t-Tests
Steps for finding pooled variance
1.) Add the sum of squares of both samples (may need to find) 2.) Add the degrees of freedom of both samples (n-1) 3.) Divide the SS 1+2/ df 1+2
2. Ways to Check the Homogeneity of Variance
1.) Check to see if variances are similar or differ (3-4x larger) 2.) Hartley's F-Max Test
Steps for computing Hartley's f-max test
1.) Compute the variances of each sample (SS/df) 2.) Select the largest sample variance and divide it by the smallest sample variance 3.) Compare the value to the F-max table critical value
Steps for cohen's d formula for an independent measures t-statistic
1.) Find mean difference 2.) Find pooled variance 3.) Square root the pooled variance 4.) Divide the mean difference by the square root of the pooled variance
Steps to finding the independent measures t-statistic
1.) Find mean difference between the two means 2.) Find the population difference between the two populations 3.) Subtract the mean dif. with the population dif. 4.) Find the estimated standard error (*pooled variance) 5.) Solve by dividing the dif. of the means (populations)/ estimated standard error
Steps for independent measures t-test estimated standard error (same sample size)
1.) Find sample variance for both means 2.) Square root the sample variance1/ samplesize1 + samplevariance2/ samplesize2
Three Facts about how we use the Sample Mean (M) to test Hypotheses for an Unknown Population
1.) M is expected to approximate its population mean (why we use M for the hypothesis) 2.) Standard error/ standard distance between the sample mean and population mean determines the reasonable difference between them 3.) Computing a z-score to compare the obtained sample (M) to the hypothesized population mean
3 Realizations after the confidence level interval for a independent measures t-statistic are....
1.) One treatment is larger than the other 2.) The sample difference is in the interval 3.) Your confidence level is how sure your t-value is in the interval (ex. 95% confident that your value is in this confidence interval)
Explain the steps for a hypothesis test with the independent measures t-statistic (regular two-tailed)
1.) Set up your hypotheses/ Choose your alpha level 2.) Locate critical region 3.) Obtain data and compute your t-statistic 4.) Make a decision
Explain the steps for a hypothesis test with the independent measures t-statistic (ONE-tailed/ directional)
1.) State hypotheses/ select alpha level 2.) Locate the critical regions 3.) Collect data and calculate your t-statistic 4.) Make a final decision
3 assumptions underlying the independent measures t-formula
1.) The observations are within each sample are independent 2.) The two selected populations must be normal 3.) The two selected populations must have equal variances
For the homogeneity of variance, why is it important for large discrepancies of the sample size?
1.) Used to accurately interpret the t-statistic with the pooled variance 2.) Leads to negating meaningful info because it is in the critical region
For either a independent measures t-statistic or a single t-statistic, what determines the t-value (boundary)
1.) df 2.) Confidence 3.) Distribution Table (ex. 95% Confident with 11=df, 5% on a two-tailed distribution lets you have the t-value of +-2.210)
Explain STEP1 for a hypothesis test with the independent measures t-statistic (ONE-tailed/ directional)
1st: State your hypotheses as sentences and then as symbols 2nd: choose an appropriate alpha level
Why do we use larger sample sizes (n>30)?
Because they better represent the population
Explain STEP3 for a hypothesis test with the independent measures t-statistic (ONE-tailed/ directional)
Calculate your t-statistic (1. Find pooled variance, 2. use pooled variance in your estimated standard error formula, 3. calculate your t-statistic with your estimated standard error)
Explain the STEP3 for a hypothesis test with the independent measures t-statistic (regular two-tailed)
Calculate your t-statistic (1. find pooled varaince, 2. use it for your estimated standard error, 3. instert that for your t-statistic and solve)
Explain step four of a one-tailed test/t-statistics
Decision to reject the null hypothesis is based on whether the t-statistic is in the critical region; more specifically if the t-statistic is greater than the t-value
What is an alternative method to measure effect size?
Determine how much variability in the scores is explained by the treatment effect ie increase/decreased scores means treatment caused it
What is the goal of Hartley's f-max test?
FAIL TO REJECT H0 (if there are no significant differences between the sample variances than you can proceed calculating your pooled variance knowing there's no bias in your independent-measures t-statistic)
Explain step 2 for t-statistics.
Figure out the degrees of freedom. Then, use it to look in the unit table based on that and whether it is one or two-tailed. Then, draw a normal distribution with your boundaries of the critical regions that you found.
