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When testing a hypothesis; how many outcomes?
Two possible outcomes regarding a null hypothesis
Two possible states of real world Thus four possible decisions: Two are incorrect ... i.e. errors
Alpha Level or the level of significance
a probability value that is used to define the very unlikely sample outcomes if the null hypothesis is true.
If sample data fall in the critical region, the null hypothesis is...?
composed of extreme sample values that are very unlikely to be obtained if the null hypothesis is true.
The boundaries for the critical region are determined by the alpha level.
If sample data fall in the critical region, the null hypothesis is rejected.
**Estimating Population Parameters from Samples
-biased or unbiased, why?
Sample mean-Unlikely to be exactly equal to population mean BUT Not more likely to be greater OR Not more likely to be less sooo sample mean is an unbiased estimate of population mean
**Sample standard deviation- biased or unbiased?
- Unlikely to be exactly equal to population standard deviation BUT More likely to be less
- -Is usually an under estimate of population parameter
- -So sample standard deviation is a biased estimate of the population standard deviation
--more likely to be smaller than population variance and Pop standard deviation --> "Degrees of freedom"
And so must correct any estimate of the population variance increase it (i.e. use "n-1" when calculating the estimate)
-smaller the P-value?
-A moderate to large P-value means?
The probability, when Ho is true, of a test statistic value at least as contradictory to Ho as the value actually observed.
- The smaller the P-value, the more strongly the data contradict Ho.
- The P-value summarizes the evidence in the data about the null hypothesis.
A moderate to large P-value means that the data are consistent with Ho. (fail to reject Null)
=>Ex. P-value .26 or .83 indicates that the observed data would not be unusual if Ho were true; However, a P-value such as .001 means that such data would be very unlikely, if Ho were true.
-The P-value is the primary reported result of a significance test. If the P-value is sufficiently small, one rejects Ho and accepts H1.
Problem: compare a sample to a population; what is method?
- 1. Use population parameters to calculate the standard error of the mean of a sampling distribution. (σm = σ ∕ √n)
2. Use the standard error of the mean to compare sample mean with population mean by calculating a z-score (Z = (M – μ) ∕ σm)
3. Use z-table to determine the probability that a random sample would yield a mean greater than the mean of the sample
**A word on the logic and requirements of the statistic
-The "uniqueness" of your sample
The "uniqueness" of your sample is the probability that another random sample of the same size would have the same mean as your sample.
Or put otherwise, is your sample mean, is what would be expected by chance, a random selection?
=>The more unique your sample, the more likely it reflects a relationship between:
Your independent variable & Your dependent measure
- Two requirements
- 1.The population is normally distributed
- 2. You know the population Mean (μ) & Standard deviation (σ)
The t statistic
-An alternative to z
==> And so an ESTIMATED standard error of the mean is: "sm = s ∕ √n"
- ==>MUST know the population mean But can estimate population standard deviation from sample data (SD is missing)
- -A sample standard deviation is given by (as you know) BY THE SS FORMULA--> Sigma/ Population SD
-And to use the estimated standard error of the mean to compare your sample to the population must make one adjustment (Adjustment is necessary to account for the fact that you are estimating)
2. S^2 =SS/(n-1)
Estimated Standard error of the mean BECAUSE the SD is missing!
(Adjustment is necessary to account for the fact that you are estimating)
- 5. Final Comparision: t = (M – μ) ∕ sm;
- M= Sample mean
- μ= Population mean
- Sm=Estimated Standard error of the mean
THIS IS IMPORTANT REGARDING T-TESTS Values!
-Relationship between T-Value obtained and T-Critical?
-when do you fail to reject?
- To reject the null?
