Ch7 - Hypothesis Testing with z Tests Flashcards
What are the 3 ways to identify the same point beneath the normal curve?
raw score, z score, and percentile ranking
What information does the z table give us?
- provides the % of scores between the mean and a given z value
- how we transition from one way of naming a score to another
- gives us a way to state and test hypotheses by standardizing different kinds of observations onto the same scale
In using the z table, we can determine:
- Percentile rank - the % of scores below the z score
- % of scores above the z score
- The scores at least as extreme as the z score, in both directions (Scores extreme enough to be below the z score or above - thus, can double the total percentage, due to the curve being symmetric)
The z table and distributions of means: shift focus from…
- the z score of an individual within a group to the z statistics for a group
- Changes to the equation: we use means rather than individual scores
What are statistical assumptions?
- Think of statistical assumptions as the ideal conditions for hypothesis testing
- More formally - assumptions: the characteristics that we ideally require the population from w
What other tests can the assumptions for the z test apply to?
- parametric tests: inferential statistical analyses based on a set of assumptions about the population
- in contrast - nonparametric tests: inferential statistical analyses that are not based on a set of assumptions about the population
Assumption 1
- the DV is assessed using a scale measure
- If it’s clear that the DV is nominal or ordinal, we could not make this first assumption and thus should not use a parametric hypothesis test
Assumption 2
- The participants are randomly selected
- Every member of the population of interest must have had an equal chance of being selected for the study
- This assumption is often violated - most likely, participants are convenience samples
- Must be cautious of generalizing
Assumption 3
- The distribution of the population of interest must be approximately normal
- Many distributions are approximately normal, but it’s important to remember that there’s exceptions to this guideline
- Because hypothesis tests deal with sample means rather than individual scores, as long as the sample size is at least 30, it’s likely that this third assumption is met
What are Robust hypothesis tests?
- produce fairly accurate results even when the data suggest that the population might not meet some of the assumptions
- Many parametric tests can be conducted even if some of the assumption are not met, and are robust against violations of some of these assumptions
The 6 steps of hypothesis testing
- Identify the populations, comparison distribution, and assumptions
- State the null and research hypothesis
- Determine the character
- Determine the critical values, or cutoffs
- Calculate the test statistic
- Make a decision
1: Identify the populations, comparison distribution, and assumptions
The 6 steps of hypothesis testing
We consider the characteristics of the data to determine the distribution to which we will compare the sample
- 1) First we state the populations represented by the groups to be compared
- 2) Second we identify the comparison distribution
- 3) Third, we review the assumptions of hypothesis testing
(The information gathered in this step helps us to choose the right hypothesis test)
2: State the null and research hypothesis
The 6 steps of hypothesis testing
- Hypotheses are about POPULATIONS, not samples
- State the null/research hypotheses in both words and symbolic notation
3: Determine the characteristics
The 6 steps of hypothesis testing
- State the relevant characteristics of the comparison distribution (distribution based on the null hypothesis)
- For z tests, we will determine the mean and standard error of the comparison distribution
- These numbers describe the distribution represented by the null hypothesis and will be used when we calculate the test statistic
4: Determine the critical values, or cutoffs
The 6 steps of hypothesis testing
- Critical values/cutoffs: the test statistic values beyond which we reject the null hypothesis (indicate how extreme the data must be, in terms of the z statistic, to reject the null hypothesis)
- In most cases, we determine 2 cutoffs: one for extreme samples below the mean and one for extreme samples above the mean
4: Determine the critical values, or cutoffs - CRITICAL REGION
- Regardless of the chosen cutoff, the area beyond the cutoff, or critical value, is often referred to as the critical region: the area in the tails of the comparison distribution in which the null hypothesis can be rejected
- These percentages are typically written as probabilities; that is, 5% would be written as 0.5
- THE PROBABILITIES USED TO DETERMINE THE CRITICAL VALUES, OR CUTOFFS, IN HYPOTHESIS TESTING ARE ALPHA LEVELS (aka p levels)
5: Calculate the test statistic
The 6 steps of hypothesis testing
- Use info from step 3 to calculate the test statistic, in this case the z statistic
- Can then compare the test stats to the critical values to determine whether the sample is extreme enough to warrant rejecting the null hypothesis
- The p value: probability of finding this particular test statistic, or one even large, if the null hypothesis is true - that is, if there’s no difference between means
6: Make a decision
The 6 steps of hypothesis testing
- Using the statistical evidence, we can now decide whether to reject or fail to reject the null hypothesis
- Reject: if the test statistic is beyond the cutoffs, OR
- Fail to reject: if the test stat is not beyond the cutoffs
- If we reject, we know that the p value associated with the test statistic is smaller than the alpha level of 0.05
When is a finding statistically significant:
if the data differ from what we would expect by chance if there were, no actual difference
Hypotheses have two possible sets:
- Directional
- Nondirectional
Directional
Hypotheses have two possible sets
- predicting either an increase or decrease
- One-tailed test: a hypothesis test in which the research hypothesis is directional, positing either a mean decrease or a mean increase in the DV, but not both (ONE OR THE OTHER), as a result of the IV
When is a one-tailed test used?
