STA 254 Final Exam
What are the two interpretations of r2?
1. The improvement of predictions compared to the baseline model of Y^ = Mean of Y. 2. Percentage of dependent variance that is shared with the independent variable's variance.
What is the probability of rolling a 4 or a 5 or a 6 when using one six sided dice?
1/2
If rolling 3 dice, what is the probability of rolling 3 (the sum of all three being three)?
1/216
When rolling two dice, what is the probability of rolling 12?
1/36
What is the probability of rolling a 4 when using one six sided dice?
1/6
If a researcher wanted to compare means from five groups, how many two-sample t-tests would be required?
10
Why does ANOVA have its own effect size, instead of Cohen's d?
ANOVA has more than 2 groups; using Glass' d or Cohen's d would result in more than one effect size
If the null hypothesis in an ANOVA is about testing group means, why is it an analysis of variance?
AVOVA is really about examining the relationship between an independent variable and a dependent variable. This is bc ANOVA is a member of the GLM, and all members investigate the relationship among variables
What is the null hypothesis for ANOVA?
All group means are equal
Explain the relationship between pearson's r and the unpaired samples t test in terms of the GLM
Both are NHST procedures that are members of the general linear model. both are correlations, but the unpaired t-test is a correlation adapted to a dichotomous nominal variable.
What is Cohen's d? What roles does Cohen's d play in the z-test model?
Cohen's d is a measure of the distance between the sample mean(X) and population mean. It quantifies the number of standard deviations that are between sample and population means. It states the z-score of the average person in the sample when compared to the population as a whole. *know formula
What is the effect size for a paired-samples t-test?
Cohen's d or Glass' delta
Samantha found that the correlation between income and self-reported hapiness is r =+0.39 she concludes that if more people were wealthy they would be happier . What is the problem with her logic?
Correlation does not imply causation. Samantha's logic would only be correct if a person's level of wealth causes them to be happy. However, if a person's happiness causes their wealth or if an unknown variable causes both variables than Samantha's reasoning would be incorrect.
In a paired-samples t-test, how do you determine whether to conduct a one-tailed test or a two-tailed test?
Determine if there is prior research or theory that can be used to predict the results of NHST. IF there is enough previous research, chose one tailed, if not, then 2 tailed
What are the four things that determine statistical power?
Effect size, sample size, alpha, and study design
In which of the three cases, would you expect the probability distribution to most closely resemble a normal distribution? Why?
I would expect three dice, because the normal distribution appears to form a smooth curve with tightly packed numbers. Rolling 3 dice has larger number intervals and would look like a smooth curve on a histogram.
After calculating your observed value and your critical value how do you know whetehr to retain or reject the null hypotheiss?
If t obs is more extreme than t crit you should reject the null. This would mean that the observed value is inside the rejection region.
How do we determine the range of a p - value for a paired samples t test?
In the same way as other t tests. compare the t observed value to the t criticla values for different alpha values that correspond to the correct degrees of freedom. If the null hypothesis is rejected when the alpha is .01 then p is less than .01. If it is rejected when alpha is .05 but not .01 then p is less than .05 but greater than .01. If it is retained at .05 then p is greater than .05
Interval data
Indicated that there is a rank order and that the distance between numbers on a scale is constant along the entire length of scale; can do everything nominal and ordinal do and also add, subtract, and divide to form averages
What is HARKing and why does the textbook say it is questionable?
It is hypothesizing after the results are known- generating or altering a research hypothesis after examining data and then presenting results as if the hypothesis was created before data was collected. It is questionable because it is inappropriate and may distort the scientific literature.
what is the correct interpretaiton of the p value
It is the probability of making a type 1 error if the null hypothesis were perfectly true.
Explain the homogeneity of variance assumption
It means that the groups have similar variance values. In order to examine whether our data violates the homogeneity of variance assumption you need to find the ratio between the larger variance and the smaller variance. If this ratio is less than or equal to 10:1 your data meets this assumption and your results will not be distorted. If one groups scores is much more variable than other group's scores then comparing the two means may be deceptive and simplistic.
Interpret the formula you would use to estimate the population standard deviation based on a sample (explain what is going on in the numerator and in the denominator and why)
Population standard deviation (estimated from the population parameter) is the square root of the sum of the squared deviation scores that have been divided by the sample size minus one. One is subtracted in the denominator to make the overall value of the fraction slightly larger to correct for underestimation (Bessel's correction).
Which sum of squares value will always be the largest?
