Lecture 9 - Single sample and repeated measures t-tests Tests used when population variance is uncertain Flashcards
what are the four possible decision outcomes in null hypothesis significance testing?
the four possible outcomes are:
- true positive (correct Rejection): The null hypothesis (HO) is false, and we correctly reject it
- false positive (Type I error): The null hypothesis is true, but we incorrectly reject it
- true negative (correct retention): The null hypothesis is true, and we correctly retain it
- false negative (Type II Error): The null hypothesis is false, but we incorrectly reatin it
How does increasing sample size affect the false positive rate and statistical power?
increasing sample size does not affect the false positive rate but increases statistical power, which is the probability of correctly rejecting a false null hypothesis
define statistical power and explain how it can be increased
statistical power is the probability of correctly rejecting a false null hypothesis (1- β). It can be increased by reducing variance, lowering the significance threshold, increasing sample size, or increasing the mean difference between groups
what are the conditions for using a single sample z-test?
the conditions for using a single sample z-test are:
- we want to test the difference between a sample mean and a known or specified population mean
- the population variance or standard deviation is known
outline the steps involved in conducting a single sample z-test
the steps involved in conducting a single sample z-test are:
1. state the statistical hypothesis (HO and H1)
2. calculate the standard error of the mean
3. calculate the z-score for the obtained mean
4. compare the obtained z-score to the critical z-score (±1.96)
5. Interpret the result
Provide an example of a single sample z-test scenario and the steps to solve it
Scenario:
Does eating an apple a day keep the doctor away?
Among all patients of a clinic over a year, the number of visits per patient had an average of 3 and a standard deviation of 1.5. A subgroup of 16 patients who reported eating 7+ apples per week had an average of 2 clinic visits. Is this statistically significant evidence that daily apple-eaters have fewer clinic visits on average than other people?
Steps:
1. State hypotheses:
- null hypothesis (HO): μ = 3
- alternative hypothesis (H1): μ ≠ 3
2. Calculate the difference scores:
- see photo
3. Calculate the z-score:
- see photo
4. Compare to critical z-score
- z obtained = −2.67 and z critical = ±1.96
5. Interpret the result:
- Daily apple-eaters have a significantly lower average rate of clinic visits than the average of all clinic patients (z = -2.67, p < .05)
When is a single sample t-test used instead of a single sample z-test?
a single sample t-test is used when the population variance is unknown and must be estimated from the sample data
Outline the steps involved in conducting a single sample t-test
- state the statistical hypothesis (HO and H1)
- Calculate the sample-estimated standard deviation
- Calculate the sample-estimated standard error of the mean
- calculate the t-score for the obtained mean
- use a t table to find the critical t value based on degrees of freedom (df)
- Compare the obtained t-score to the critical t value
- 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 325 ppm. In 2000, the average level from 25 air tests was 360 ppm. Did the average CO2 level change?
Steps:
1. State hypothesis:
- 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
Since t_obtained > t_critical, we reject H0.
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 hypothesis (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 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), critical = ±2.571.
7. Compare the critical t value:
t obtained = 3.03 and t critical = ±2.571
since obtained > 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
