Psych 210 Exam 2
What happens if we set alpha larger?
- the size of the critical region increases -we are more likely to make a type 1 error because when alpha increases and is larger, the critical region increases. we are more likely to make assumption of type 1 error
What happens if we set alpha smaller?
-size of critical region decreases -we are more likely to make a type 2 error because when alpha decreases the critical region decreases
What is the range of values for probability? Why do they have this range?
0-1 because it is a proportion or decimal
What will you get if you add the "tail" and the "mean to z" on the same side of the distribution?
0.50
4 steps for working with the Unit Normal Table
1. Draw the curve 2. Label known scores and proportions 3. Indicate what you are looking for- a proportion or score 4. Use unit normal table to find unknowns
What are the 2 requirements for a random sample?
1. Every item/individual must have an equal chance of being selected 2. If more than one item/individual is to be selected for the sample, the probability must stay constant for each and every selection. The selection of the items is independent of previous selections.
What is the distribution of sample means?
distribution of average scores from the population which the scores were sampled from
What would happen if probability was calculated using a biased sample?
if the probability was calculated using a biased sample, it would lead us to over-estimate and/or underestimate the corresponding population probabilities
Power
if the sample size was small then even a fairly large difference in sample means might not be significant -if the difference is not significant, then no strong conclusions can be drawn about the population means -when an effect is not significant, the result is inconclusive -power is defined as the probability of correctly rejecting a false null hypothesis -the probability of failing to reject a false null hypothesis is often referred to as B. Therefore, power can be defined as power-1-B
Why can't cohen's d be negative?
it is a measure of standard deviation and you can't have negative standard deviation
Why is the SDOM important?
it is critical for hypothesis testing and inferential statistics because it links probability to samples. with the SDOM we can calculate the probability of obtaining a sample mean and determine if a sample mean is high probability and expected or low probability and unexpected.
Type 1 error
occurs when a significance test results in the rejection of a true null hypothesis -if probability value is below .05, then the null hypothesis is rejected -another convention is to reject the null hypothesis is to reject the null hypothesis if the probability value is below .01 -threshold for rejecting the null hypothesis is called the alpha(a) level. aka significance level -the lower the alpha level, the lower the type 1 error rate -if the null hypothesis is false, it is impossible to make type 1 error
Critical region
the area of a normal curve that contains extreme sample values that have a low probability of occurring if the null hypothesis is true -area under the curve that allows you to reject or fail to reject the null hypothesis. contains sample means that are not likely to be picked from the SDOM if the null hypothesis were true -allows us to make decision about the null hypothesis. if it is a random sample mean from SDOM it is unlikely that the sample would be accepted if the null hypothesis is true
What does the standard error of the mean measure or tell you about the distribution of sample means?
the average error, or distance that is expected between sample mean and population mean
Critical value
the boundary of the critical region. it corresponds to the chosen alpha level
Probability
the chances of an event occurring(how likely it is)
Formula for calculating z score for a group or sample
z score= (sample mean-population mean)/standard error standard error= standard deviation/square root of sample size
Formula for calculating z-score of sample mean(test statistic)
z= sample mean-population mean/standard error standard error= standard deviation/square root of sample size
5 things that influence power
Beta(as beta increases, power decreases) Effect size(as effect size increases, power increases) Alpha(increase, power increases) N- sample size(increase, power increases) Standard deviation(increase, power decreases)
Do we ever accept the null or alternative hypothesis?
No. We NEVER accept or disprove the null hypothesis -we can only talk about likelihood that the null hypothesis COULD BE correct -this is the p-value which tells us the likelihood/probability of obtaining our sample from the sample distribution of means if the null hypothesis was true -if the likelihood of obtaining that sample from a sample distribution is really low, then we reason that it is very unlikely that the null hypothesis is correct and we can reject the null hypothesis
Does a one-tailed or two-tailed test predict direction?
One-tailed: direction Two-tailed: doesn't predict direction
Does probability increase or decrease when drawing with sampling w/o replacement?
Probability increases when drawing w/ sampling without replacement
If there is enough evidence, what does the researcher conclude and do(in a hypothesis test?