Explain step one of a one-tailed test/ t-statistics
First, as sentences and then as symbols. Next, choose your alpha level
For pooled variance, how would the result be different for two separate sample sizes v. two same sample sizes?
For different sample sizes, it wouldn't be halfway and be pulled towards the larger sample. For same sample sizes, it would be halfway and not pulled towards any sample
Describe how to know whether the effect is significant for hypothesis test with a confidence interval from a independent measures t-statistic
If 0 (population difference) is outside of the interval then it is significant and can reject the null hypothesis. If 0 is inside the interval then it is NOT significant and can fail to reject the null hypothesis.
Explain STEP4 for a hypothesis test with the independent measures t-statistic (regular two-tailed)
If your t-statitic is in either of the critical region then it would greater than the difference between the population (reject the H0 and that there is a significant error between a treatment v. the other)
What would a large sample variance do to the likelihood of rejecting the H0 and measures of effect size?
It would decrease both the likelihood of rejecting the H0 and measures of the effect size.
Explain STEP2 for a hypothesis test with the independent measures t-statistic (ONE-tailed/ directional)
Locate the critical region with (df=n-1 + n-2) BUT the alpha level must be one-tailed in the direction of your choice
Explain STEP2 for a hypothesis test with the independent measures t-statistic (regular two-tailed)
Locate the critical region with the degrees of freedom of both samples (df= n-1 + n-1). Then match the degrees of freedom to the appropriate alpha level and you have your boundaries
What does the numerator/ denominator in estimated Cohen's d represent?
Magnitude of the treatment effect/ Standard Deviation Units (ex. 1.00 standard deviation means that the size of the treatment effect was 1)
Solution to an independent measures study with a large sample variance
Mann-Whitney Test
What is the purpose of the estimated standard error, regardless of t-test or independent measures test?
Measures a reasonable error between the sample and population (how well the sample approximates the population)
While a large sample size increases the chances of rejecting H0, it does doesn't affect what? (independent measures design)
Measures of effect size
What is an advantage for confidence intervals for hypothesis test?
Not only describes the effect size but its significance as well (reject or fail to reject the H0)
What will lead to getting a small t-value (fail to reject the H0)?
Small numerator (small difference of sample-population)
Explain STEP1 for a hypothesis test with the independent measures t-statistic (regular two-tailed)
State your hypotheses and choose an appropriate alpha level
Explain step one for t-statistics.
State your null hypothesis (no treatment effect) and alternate hypothesis (treatment effect). Then, choose an alpha level (0.05, 0.01, 0.001). Also, identify if it is one or two-tailed.
To remove the effect, you would do what to the scores?
Subtract or add points until they are the represented total mean
Also, you can use what other method to find the reduced treatment effect?
Subtraction of the Sum of Squares Method
For the specific variance, sample size, and sample mean to create a large t-statistic, what is the reason?
To approximate the population
When should you use Hartley's f-max test?
Use it prior to calculating your independent measures t-statistic AND pooled variance
The goal of t-statistics is the same as hypothesis testing how?
Using Sample Data from the Treated Sample to Prove a Hypothesis for an Unknown Population (prove there is a treatment effect)
Explain STEP4 for a hypothesis test with the independent measures t-statistic (ONE-tailed/ directional)
You make a decision based off of whether it is outside of the critical region and is in the predicted direction of your hypothesis (bigger than your t-value boundary)
Why would a large sample variance decreased the likelihood of rejecting the H0 and the measures of effect size?
You would not be able to tell a significant difference between the two samples (and their conditions) because they are blended together now that the variability has increased. So, the measures of effect size have substantially decreased and you would have to fail to reject the H0 because there is not a significant difference.
Difference between the variances of z-scores and t-statistics
Z-scores use the actual population variance while t-statistics use the sample variance because the population values are unknown
A small sample size means....
a large error because score are less constant (will vary more-won't approximate the population) (creates a small t-statistic that will less likely produce a treatment effect/ decreases the likelihood of rejecting H0)
The r2 formula for an independent measures t-statistic/ test is the same as...
a single t-statistic (t^2/ t^2 +df)
pooled variance
a weighted average of separate (different) sample variances; ultimately allows the bigger sample to carry more weight
estimated standard error
an estimate of the standard error that uses the sample variance (or standard deviation) in place of the corresponding population value because the population is unknown
A large sample size influences the width of a confidence level by...