- --> In order to reject the null (There's a change), the T-value obtained (from the formula)must be greater than the T-critical (obtained from the table values); meaning SPECIAL (Less than .05 -alpha level)
- -TObtained>TCritical (change)
- -->Failing to Reject: (No change-confirm the Null) T-Critical is greater than Obtained T value
- TObtained<TCritical (change)
**THREE STREPS TO LOOKING UP VALUES OF T-CRITICAL:
- 1. Two Tails by Default
- 2. degrees of Freedom (from your sample--> n-1)
- 3. Alpha level: .05
Directionality of Statistical Tests
- Statistical tests have a property called "directionality"
- Nondirectional, called "two-tailed" tests
Directional, called "one-tailed" tests
**Looking up t-critical on table-
-reject null when?
-moving left to right?
-Directionality? two tailed vs one tailed??
reject when t crit is less than t obtained
-moving left to right, t values get smaller, and t critical gets bigger
-two tailed test: 2.04 is smaller than 2.262
-directionality: prior knowledge, predict outcome from prior knowledge (classical music makes more milk) meaning now i should perform a one tail test... more likely to reject the null because the event is much more different and chance of rejecting is larger. -so using one tail: more likely to reject null because you increase the area
-Your ability to predict an outcome means that you are better able to determine whether an event is a chance occurrence;More likely to reject Null hypothesis
-In statistical terms the region of the sampling distribution indicating that an event is something different than what would be expected by chance is larger
Nondirectional vs Directional
-Non-directional (two-tailed), where rejection of the sample mean is either above or below hypothesized population mean.
-Directional (one-tailed), where rejection of the sample mean is determined prior to experimentation.
**Compared a Sample to a population:When population parameters are known…..
**Compared a Sample to a population: When population parameters are unknown….
- When population parameters are known…
- assume a normal population and known standard deviation
- Z = (M – μ) ∕ σm
- When population parameters are unknown…
- -Sample to population: assume a normal population and unknown standard deviation
- t = (M – μ) ∕ sm
TYPE 1 and TYPE 2 ERRORS (Hypothesis Conclusions)
- ==> Type 1 Errors: When you reject the null hypothesis when in actuality, there was no effect from the treatment;
- it is a false report-determined by the alpha level: the probablility that the test will lead to a type 1 error if the null hypothesisis true; it determines the probability of obtaining sample data in the critical region even though there is no treatment effect
- -You can directly "set" this It is the chance (probability of the making the error) you are willing to accept when you test your hypothesis.
- ==> Type 2 Errors: occurs when a researcher fails to reject a null hypothesis(saying there is no effect) that IS REALLY FALSE, meaning the treatment did effect the sample but the hypothesis test failed to detect it (fail to detect the actual effect or change that resulted).
- -likely to occur when the treatment effect is very small; so a study is more likely to fail to detect the effect.
- -You cannot directly "set" this You can attempt to control it through good experimental design.
Increasing Sample size increases the ...
z-score and produces a smaller standard error and a larger value for z-score,
Higher variability can reduce ...
the chances of finding a significant treatment effect, increases the standard error and bring the z-score closer to the mean/center of distribution making it more likely to be concluded as a "fail to reject the null" (no change occurs/not special)
Directional hypothesis Test/One Tailed:
the stat hypotheses (both null and H1) specify either an increase or a decrease in the population mean
Power of a statistical test is the probability that the test will correctly reject a false null hypothesis. Meaning power is the probability that the test will identify a treatment effect if one really exists.
-Alpha level: Reducing alpha level reduces power of test; there would be lower probability of rejecting the null hypothesis and a power value of test because area increases?
One Tail vs Two tailed tests
Going from 2 tails to 1 Tail increases the power of the hypothesis test.
the natural discrepancy, or amount of error, between a sample statistic and it's corresponding population parameter.
EXTRA CREDIT: Remember to write this...
A T-Test was performed and revealed a significant/non-significant difference between the sample mean and population, t(df)=tOBT, P<.05
- Standard Deviation of Sample distribution or Standard Error of Mean (sample vs population)
- -Find by: σ/
- n= Number of samples
- Estimated Standard Error of the mean (SD is missing)
- -find by: s/