Ch. 9-10: One-sample and paired t tests
These conditions describe the appropriate time to use a one-sample t test.
What are ...?
1) We have data from a single sample with only one set of observations from each subject
2) We want to compare the sample mean against a known population value (or value expected by chance)
3) We don't know the population standard deviation (otherwise we use a z test!)
These conditions describe the appropriate time to use an independent-samples t test.
What are ...?
1) We have data from two samples (i.e. two groups) with one set of observations for each subject
2) We want to determine whether there is a difference in the means between the two groups
3) We don't know the population standard deviation (otherwise we use a z test!)
These conditions describe the appropriate time to use a one-way ANOVA test.
What are ...?
1) We have data from multiple (at least 2) groups with one set of observations from each subject
2) We want to determine whether there is a difference in the means between the groups
3) We don't know the population standard deviation (otherwise we would use a z test! Or even a "ZANOVA")
This is the difference between a one-way ANOVA design and a one-way repeated measures ANOVA design.
What is whether you have multiple observations from each subject? (i.e. one-way ANOVA = multiple groups, 1 data point per subject; repeated measures = multiple groups, multiple data points per subject)
This is the term for the experimental phenomenon in which measurements taken in a repeated-measures design are dependent on which condition is presented first, second, etc.
What are order effects?
These conditions describe the appropriate time to use a paired-samples t test.
What are ...?
1) We have data from a single sample with only two sets of observations from each subject (e.g., before and after some treatment or training)
2) We want to determine whether there is a difference in mean values between the two sets of observations
3) We don't know the population standard deviation (otherwise we use a z test!)
This is the meaning/definition of "degrees of freedom".
What is the number of freely varying parameters once a statistic is known?
In other words, if we have 3 numbers as well as the mean of the numbers, once we know two of them, the third value is automatically fixed
M = (n1 + n2 + n3)/3 => n3 = 3*M - n1 - n2
These conditions describe the appropriate time to use a post-hoc test (e.g. Tukey's HSD)
What is ...?
1) I have finished running an ANOVA (or other multi-group statistical test) and gotten a significant F-value, indicating at least one group mean differs from the other groups?
2) I want to look more carefully at the pairwise relationships between my groups to figure out which one are different?
X = (Between-groups variance)/(Within-groups variance)
X is this test statistic.
What is the F statistic?
This is a statistical practice in which conditions are presented in different orders to different subjects.
What is counter-balancing?
Suppose I collect data from n=78 students to run a single-sample (or paired) t test. This number is the degrees of freedom for my study.
What is df = n - 1 = 78 - 1 = 77?
A researcher collects data from 100 subjects using 2 samples of equal variance. The researcher reports the results as following:
t(99) = 3.2, p < 0.001
What is the error?
What is incorrect reporting of degrees of freedom (should be n - 2 = 98)?
(df=n-1 is for the calculation for single-sample and paired t-tests only)
This is the name of the measure of effect size when using an ANOVA statistical test.
What is R2 (οr η2 or ω2)?
A researcher collects data on reading speed from 4 local high schools to compare against the state-average reading speed. These are the null and alternative hypotheses for this experiment.
What is:
H0: The mean reading scores across all schools are the same as the population mean reading score (i.e. all groups are equal)
H1: At least one group mean reading score differs from the population mean reading score
Draw the 'latin squares' for four groups: G1, G2, G3, G4.
What is ... ?
[[G1, G2, G3, G4],
[G2, G3, G4, G1],
[G3, G4, G1, G2],
[G4, G1, G2, G3]]
I sample 7 subjects on the number of jumps they can get in a row without stopping with a jump rope. After recording their initial attempt, they attend my world-renowned two-week intensive jump-rope training summer course. Once they complete the training I test them again. The scores are as follows:
Time 1: [7, 14, 31, 11, 9, 3, 12]
Time 2: [19, 31, 62, 11, 17, 2, 15]
Using SPSS, compute and report the test statistic and effect size in APA format.
What is:
The training worked! Subjects reported more skips without stopping after the training (M = 22.86) than before (M = 12.43), t(6) = 2.52, p = 0.023, d = 0.951
Suppose I have data from two samples;
Group 1: [12, 17, 15, 19, 23, 25, 11, 18, 15]
Group 2: [14, 18, 25, 30, 27, 26, 18, 15, 19, 21, 23]
Using SPSS, assuming equal variances between groups, report and interpret the 95% confidence interval of the difference.
What is [-9.96,1.15]? Since this interval includes 0, we cannot conclude these groups are significantly different; i.e. we fail to reject the null hypothesis (assuming two-tailed)
A researcher gathers data from 4 age groups to look at the relationship between reading and taking naps over adolescence development. The researcher records the number of times each subject yawns while being read a children's story. She collects data from 31 subjects in each group, runs a between-groups ANOVA and gets a test statistic of 4.61 (significant with p < .01). How should this researcher report her findings (in APA format)?