Hypotheses have two possible sets
Only used when the researcher is absolutely certain that the effect cannot go in the other direction or the researcher would not be interest in the result if it did
Nondirectional
Hypotheses have two possible sets
- Predicting a difference in either direction
- Second set of hypotheses (the null) - nondirectional
- Two tailed test: in which the research hypothesis does NOT indicate a direction of mean difference or change in the DV, but merely indicates that there will be a mean difference (very simplified, will there or won’t there be a mean difference)
2 recommendations for reporting results:
Data ethics
- Distinguish clearly between the original hypotheses and those developed after seeing the results
- Report all variables, experimental conditions, and analyses
p-hacking
Data ethics
the use of questionable research practices to increase the chances of achieving a statistically significant result
- Combines the symbol p with “hacking”
- EX: create criteria for removing scores - like outliers - after initial analyses have been performed, rather than before
- Can explain the push to preregister studies
LECTURES
Can we use this same distribution (the normal distribution) to judge whether a sample group mean comes from a population? WHY?
- YES
- Central limit theorem: as N= a min. 30, sampling distribution of the mean is normally shaped even if actual population distribution isn’t normally shaped
- Therefore, can use the normal distribution to model the sampling distribution of the mean
- Helps us find out probabilities of getting various sample means under the null hypothesis, where the population mean is ZERO!
What are two ways we can use the normal distribution?
- Raw scores => z scores => percentile
- Raw group mean => z statistic => percentile for the mean
What model can we use to solve: “did this sample come from the same population as before? (the null hypothesis, average/”no effect”, population)? Or does it differ (AKA, is it an extremity/outlier)?”
- Use the normal distribution to model the null hypothesis sampling distribution of the mean
- This is where we can start plotting our p values to determine our answer - if the sample came from the null hypothesis, or extends into the critical regions (where we can then say it is NOT from the same population as before)
Two-tailed null hypothesis, H0: μnew = μoriginal, means…
That the sample is just another random sample from the same parent population
Two-tailed research hypothesis, H1: μnew DOES NOT EQUAL μoriginal, means…
That the sample is NOT just another random sample from the same parent population; can prove as statistically significant
How do we know if we can reject the null hypothesis?
- Alpha level: The proportion of scores that allows us to argue that a sample didn’t come from the null hypothesis population
- THAT IT DIFFERS, allowing us to reject the null hypothesis
Alpha of rejection cuts off what % of values?
2.5%
To calculate p value:
Can look at the z table at the “area beyond z”, and the p value is the number listed
- Two-tailed tests - double the amount that’s beyond a certain z score
- One-tailed - keep the amount that’s beyond a certain z score
Difference between p value and alpha level:
- Choose an alpha level. For example, if you’re 95% confident, your alpha level is 0.05.
- Run the hypothesis test and calculate the p-value.
- Compare the p-value to the alpha level.
- If the p-value is less than the alpha level, reject the null hypothesis.
- If the p-value is greater than the alpha level, do not reject the null hypothesis.
NHST: determine alpha -> calculate z statistic -> using z, look up p value -> compare p value to alpha level to determine statistical significance
Statistical significance:
- A result that’s this extreme or more extreme is unlikely to occur by chance alone, if the null hypothesis is true
- PASS/FAIL
What are the only two outcomes of NHST?
pass/fail test - you either reject or retain the null ONLY
- Null hypothesis: this sample came from the same population
- Two-tailed research hypothesis: this sample came from a different population that is either shorter or taller than the existing population
When do we choose the z test?
- When we have a sample that we are comparing to a population for which we know the mean and SD
- U and o are population parameters
- z tests are a parametric test!
- M and s are sample statistics