Reject the null hypothesis or fail to reject if there is not enough evidence--> fail to reject
Hypothesis Test
assesses possibility of a hypothesis using a sample -statistical method that uses sample data
How does hypothesis testing begin?
begins with the assumption that there is no treatment effect -the study gathers evidence in form of data with the hope that the data will prove or disprove the hypothesis -if there is enough evidence, the researcher rejects the null hypothesis or not(alternative) -if there is not enough evidence, the researcher fails to reject the null hypothesis
The Alternative Hypothesis(Ha)
claim about the population that is contradictory to the null and what we conclude when we can't accept the null. This is usually what researcher is trying to prove(must win with significant evidence to overthrow the null)
What happens to standard error of mean as sample size increases?
As sample size increases, the standard error of the mean decreases.
7 steps for hypothesis testing
1. Identify Variables and number of tails -independent variable, dependent variable, tails(one or two) -one-tailed--> specify the direction of the difference or effect(null always contains <>(or equal to) or = -two-tailed-->focus on extremity of scores(above or below mean) 2. State hypotheses -null hypothesis: there is no difference/effect/change -alternative hypothesis: there IS some sort of difference/change/effect 3. Determine decision criteria- draw distribution and mark critical regions -scientific reasoning starts with the assumption of no effect(null is true) -the demonstrate that we have an effect, we need to show that our result would be very unlikely if the null were true -alpha level- probability value that defines "unlikely", aka significance level -critical region- area of a normal curve that contains extreme sample values that have a low prob of occurring if the null is true -critical value- boundary of critical region; corresponds to chosen alpha level 4. Compute test statistic(if appropriate) -test statistic is always a ratio(a ratio comparing the obtained difference between sample mean and pop mean to the difference expected by chance -z= sample mean-pop mean/standard error -standard error= standard deviation/square root of sample size 5. Make a decision -reject null hypothesis- test statistic more extreme than the critical value, does fall in critical region--> null is not supported= significant differences -fail to reject null hypothesis- test statistic less extreme than critical value, does NOT fall in critical region-->null IS supported, no significant differences *NEVER accept the null hypothesis and NEVER accept the alternative hypothesis. The only options are to reject or fail to reject 6. Compute effect size -effect size: a measure of the magnitude/size of the treatment effect. NOT a measure of how significant -significant results= present effect size -Cohen's D: standardized mean difference. d= sample mean-pop mean/ standard deviation(ALWAYS A POSITIVE VALUE because its a form of standard deviation and standard deviation can't be negative) -if failed to reject--> write N/A 7. Interpretation -explain the difference simply and clearly, include direction of effect -include means and SD';s -include obtained test statistic, significance level, effect size(when significant)
3 Characteristics of the Central Limit Theorem
1. Mean of sample means equals the population 2. Sample distribution of means(SDOM) will be normal if the pop. is normal or n is greater than or equal to 30 3. Standard error= standard deviation/square root of sample size
Four possible outcomes of Type 1/Type 2 Errors
1. The decision is cannot rejected the null when the null is true(correct decision) 2. The decision is cannot accept the null when the null is true(incorrect decision known as Type I error- "rejecting a good null") 3. The decision is cannot reject the null when the null is false(incorrect decision known as Type II error- "accepting a false null") 4. The decision is cannot accept the null when the null is false(correct decision)
What will you get if you add the "body" and the "tail" for any one z-score?
1.0
What percent of scores are within 2 standard deviations in a normal distribution?
95%
Alpha and beta are correlated to which errors?
Alpha--> type 1 Beta--> type 2
Why does each event need to be independent for probability to be accurate?
If it is dependent on something else, the probability will change.
Do we ever test our alternative hypothesis?
No- but the alternative hypothesis is what we want to know -instead we examine the probability/likelihood that the null hypothesis might be retained
What is the sample distribution of means?
