Statistics Exam 4
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. What would you like to do?
- Discover the relationships that exist between events/things
- -Accomplished by:Asking questions and Getting answers =>In accord with certain rules ... the scientific method.
- -Question is a hypothesis & Answer is obtained by testing the hypothesis Which gives the general model………
Some IMPORTANT restrictions about How the hypothesis is formed.
- How the hypothesis is tested…
- Forming hypotheses is an "everyday-everybody" activity
- ex: I will do better on examinations if relax the night before--> Is a OK "hypothesis" ... a statement of a relationship OK, BUT NOT a scientific hypothesis
- A scientific hypothesis must be: Specific,Empirically testable,Strictly related to some experimental procedure
- -actually consists of two separate mutually exclusive hypotheses: 1. A null hypothesis and/or 2. An alternative hypothesis
A statement reflecting the possibility that there are no differences between the objects and/or events that are being observed
In formal terms: Ho: µ1 = µ2Where: µ1 and µ2 are the mean or average of several observations
A statement reflecting the possibility that there are differences between the objects and/or events that are being observed
- In formal terms:
- H1: µ1 <> µ2 or H1: µ1 < µ2 or H1: µ1 > µ2
Testing between the null and alternative hypothesis Accomplished through
collection of data that must be scientifically acceptable, i.e. Observable, Public, Replicable
The test concentrates on the null hypothesis which you either Reject or Fail to reject
If there are no differences between your observations you
Fail to reject the null hypothesis and Disregard the alternative hypothesis
If there are differences between your observations you
Reject the null hypothesis and Accept the alternative hypothesis
Some things to note about hypothesis testing
Failing to reject the null hypothesis does not mean that the null hypothesis is TRUE
The null hypothesis can never be proven ;You can only fail to reject it
Rejecting the null hypothesis Means you accept the alternative hypothesis It does not establish the validity of a relationship
Validity is a function of experimental design
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
Type I error
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 II error
You cannot directly "set" this You can attempt to control it through good experimental design.
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)
Tests that do make assumptions and test hypotheses about population parameters.
-z & t, ANOVA, and F test=>Involves an assessment of whether your observed data is related to your independent variable
-Or is simply what might be expected by chance random sampling (i.e. no relation between Independent variable and Dependent measure)
- Requires knowing or estimating population parameters
- Mean: (μ)
- Standard deviation: (σ)
- Assumption of normality
**For example: consider Pat (individual score) = 64 Population mean (μ) = 50 Population standard deviation (σ) = 8
- ==>And if population is normal, then you know
- ~68 (68.26) % data points between + 1
- ~95 (95.44) % data points between + 2
- ~99 (99.74) % data points between + 3
- And remember: these are percentages, not absolute values
- Areas under the normal curve
- ==>Remember the z-distribution?
- Provided areas (proportion of scores) under a normal distribution according to z-equation (x-u/Pop SD) =1.76
==>And if convert Pat's raw score to a z-score And then look up Pat's z-score in the Z-table Meaning that ~96% (1.00 - .0401 = .9599) of scores in distribution are below Pat. (page 699, G&W).
- =>OR PUT OTHERWISE: If we were to randomly select a score from Pat's distribution The probability that the score would be greater than Pat's would only be 4 in 100And we have done what we set out to do--Accomplished a statistical test (comparing Observed data and What would be expected by chance)
- And thus, Pat's score is significant at p < .05
-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.
Standard Error of the Mean (ch 7)
-When comparing an individual to a population needed to know two things about the population
-the measure of "variability"?
- When comparing an individual to a population needed to know two things about the population
- Mean: (μ)
- Standard deviation: (σ)
- Only slightly different when comparing a sample to a population
- BUT NOW--a SAMPLE: Since you are not concerned with a single individual but with a sample of individuals;
- The "population" of interest is not A population of individuals but rather A population of samples, i.e. a SAMPLING DISTRIBUTION
- And the measure of "variability" is not "The standard deviation" of a population but rather "The standard deviation of a sampling distribution", IE...STANDARD ERROR OF THE MEAN
The calculation details of Standard Error of Mean: (2 steps)
- Step 1: The standard error of the mean
- σm = σ ∕ √n; where: n = sample size
- Step 2:The z comparison
- Z = (M – μ) ∕ σm
- M=Sample Mean
==>Which is not really different than what we did when comparing an individual to a population
** Herd Of Cows Example:
Suppose a Herd of 10 cows (n=10); Mean milk production is 1.8 gallons/cow
Question: How unique is this herd?
Given: μ=1.5 (population mean), σ (population)=.55, n=10, M (sample mean)=1.8
- => Calculate the standard Error of Mean using:
- Z = (M – μ) ∕ σm
- Z=.1724; using Z table: .0427 which is less than .05 (means our cows are special :D )
- =>Thus herd is pretty unique since the likelihood that any random sample of 10 cows from the population would produce more milk is less than 5 times in 100 (study!)
