Lecture 10 - Single sample and repeated measures t-tests, and standardised “effect size” Flashcards
Why do we need to estimate population variance for t-tests?
population variance is often unknown, so we estimate it from the sample scores. Estimating the variance comes with two problems: sample variance tends to be smaller than population variance, and sample variance is subject to sampling error.
How do we adjust for the tendency of sample variance to underestimate population variance?
We use a different formula when estimating population variance from a sample: see photo
Dividing by N - 1 instead of N corrects the underestimation
what is a t distribution and why do we use it?
A t distribution accounts for the uncertainty in the variance by making the tails thicker for smaller samples. This makes the test harder to pass when results are uncertain. We use a t table to find critical values based on the significance threshold (α) and degrees of freedom (df).
what are the conditions for using a single sample t-test?
the conditions for using a single sample t-test are similar to the z-test, but the population variance is not known and must be estimated from the sample
Outline the steps involved in conducting a single sample t-test
the steps involved in conducting a single sample t-test are:
1. State the statistical hypotheses (HO and H1)
2. Calculate the sample-estimated standard deviation
3. Calculate the sample-estimated standard error of the mean
4. Calculate the t-score for the obtained mean
5. Use a t table to find the critical t value based on df
6. Compare the obtained t-score to the critical t value
7. Interpret the result
Provide an example of a single sample t-test scenario and the steps to solve it
Scenario:
Did atmospheric CO2 levels rise between 1970 and 2000? In 1970, the CO2 level was 325ppm. In 2000, the average level from 25 air tests was 360ppm. Did the average CO2 level change?
Steps:
1. State the hypotheses:
- null hypothesis (HO): μ = 325
- alternative hypothesis (H1): μ ≠ 325
2. Calculate the sample-estimated standard deviation:
- see photo
3. Calculate the sample-estimated standard error:
- see photo
4. Calculate the t-score:
- see photo
5. find the critical t value:
- for df = 24 and α = 0.05 (two-tailed), t_critical = ±2.064.
6. Compare to critical t value
- t obtained = 3.18 and t critical = ±2.064
7. Interpret the result:
- the average level of atmospheric CO2 in 2000 (360 ppm) was significantly higher than in 1970 (325 ppm), t(24) = 3.18, p < .05
What is a repeated measures t-test and when is it used?
a repeated measures t-test is used to compare the means of two conditions within the same participants. It tests whether the mean difference score between the two conditions is significantly different from zero.
Outline the steps involved in conducting a repeated measures t-test
The steps involved in conducting a repeated measures t-test are:
1. state the statistical hypotheses (HO and H1)
2. Calculate difference scores for each participant
3. Calculate the sample-estimated standard deviation of difference scores
4. calculate the standard error of the mean difference
5. Calculate the t-score for the obtained mean difference
6. Use a t table to find the critical t value based on degrees of freedom (df)
7. Compare the obtained t-score to the critical t value
8. Interpret the result
Provide an example of a repeated measures t-test scenario and the steps to solve it
Scenario: can you tickle yourself? “A tickling robot” tickled each participant’s right hand in two conditions: self-controlled and experimenter-controlled. The time from start of tickling to pulling the hand away was recorded
Steps:
1. State the hypotheses:
- null hypothesis: (HO): μD = 0
- alternative hypothesis (H1): μD ≠ 0
2. Calculate difference scores:
- calculate the difference between self-tickled and experimenter-tickled times for each participant
3. Calculate the sample-estimated standard deviation of difference scores
- see photo
4. Calculate the standard error of the mean difference:
- see photo
5. calculate the t-score:
- see photo
6. Find the critical t value:
- for df = 5 and α = 0.05 (two-tailed), t_critical = ±2.571.
7. Compare to critical t value:
- t obtained = 3.03 and t critical = ±2.571
- since t_obtained > t_critical, we reject HO
8. Interpret the result:
- There was a significant difference in ticklishness between the self-controlled and experimenter-controlled conditions, t(5) = 3.03, p < .05
Why is it important to assess the size of a difference, association, or effect?
Statistical significance is important for reliability, but a statistically significant result can still be unimportant. Assessing the size of the effect helps determine its practical significance
How can we express the size of an effect in standard deviations?
The size of an effect in standard deviations can be expressed using Cohen’s d:
- see photo
This formula expresses the difference between means in standard deviations
Provide an example of calculating Cohen’s d
Scores on a test have a standard deviation of 5. A specially trained group gets a score of 20 and a control group gets a score of 23. the standardized effect size (Cohen’s d) of this difference would be:
- see photo
Outline the complete steps for conducting a single sample t-test with an example
Scenario:
Did atmospheric CO2 levels rise between 1970 and 2000? in 1970, the CO2 level was 325 ppm. In 2000, the average level from 25 air tests was 360 ppm. Did the average CO2 level change?
Steps:
1. State hypotheses:
- null hypothesis (HO): μ = 325
- alternative hypothesis: μ ≠ 325
2. Calculate the sample-estimated standard deviation:
- see photo
3. Calculate the sample-estimated standard error:
- see photo
4. Calculate the t-score:
- see photo
5. Find the critical t value:
- for df = 24 and α = 0.05 (two-tailed), t_critical = ±2.064.
6. Compare to critical t value:
- t obtained = 3.18 and t critical = ±2.064
- since t_obtained > t_critical, we reject HO
7. Interpret the result:
- the average level of atmospheric CO2 in 2000 (360 ppm) was significantly higher than in 1970 (325 ppm), t(24) = 3.18, p < .05
Outline the complete steps for conducting a repeated measures t-test with an example
Scenario: Can you tickle yourself? A “tickling robot” tickled each participant’s right hand in two conditions: self-controlled and experimenter-controlled. The time from start of tickling to pulling the hand away was recorded.
Steps:
1. State Hypotheses:
- null hypothesis (HO): μD = 0
- alternative hypothesis (H1): μD ≠ 0
2. Calculate the difference between self-tickled and experimenter-tickled times for each participant
- see table in photo
3. Calculate the sample-estimated standard deviation of difference scores:
- see photo
4. Calculate the standard error of the mean difference:
- see photo
5. Calculate the t-score:
- see photo
6. Find the critical t value:
- for df = 5 and α = 0.05 (two-tailed), t_critical = ±2.571.
7. Compare to critical t value:
- t obtained = 3.03 and t critical = ±2.571
- since t_obtained > t_critical, we reject HO
8. Interpret the result:
- there was a significant difference in ticklishness between the self-controlled and experimenter-controlled conditions t(5) = 3.03, p < .05
Why is it important to assess the size of a difference, association or effect?
statistical significance is important for reliability, but a statistically significant result can still be unimportant. Assessing the size of the effect helps determine its practical significance.
How can we express the size of an effect in standard deviations?
The size of an effect in standard deviations can be expressed using Cohen’s d: see photo
- this formula expresses the difference between means in standard deviations