SOS total
Explain the differences between a statistical model and a theoretical model
Simplified, statistical models use a limited number of specific variables that are expressed as mathematical equations. They often result from quantitative research. Simplified, a theoretical model isn't described in terms of numbers or mathematical equations, but it is a causal explaination of reality.
What does skewness measure? Explain how the numerical measurement of skewness represents this
Skewness measures the degree to which a histogram or distribution is symmetrical. If skewness is equal to zero, then the distribution is symmetrical. If it is less than zero, then it is negatively skewed. If greater than zero, it is positively skewed.
What is the difference between the mean deviation and the standard deviation? Why is the standard deviation preferred?
Standard deviation is the typical difference between the mean and the deviation scores in the dataset. Mean deviation takes the absolute value of deviation scores before summing them. It is a true average difference between the mean and the scores in the data. The standard deviation is preferred because when a variable is normally distributed, the mean deviation produces less stable results than standard deviation. Also, the standard deviation is required for most statistical procedures discussed in the book.
Define statistical power and expain why it is important in social sicence research?
Statistical power is the ability of an NHST to reject the null hypothesis.
Is it better in a study to have high statistical power or low statistical power
Statistical power is the probability that the NHST wil reject the null hypothesis when the null is false. Studies with high statistical power have a lower probability of type two error. This is a good thing.
What are statistics?
Statistics can separate good research from bad, help you to evaluate conclusions, report findings to others, and improve practice. They can be qualitative or quantitative and use visuals to aid in understanding.
Which correlation coefficient is the strongest a)-0.21 b)-0.62 c)+0.04 d)+0.49
Strongest is -0.62 and the weakest is +0.04
Which correlation is the strongest and which is the weakest? a)+0.42 b)-0.87 c)+0.59 d)-0.34
Strongest is -0.87 and the weakest is -0.34
Interpret the formula for the observed value in a one-sample t-test.
T observed is equal to the sample mean minus the population mean divided by the standard deviation, divided by the sample mean
What is alpha? what does it mean when alpha is 0.05?
The default alpha is .05 this means that the rejection region is 5% of the area of the sampling distribution.
How do strictly ordinal and strictly interval data differ?
The distance between any pair of adjacent numbers will be equal, no matter which pair you examine. You can add, subtract, and compute averages, and you can do everything that ordinal and nominal do.
What is the effect size for an unpaired two sample t test? How is this calculated?
The effect size states how far apart the means are. This is calculated using cohen's d which is equal to the mean of sample 1 - the mean of sample 2 / s pooled
How do the mean and median balance the distribution differently (make reference to deviation scores and outliers)
The mean is a central balance point where the difference between each score and the mean cancel out. The difference is called a deviation score. The mean is highly sensitive to outliers. The median is not effected much by outliers. It balances distributions by either taking the middle variable when ordered from smallest to largest or vice versa. If scores are even, you find the 2 middle scores, add them, and divide by 2.
What is the null hypothesis when conductin an NHST on a pearson's r value
The null hypothesis is always that there is no difference between groups or that there is no relationship between independent and dependent variables. When there is no relationship then r = 0 so the null hypothesis when conducting an NHST on a Pearson's r value is: r=0
What is always the null hypothesis in an unpaired samples t test
The null hypothesis is that the mean of the first group is equal to the mean of the second group.
Explain the relationship between the pooled standard deviation and the pooled variance
The pooled standard deviation is the square root of the pooled variance
What is the pooled standard deviation?
The standard deviation of the combined data from the two groups
What are the types of kurtosis and what do they look like?
There are three types: mesokurtosis, platykurt, and leptokurtosis. These describe the shape of a bell graph depending on where their given mean and individual values are. A mesokurtic shape is normal. Platykurtic is flat. Leptokurtic is taller with short tails.
what does the phrase "unpaired data mean"?
There is no relationship between scores in the different groups (unlike the paired samples t test0
How do the effect sizes Glass' and Cohen's d differ?
They differ in their denominators. The denominator for Cohen's d is the pooled SD of both scores; the denominator for Glass's delta is the SD of the control group
What does it mean to operationalize a variable?
To create a definition of the variable that can allow them to collect numerical data that allows a social scientist to easily study it
Define the Type 1 and Type 2 error for a paired samples t-test?