What is ...?
F(3,27)=4.61, p < 0.01, R2=?
Generally:
F(dfB,dfW) = F_val, p < α
where dfB = k - 1 (number of samples - 1)
and dfW = N - k (number of subjects - number of samples)
When we only have 2 samples, we can still run a one-way ANOVA. In this case, the F distribution relates to the t distribution according to this equation.
What is F = t2?
(The F distribution is generalization of t tests designed to work for any number of groups -- if you only have two groups, you can still run it, it's just the same as running a t test!)
This is a definition for demand characteristics.
What is when the expectations of the researcher are made known to and influence the measurements of the subject?
E.g., if the researcher knows that they are giving somebody a placebo, they might interact with the subject differently than if they know it is a legitimate drug being tested; if the subject can detect this difference, their expectation can change the course of their treatment
Suppose I collect a poll of my students' (n=30) daily screen time and find a mean reported "cell phone usage during class" of 46.5 and a standard deviation of 6 minutes. I hypothesize that I am a more engaging statistics teacher than the national average (whose students' mean screen time = 49.5 minutes).
This is the 95% confidence interval for my students' population mean in-class screen time and the appropriate statistical interpretation.
Daily double: Does your interpretation change if I use a 99% confidence interval instead?
What is:
[Check t-table for one-tailed p < 0.05 with df = 29; t = +/- 2.045]
[46.5 +/- 2.045 * (6 / sqrt(29)]
= [sample mean +/- (t threshold) * standard error]
= [44.22, 48.78]
minutes? Since this interval excludes the population mean 49.5 we can reject the null hypothesis and conclude that I am (slightly) more engaging than the average stats TA. Yay!
Daily double:
[46.5 +/- 2.757 * (6 / sqrt(29)]
= [sample mean +/- (t threshold) * standard error]
= [43.43, 49.57] -- The 99% confidence interval includes the population mean of 49.50, so we cannot say with 99% confidence that I am more engaging than the average stats teacher!
Suppose I have data from n=30 subjects each across two samples of people who are allergic to dogs (group 1) or cats (group 2).
The sample means are ndogs = 32 sneezes per hour and ncats = 61 sneezes per hour while exposed to the animal that they are allergic to (mean experiment!). The t test is comes out significant with t(29) = 3.0. Compute the sample standard error (assuming equal sample variances).
What is 9.67 sneezes per minute?
t = (mean1 - mean2) / (standard error)
3.0 = (61 - 32)/(SE)
3.0 * SE = 29
SE = 29/3 ~= 9.67
Suppose you collect data from 3 groups of skateboarders across different cities, recording how many gnarly tricks each subject can do per minute. We sample 5 skateboarders per group;
Group 1: [2.5, 4.1, 3.7, 5.1, 3.3]
Group 2: [2.3, 6.1, 4.1, 3.2, 2.0]
Group 3: [1.8, 1.1, 0.6, 0.0, 1.3]
Using SPSS, compute and report the test statistic, p value, effect size, and explain the results of the post-hoc Tukey test.
What are ...?
F(2,12) = 8.74, p = 0.005
(Reject the null hypothesis: at least one skater is different)
Effect size η2 = 0.529 (large effect size)
Post-hoc Tukey reveals no significant difference in mean gnarly tricks per minute between skaters 1 (M=3.74) and 2 (M=3.54), but a significant difference between skater 3 (M=0.96) and the others (p = 0.07 vs #1, p = 0.12 vs #2).
A researcher gathers data from two cities (Boston and Philadelphia). They collect blood pressure data from a single group before and after ingesting a new heart medicine. They run a statistical test and find a significant difference between the cities.
Afterwards, they collaborate with another researcher to collect data from subjects in a third city (San Francisco), and they run another statistical test using data from all 3 groups (cities).
These are the two statistical tests the researchers used.
Daily double: This is the statistical mistake the researchers made.
What are an independent-samples t test and between-groups ANOVA?
Daily double: Multiple comparisons issue! Once you run a statistical test, you can't just add more data and rerun -- knowing the result of the first test influences the result of the second test. (To fix this, either wait to run your test until all data are collected, or add a post-hoc correction to your test)
Suppose you have data from 5 groups; A, B, C, D, and E. You perform a planned-comparisons test (or contrast) to find whether A differs from C & D and whether B, C, and E differ from D. These values are the planned comparisons coefficients you would use for these tasks.
What are... ?
A vs C + D: 2 0 0 -1 -1
B + C + E vs D: 0 1 1 -3 1