- Or in "statistics": The probability of selecting a herd of greater milk producers is p < 0.05
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)
Comparisons When Estimating Population Parameters & T-Distribution
-The adjustment part: Sm (estimated Standard Error of the mean)
-Estimating the standard deviation requires a different sampling distribution which is the t-distribution==>The t-distribution :Normal distribution, More platokurtic than z-distribution, Tails more elevated
==>And comparison becomes ==> step 2 of T-Test after SS and finding sm (step 1) t = (M – μ) ∕ sm
Thus: Because use estimate of population standard deviation to estimate standard error of mean , Must use t-distribution to get probability of randomly selecting a sample with a mean similar to the mean of your sample,
But not conceptually different -- just an adjustment
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
What does that "t-value" mean ?
-What is the probability of a random sample of 10 cows being like your cows
-To find out, consult a t-table
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)
Z vs T Tables
Z-table gives exact probabilities
t-table gives ranges of probabilities
**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
**Degrees of freedom:
- Number of values in a calculation that are free to vary
- That is: The degrees of freedom for a mean of 10 values is 9 ... because
- -If the mean of 10 numbers is, for example, 5 -Nine of the numbers "free" to be any value but when these are established, the 10th number is determined if the mean is to be 5
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
Review:Z-test a statistical test
-used to decide if a sample mean does or does not come from a specified population-when the standard deviation of the population is known.-When the standard deviation of the population is unknown then a t-test is performed.
the goal is to decide whether to reject the null hypothesis.
traditionally set at .05 where, also, the acceptance and rejection regions are determined.
the absolute value of that defines the rejection region(s).
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
Hypothesis Test": .
- a statistical method that uses sample data to evaluate a hypothesis about a population
- 1. We state a hypothesis about a population
2. Use the hypothesis to predict the characteristics that the sample should have (meaning sample should be similar to the population; along with a certain amount of error)
3. Then we obtain a sample randomly from a sample and measure the changing variables.
4. Compare the obtained sample data with the Hypothesis Prediction. If sample is consistent, then hypothesis is reasonable. BUT if there's a big discrepancy, we must conclude the hypothesis is wrong.
there are different versions of hypothesis testing; but the simplest version is using a sample mean to test a hypothesis about a population mean
The Unknown Population:
Reseearch begins with a known population with a set of individuals that exists BEFORE treatment; the purpose of the reasearch is to determine the effect of a treatment on the indivdiuals in the population.
-assume that if treatment has any effect, it is simply to add a constant amount or subtract from the individual's scores (which wouldn't change the shape of the distribution or SD)
-unknown population (after treatment) is the focus of the research question
**THE FOUR STEPS OF A HYPOTHESIS TEST:
- ==> STEP ONE: STATE THE HYPOTHESIS <==
- -state a hypothesis about the unknown popultion; there are TWO OPPOSING hypothesis stated in terms of population parameters (mean)
- -Null & Alternative: (directionality)
- ==> STEP 2: SET THE CRITERIA FOR A DECISION:
- => Alpha level/Level of significance: The boundaries that sepearate the high0probablility samples from the low probability samples; if a=.05 (5%) then 95% lies in the center and the tail ends are the "critical regions"(very unlikely)
- => Critical Regions: composed of the extreme sample values that are very unlikely (as defined by the Alpha level) to be obtained if the null hypothesis is true.-boundaries for the critical region: use the alpha-level probability and the unit normal table; 5% of the critical regions are divided into two tails of the distribution so that the Z-score boundary is 1.96 or -1.96. (column c)
- **Whenever athe data froma research study produce a sample mean that is located in the critical region, we can conclude that the data are NOT consistent with the Null and we REJECT the null.
- ==> STEP 3: COLLECT DATA AND COMPUTE SAMPLE STATISTICS
- -compare sample mean (data) to the Null hypothesis (heart of the hypothesis test)
-comparisiion computed by computered a z-score that describes exactly where the sample mean is located realtive to the hypotheisized population mean from Null/H0.-z= sample mean subtract the Value of the mean from Null Hypothesis and divde by "standard error between M and U" (pg 211)
- ==> STEP 4: MAKE A DECISIONUse the z-score value to make a decision about the Null Hypothesis according to the criteria established in step 2 with two possible outcomes:
- -Final decision is made by comparing treated sample (those who get treatment) with the distribution of sample means that would be obtained for untreated samples (the majority)
TYPE 1 and TYPE 2 ERRORS (Hypothesis Conclusions)
==> 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
- ==> 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
(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.
if it is very unlikely to occur when the Null Hyp is true; meaning result is significant enough to reject the null hypothesis. So treatment has a significant effect if the decision from the hypothesis is to reject the null
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 discrpancy, or amount of error, between a sample statistic and it's corresponding population parameter.
Distribution of sample means:
- the collection of sample means for all of the possible random smaples of a particular size (n) that can be obtained from a population; contains all the possible samples.
- -Necessary to have all the values in order to compute probability
is a distribution of statistics obtained by selecting all of the possible samples of a specific size from a population (M-sampling distribution); the larger the sample, the closer to the Population mean
Standard Error of the Sample mean:
describes the distribution of sample means, shows how much diff is experceted from one sample to another; also how well an individual sample mean respresents the entire distribution
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/
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