Type I occurs hen the researcher rejects a null hypothesis that is actually true. Type II occurs when the researcher retains a null hypothesis that is actually false.
what are the guidelines for ensuring that the data in an unpaired 2 sample t test meet the assumptions of the homogeneity of variance and similar sample sizes
Variance ratio of 10:1 or less and a sample size of 4:1 or less
Explain the third variable problem
With only a correlation between two variables, it is possible that a third variable that is not part of the model is the cause of the relationship between the original two variables.
Is it possible to artificially create the groups (e.g experimental group and a control group) in an unpaired samples t-test? why or hwy not?
Yes it is also possible for researchers to assign study participants to artificial groups such as an experimental or control group. The way that these groups are created does not change the statistical procedures.
How do you know if you should use a one tailed or a two tailed test in an unpaired sample t test?
You decided beaised on whether the there is enough theory or previous data to permit an educated guess about hte results of the study.
Explain the difference between the information table and the AVOVA table
You must use both of these to calculate the observed value, however you need the information table first. The information table includes the sum of the scores on the dependent variable, the sum of the squared scores on the dependent variable, the number of the sample members (n), and the mean score for dependent variable. The ANOVA table divides the dependent variable variance of the data into two portions. One portion is variance BETWEEN the independent variable groups, the other is variance WITHIN the independent variable groups. These two add up to the total variance.
How is a paired-samples t-test similar to a one-sample t-test? How are the two different?
a paired sample t test is a one sample t test using difference scores, so the same characteristics of the GLM apply
Why do the one-sample t-test and the paired samples t-test share so many similarities?
a paired samples t-test is just a one-sample t-test using different scores
what is the correlation coefficient
a statistic that measures the strength of the relationship between two variables. If two variables have a relationship between them we would say they are correlated.
What is the connection between the unpaired two sample test and ANOVA?
an unpaired 2 sample t test is a special case of ANOVA; they are both members of the GLM
In an ANOVA the null hypothesis is that all of the group means are equal to one another. If the null hypothesis is rejected, how do we determine which group mean differs from other means?
by conducting a post hoc test
How do you find a critical value in an ANOVA?
by finding df between groups and the df within groups and using the F-table. to identify the proper F-critical value that corresponds to the predetermined alpha value
After finding the critical value and the observed value, how do you determine whether to reject or retain the null hypothesis?
calculate an effect size by using cohen's d and find p.
How do you calculate the pooled standard deviation?
combine the 2 groups of scores together and find the standard deviation of all of them as if they were all one sample
How do you calculate the number of degrees of freedom between groups?
df between= k-1, df within= n total-k, df total=n total-1; k is the number of groups and n total is the total sample size
How do you calculate the effect size for ANOVA?
eta-squared; n2= SOSbetween/SOStotal
Ratio data
highest level where data can form mutually exclusive and exhaustive categories, have rank order, and possess equal intervals between numbers across entire length of scale and zero represents complete absence of quality being measured; can do everything the other types can and also physical measurements, most measures of time (age), cannot measure temperature
When is it appropriate to raise alpha?
if getting a large sample size is difficult or impossible or if a study is very time consuming or expensive. Also, in situations where the costs of Type I error are much lower than those of type II error.
Why is it always necesarry in an unpaired two sample t test to worry about either type 1 or 2 errors?
if we reject the null hypothesis we have to worry about a type one error but if we retain the null hypothesizes then we would have to worry about a type two error. We never really know whether the null hypothesis is true or false so the possibility of both of these errors is something we should keep in mind.
Explain the difference between test wise Type I error and experimentwise Type I error
in testwise, the type 1 error is equal to the alpha of each NHST procedure; experimentwise is the probability of making a type 1 error in any one of the series of null hypothesis tests
What is the statistic that the sampling distribution in a paired-samples t-test consists of?
it compares two population means
How do we determine the range of a p-value for paired-samples t-test?
it is the same as one sample t-test; compare t-observed value to the t-critical values for different alpha values that correspond to the correct degrees of freedom. if the null hypothesis is rejected when a=.01, then p<.01. If the null hypothesis is rejected when a=.05, but not when a=.01, then .05>p>.01. If the null hypothesis is retained when a+.05, then p>a.