creating a smaller interval (what you want) and smaller standard error
In general the confidence level comes from...
df (assumes the t-statistic)
Overall, the independent measures t-statistic describes what?
difference of the sample data - difference of the population data/ estimated error of the sample data
Cohen's d formula for an independent measures t-statistic/ test
estimated d= M1-M2/ square root of pooled variance
Estimated Cohen's D formula
estimated d= sample mean- population mean/ sample standard deviation
estimated standard error independent measures t-test (different samples)
estimated standard error= square root of pooledvariance/ samplesize1 + pooledvariance/ samplesize2 *pooled variance is the same for both
formula for estimated standard for independent measures t test (same sample size)
estimated standard error= square root of sample variance1/ sample size1 + sample variance2/ sample size2
Hartley's F-Max test FORMULA
largest sample variance/ smallest sample variance
A narrow width means it is....
more specific
degrees of freedom
n-1; determines scores that are independent/ free to vary because the sample mean places a restriction on values of one of the scores
Confidence interval formula for an independent measures t-statistic/ test
population mean1- population mean2= Mean1- Mean2 +- t(estimated standard error)
confidence interval formula
population mean= Sample mean +- t-value(estimated standard error)
pooled variance formula
s2p= SS1 + SS2/ df1 + df2
repeated measures design (within subjects design)
same group of participants (before and after method with one group but two different results)
Like the confidence level for a single t-statistic, The confidence level for the independent measures t-statistic is still affected by ...
same size and size of confidence interval (large sample size and small confidence level for a narrow width)
Cohen's D and r^2 aren't affected by...
sample size (r^2 may be slightly changed though)
Hartley's F-max Test
simplest way to check the homogeneity of variance by comparing two independent samples' based on the principle that the sample variance is unbiased because both population variances are the same
homogeneity of variance
the assumption that the variances are equal for the two (or more) groups you plan to compare statistically
IF you compute a small value from the Hartley f-max test and smaller than the f-max table critical value, then....
the homogeneity of variance is not violated
IF you compute a large value from the Hartley f-max test and larger than the f-max table critical value, then....
the homogeneity of variance is violated
High variance will reduce
the measure of the effect size and reduce the probability of rejecting the null hypothesis
Mann-Whitney test
the nonparametric version of the Independent samples t-test for ranked scores (test for ordinal data)
r^2
the percentage of variance accounted for by the treatment effect; can be found based on the outcome of a t-test
M (sample mean) represents the
treated sample
2 assumptions of the t-test
1.) There must be two independent observations (the first event will not affect the second event's probability because they're not predictable). 2.) The population sampled must be normal but it can be violated with a large sample because it doesn't affect the validity of the hypothesis test (small samples mean the normal population distribution is important)
Cohen's d for population
Cohen's d= population mean- population mean/ population standard deviation
t-distribution unit table
Column 1: listed degrees of freedom for t-statistics (ex. df=3) Row 1: proportion of one tail (p= 0.05) Row 2: proportion of two tails (p=0.10) Body: t-values that determine the boundaries that separate from the main body (t= +2.353 for one tail, t= +- 2.353 for two tails)
Describe the Relationship between the degrees of freedom and t-statistics
Degrees of freedom are used to determine how well the sample variance approximates the population variance AKA if the t-statistics is a good approximation to the z-score (higher df= higher chance t-statistics are a good approximation of the z-score)
What is the goal of the hypothesis test?
Determine whatever the obtained difference between the sample mean and population mean is greater than expected by chance (extreme because there's a treatment effect)
Why can't you use the Cohen's d population formula?
Do not know the population values so you would use sample values like the sample variance (estimated Cohen's d)
Con of Hypothesis Testing
Doesn't really evaluate the size of a treatment effect (only determines that the difference between the null hypothesis and mean is greater by chance-treatment effect)
How can t-statistics be used?