a collection of all possible sample means of a given size from every possible combination of individuals in a population, the resulting distribution of sample means would be the sample distribution of means
Significance Testing
a low probability value casts doubt on the null hypothesis -conventional to conclude the null hypothesis is false if the probability value is less than .05 -when the researcher concludes that the null hypothesis is false, the researcher is said to have rejected the null hypothesis -the probability value below which the null hypothesis is rejected is called the alpha(a) level. AKA significance level -when the null hypothesis is rejected, the effect is said to be statistically significant -when an effect is significant, you can have confidence that the effect is not exactly zero -finding that an effect is statistically significant signifies that the effect is real and not due to chance -if the probability value is below .05 but larger than .01, than the null hypothesis is typically rejected, but not with as much confidence as it would be if the probability value were below .01 -higher probabilities provide less evidence that the null hypothesis is false
Type 2 error
occurs when we fail to reject a false null hypothesis(not really an error) -when a statistical test is not significant, it means that the data don't provide strong evidence that the null hypothesis is false -lack of significance doesn't support the conclusion that the null hypothesis is true, therefore a researcher shouldn't make the mistake of incorrectly concluding that the null hypothesis is true when a statistical test is significant -researcher should consider the test inconclusive -a type II error can only occur if the null hypothesis is false -if the null hypothesis is false, then the probability of a type II error is called Beta(b) -the probability of correctly rejecting a false null hypothesis equals 1-B and is called power
Two-tailed tests
probability calculated in both tails of a distribution -the null hypothesis for the two tailed is n=.5 -we reject the two-tailed hypothesis if the sample proportion deviates greatly from .5 in either direction
One-tailed tests
probability calculated in only one tail of the distribution -null hypothesis for a one tailed test: n is less than or equal to .5 -the one tailed hypothesis is rejected only if the sample proportion is much greater than .5
Beta(b): Probability of a Type II Error
probability of not rejecting the null hypothesis when the null hypothesis is false
Alpha(a): Probability of Type 1 Error
probability of rejecting the null hypothesis when the null hypothesis is true(rejecting a good null)
How is probability calculated?
probability= number of favorable outcomes/number of possible equally-likely outcomes
Hypothesis tests gather evidence in the form of data with the hopes that it will do what?
prove or disprove the hypothesis
Power is the ability to what?
reject a false null hypothesis correctly
Why do researchers need to conduct a formal hypothesis test?
researchers need to determine if there is a significant difference between something or if it is just due to change/standard error/sampling error
What is the standard error of the mean?
standard deviation of the sample distribution of means. the standard error of the mean measures the average distance that a sample mean is from the population mean. it is a measure of "error" in terms of how close our sample mean is to population mean. with small sample sizes, standard error is large, indicating more error around your samples estimation of the population
What is the logic that allows us to make decisions about the null hypothesis based on whether or not a test statistic falls in the critical region?
the critical region allows us to make a decision about the null hypothesis because if we pick a random sample mean from the SDOM ,it would be rather unlikely that we would obtain or pick a sample that falls in the critical region if the null is true. However, if you do obtain a sample mean that is in the critical region, you can interpret this as evidence against the null hypothesis because the likelihood of obtaining that sample mean by chance is so small and you can reject the null hypothesis.
Is critical region bigger or smaller with one-tailed test vs two-tailed test?
the critical region is bigger with a one-tailed test
The Null Hypothesis(H0)
the hypothesis that an apparent effect is due to chance, statement of no difference between the variables--> they are not related(considered status quo) -if the null is rejected, it is highly unlikely that we obtained our sample by chance and we favor the alternative hypothesis -if we fail to reject the null hypothesis, it is pretty likely that we just obtained this sample by chance and we retain the null hypothesis -typically opposite of researchers study -typically put forward with hope that it can be discredited and therefore rejected -the direction of the sample means determines which alternative is adopted(e.g. Mobese<Maverage or Mobese>Maverage)
Law of large numbers
the larger the number of individuals that are randomly drawn from a population, the more representative the resulting group will be of the entire population -increased sample size--> more representative data of the population
Effect size indicates practical significance. What is practical significance?
the magnitude of the difference in real life. it helps us find effect size and it is important to note that it doesn't help us make a decision about the null hypothesis.
What is probability value?
the probability of an outcome given the hypothesis. -not the probability of the hypothesis given the outcome
Type 1 error vs type 2 error
type 1: probability of rejecting the null hypothesis if the null is true type 2: failed to reject the null hypothesis when the null hypothesis is false
Inferential Statistics
using sample data to make some sort of conclusion about the population
Interpreting Significant Results
when a probability value is below the alpha level, the effect is statistically significant and the null hypothesis is rejected -if the null hypothesis is rejected, than the alternative hypothesis is accepted
Interpreting Non-Significant Results
when a significance test results in a high probability value, it means that the data provides little or no evidence that the null hypothesis is true -concluding that the null hypothesis is true is called accepting the null hypothesis -DO NOT ACCEPT THE NULL HYPOTHESIS WHEN YOU DO NOT REJECT IT -often non-significant findings increase one's confidence that the null hypothesis is false
