding that we reject the null hypothesis.
How are μM and μ related (in words -not math) and why?
they are both the same according to CLT
What are Z tests comparing
theyre done when we have one sample and know the mean and st dev of population comparing samples score to the mean of a comparison distribution
Why do we used two tailed rather than one tailed test
two tailed test does not have directionality in it, one tailed does. it is only used when the researcher is absolutely certain the effect cannot go in the other direction
If we calculate the confidence interval around the sample mean difference used for a t test, and it includes the value of 0, what can you conclude?
you can conclude that we fail to reject the null hypothesis
What happens if Z test is in critical region
you can reject the null hypothesis
What does a z statistic tell us about a sample mean? How are z scores different from z statistics?
z score = standardized version of raw score, usually calculated based on distribution of means rather than distribution of scores z score = distance that a score is frmo the mean of its distribution in terms of st dev z statistic = means, z score = scores The z-statistic tells us how many standard errors a sample mean is from the population mean.
What do each of the symbols stand for in the formula for the regression equation: zŶ = (rXY)(zX)?
z score for dependent variable, pearson correlation coefficient, z score for independent variable
What does the symbol σM stand for?
σM = standard error (aka standard deviation of a distribution of means, the typical amount that a sample mean varies from the population mean)