In situations where you don't know the population (wishful thinking, logical prediction, theory)
A large sample variance means...
a larger error because scores are more scatted (less likely to obtain a significant treatment effect because they're inconsistent /decreases likelihood of rejecting H0)
A small sample size influences the width of a confidence level by...
creating a larger interval and larger standard error
Estimated Cohen's d
estimate of effect size that is most often used with a t test; substitute test statistics instead of population parameters
The goal of an independent measures hypothesis test is to
evaluate the mean difference between two populations using two samples (does one treatment work than the other)
The difference between the sample mean and population mean represents
how the scores have shifted/ increased or decreased deviations (ex. 44-40 has a 4 point difference that means that scores away towards the right by 4 deviations)
The t-statistic uses the H0 as a....
hypothesis based on logic because the population mean doesn't need to be known
independent measures t-statistic formula
t= (M1-M2)- (pop1. -pop.2)/ S (M1-M2)
r^2 formula
t^2/t^2+df (will be the same as method one)
t-distribution
the complete set of t values computed for every possible random sample for a specific sample size or a specific degrees of freedom; approximates the shape of a normal distribution
4 Steps for t-statistics
1.) State Hypotheses and Select an Alpha Level 2.) Locate Critical Regions 3.) Calculate the t-statistic 4.) Make a Final Decision
Steps for using a one-tailed test and t-statistic
1.) State hypotheses/ select alpha level 2.) Locate critical region 3.) Calculate t-statistic 4.) Make a decision
What accounts for a small treatment effect, medium effect, large effect for r^2
1.) 0.01 2.) 0.09 3.) 0.25
3 points to consider for estimated standard error (independent measures t-test)
1.) 2 sources of error because the first mean approximates the first population/ the second mean approximates the second population 2.) Both estimated standard errors of mean using this formula (estimated error= square root of sample variance/ sample size) 3.) Final formula for the estimated standard error would be the square root of both sample variances/ sample sizes of the two means
A t-distribution will be alike to a normal distribution in what three ways?
1.) Bell-shaped 2.) Symmetrical 3.) Mean is equal to 0
2 Ways to Measure the Reduced Treatment Effect When it is Removed
1.) Compute Sum of Squares for each set of scores (treatment effect and removed treatment effect) 2.) Use r2
Without the treatment effect, what two things happen?
1.) Deviations aren't as far 2.) Scores are closer to the mean
Explain steps for method one to measure the reduced treatment effect
1.) Find SS for each set of scores (X-M)^2 2.) Find difference between the sets of scores SS 3.) Divide difference by SS of the treatment effect
Steps to find estimated Cohen's d
1.) Find sample variance (SS/n-1) 2.) Find the sample standard deviation (s= square root of sample variance/ df) 3.) Substitute sample standard deviation into estimated cohen's d formula and solve (sample mean-population mean/ sample standard deviation)
What do you do if there are large degrees of freedom that aren't in the unit table?
1.) Find t-statistics for both degrees of freedom between the given one 2.) Then, use the larger t-value (if sample t-statistics is greater than larger t-value then it is in the critical region automatically/ can reject the null hypothesis
Explain steps for calculating your t-statistic (Step 3)
1.) Find variance (SS/n-1) 2.) Find the estimated error with the variance (square root of variance/ sample) 3.) Find the t-statistic with the estimated standard error (t= sample-population/ estimated standard error)
How does a t-distribution differ from a normal distribution?
1.) Flatter/ more spread out than a normal distribution because it has more variability
Difference of Population between Hypothesis testing and t-statistic
1.) Hypothesis testing requires a known population mean to know if the treatment affects the mean 2.) t-statistic doesn't require a known population mean (or population variance) to serve as a standard to know if there's an effect
What are two factors that can determine a critical region for a one-tailed test?
1.) If the sample mean is in the direction of your hypothesis (bigger or smaller than hypothesized population) 2.) IF the t-statistic is greater than the critical region boundary (if it is bigger then you would reject, but if it is smaller then you would fail to reject)
2 Categories for Research Designs Used to Obtain 2 Sets of Data
1.) Independent Measures 2.) Repeated Measures
What factors affect the width of interval to be narrow?
1.) Large sample size 2.) Small confidence level
Explain how the degrees of freedom change the shape of the t-distribution *what are the results?
1.) Larger degrees of freedom means that it is closer to the normal distribution shape because there is less variability (narrow/less spread out) 2.) Smaller degrees of freedom means that is is not close to the normal distribution shape because there is more variability (flatter/more spread out)
What do you need for t-statistics?
1.) Null hypothesis 2.) Sample from the unknown population
The estimated standard error is affected by what two factors?