Why is labeling effect sizes as small, medium, and large without any further interpretation problematic?
labels ignore context, also a label on an effect size is not a full interpretation because the standards of a small or large effect size are not consistent. the labels are also subjective
When is it appropriate to lower alpha for a null hypothesis test?
lower for a large sample size, because as a sample grows, it becomes easier to reject the null hypothesis. Also, lower alpha if there is a large number of null hypotheses in the study or in exploratory research.
what are the three possible alternative hypotheses is an unpaired two sample t test?
mean of group one is less than mean of group 2 mean of group one is greater than the mean of group2 Mean of group one is different than the mean of group 2
Nominal data
mutually exclusive and exhaustive categories that are assigned a number based on the category they belong to; labels, classifying, counting, calculating percentages
How do we measure overall precision accuracy in ANOVA?
n squared
What is the formula for degrees of freedom in an unpaired 2 sample t test?
n total - 2
Explain the difference between a natural pairing and an artificial pairing of scores
natural pairing occurs when the pairs are naturally occurring outside the study (siblings), artificial occurs when the researcher creates the pairing
What level of data is the independent variable in ANOVA?
nominal level
What does nD symbolize in a paired-samples t-test?
number of difference scores (pairs in the data)
Ordinal data
numbers are assigned to mutually exclusive and exhaustive categories that rank order, showing that some groups possess more of the variable than other groups; everything nominal does and can rank
Why is ANOVA always a one-tailed test of the null hypothesis?
observed values are always at least zero for ANOVA. we never get a negative value in the f-distribution, so we never have to worry about the negative side of the number line. The rejection region is always on the right side of the f-distribution.
What is the benefit of calculating group means in the information table?
the group means are the predicted dependent variable values for group members
What is the difference between a post hoc test and a planned contrast?
planned contrast is driven by theory, instead of by the null hypothesis being rejected. They involve fewer statistical significance tests, and thereby are not as susceptible to risks of type I error. Sometimes, they risk missing a statistically significant group because it was not in their pre-planned data analysis scheme.
what is the difference between the appearance of scatterplots that represent positive and negative correlations?
positive pearson's r values produce a pattern of dots on the lowere left to the upper right portions. Negative is the upper left to the lower right.
What are paired samples? What is an example of a type of data that would have a paired structure?
samples in which natural or matched couplings occur; Statistical difference between two time points.
How do we measure individual precision accuracy in ANOVA?
residual
How do you find the p-value in a paired samples t-test?
same as one sample; if null hypothesis is retained when a=.05 then p>0.05, if rejected when a=.05, but retained when a=.01 then .01<p<.05, if rejected when a=.01, then p<.01
In a paired-samples t-test, how do you find the degrees of freedom?
subtract one from the number of pairs
How do you calculate difference scores for a paired-samples t-test?
subtract one score in the pair from the other score in the pair
What is a null hypothesis for a paired-samples t-test?
the average difference between a subject's scores in group 1 and group 2 is zero
Define the grand mean
the average of means of several subsamples as long as the subsamples have the same number of data points.
What are two interpretations of ANOVA's effect size?
the improvement of predictions compared to the baseline model and the percentage of dependent variance that is shared with the independent variable's variance
What is a Type I error inflation, and why is it a problem when conducting multiple NHSTs?
the increase in the probability of making a type 1 error as the number or statistical significance test grows
What is residual?
the individual accuracy is the difference between the person's predicted dependent variable score and their actual dependent variable score
Some of the formulas in this chapter use the abbreviation k. What does k signify?
the number of groups
What are the two shortcomings of using multiple unpaired-samples t-tests to compare means from more than two groups?
the number of t-tests can become very large and a series of t-tests has a greater type one error rate than the type one error rate of any single t-test
What influences how spread out the dots are in correlation
the strength of the correlation. (numerical value of pearsons r) influences how spread apart the dots are. Weaker correlations with values closer to zero represents scatterplots with very widely spaced dots. Stronger correlation values with values closer to +1 or -1 produce more closely spaced dots.
What is the purpose of conducting a post hoc test?
to determine where differences among means exist
What is the purpose of a Bonferroni correction?
to prevent type 1 error inflation by decreasing the testwise alpha level. this keeps the experiment alpha low.
What should you do if appendix A2 does not have a row for the exact number of degrees of freedom in your data
use the row that has the highest number of degrees of freedom that is less than your degrees of freedom
When comparing a Pearson's r value to the r crit value, how do you know whether to reject or retain the null hypothesis
we should reject the null hypothesiss if the observed value falls within the rejection region.
After conducting an ANOVA, when should you use a post hoc test?
when the null hypothesis is rejected
When do you use pearson's r?
when you want to privede the measure of the strength of the relationship between two variables.
How do you determine whether to reject or retain the null hypothesis in an ANOVA?
you reject it when there is sufficient evidence to conclude that not all of the means are equal.