1.) Sample variance 2.) Sample size
Two Reasons Why we use variance instead of standard deviation for t-statistics
1.) Sample variance is an unbiased, accurate estimate of the population variance 2.) We use sample variance for all different t-statistics to always be computed as SM= square root of the sample variance/ n
2 Factors that Affect the Width of a Confidence Interval
1.) Size of Confidence Level (larger t-value v. smaller t-value) 2.) Sample Size (large sample v. small sample)
Overall Characteristics that Will Increase the Likelihood of Rejecting the H0/ Finding a Treatment Effect (what will get you a large t-statistic)
1.) Small variance 2.) Large sample size 3.) Small sample mean
How do you construct a confidence level?
1.) Use n, df to estimate the t-value (ex. n=9, df=8) 2.) Look in t-table to find the confidence levels AKA critical regions (df=Confidence level: 8=80%; and because it measures 100% for the entire distribution so subtract the confidence level to find boundaries: 100-80 20% would be the extremes where the boundaries: +- 1.397) 3.) Use confidence interval formula (population is estimated to be between the positive and negative outcome)
Two things to remember for independent measures t-statistic
1.) has the same basic t-test structure 2.) Doubles all elements v. single sample t-test
Explain step 3 of t-statistics.
Calculate the t-statistic because it will act as a z-score (in or not in the critical regions).
Explain step three of a one-tailed test/t-statistics
Calculate your t-statistic as you would for a regular two-tailed test (Calculate sample variance, calculate estimated error, then calculate t-statistic)
Explain step two of a one-tailed test/t-statistics
First, find the df (n-1). Then, locate the t-value (boundary) when you df matches to the one-tailed row (0.01).
How does the t-distribution shape compare to the normal distribution?
Flatter/more spread out v. narrow/ less spread out
Explain step four for t-statistics
If your t-statistics is in the critical regions, then you can reject H0. If you t-statistics isn't in the critical regions, then you fail to reject H0.
For the independent measures t-statistic formula, why would you (could) get rid of the population difference? What does that mean for the equation?
It automatically equals zero so it would just be measuring the sample difference/ estimated standard error for two means
What will lead to getting a large t-value (reject the H0)?
Large Numerator (large difference of sample-population)
What is the issue with using z-scores?
Requires information about the population's variance/ standard deviation even though you don't know them; you need it for the standard error
estimated standard error formula
SM= square root of s^2/ n
Explain the difference between the estimated standard error and the actual standard error
The sample variance changes so the estimated standard error changes as well (both numerator and the denominator). However, the population variance doesn't change so the actual standard error won't change (only the numerator changes)
The degrees of freedom for a t-distribution determines what?
The shape of the distribution
Why do we use a t-statistic?
Using sample values to know the sample variance/ standard deviation
A small sample variance means....
a small error because scores are less scattered (more likely to obtain a significant treatment because they're consistent/ increases likelihood of rejecting H0)
A large sample size means...
a small error because scores are more constant (won't vary as much-approximates the population) (creates a large t-statistic that will more likely produce a treatment effect/ increases the likelihood of rejecting H0)
One-tailed test (directional test)
a test that rejects extreme outcomes in one specified tail of the distribution (ex explanatory investigations, pilot studies, priori studies)
A large confidence interval causes
a wider interval or thick width (ex. 95% confidence level increases the width)
Cohen's d
an inferential statistic for measuring effect size
Confidence level is within the...
bell curve
The confidence level is constructed around the...
mean
While Cohen's d measures effect size, the t-statistic measures...
significance
A small confidence interval causes a
smaller interval or narrow width (what you want!)
the estimated standard error is equal to the....
square root of the sample variance/ the sample size
Sample size not only affects the width of a confidence interval but also the
standard error
t statistic formula
t = (M-u) / Sm
Sample size is not an adequate substitute for calculating...
the measure of the absolute effect size (cannot be used like cohen's d or r^2)
confidence interval
the range of values (based off a sample statistic) within which an unknown population parameter is estimated to lie; another way of describing treatment size
Population mean represents the...
untreated population from the H0
t statistic
used to test hypothesis about an unknown population; uses the estimated standard error (SM)
Independent measures design (between subjects)
uses 2 completely separate, different groups of participants for comparison between two methods (ex. sample of men v. sample of women)
What does the numerator/denominator and result for method one of measuring the reduced treatment effect represent?
variability accounted for/ total variability= removed treatment effect (how much the variability reduces)
If the measure of an effect size is not significant, then....
you fail to reject the null hypothesis (no change)
If the measure of an effect size is significant, then....
you reject the null hypothesis (especially for smaller treatment effects- change)