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20160406021853class_5_inferential_statistics._3.21.16.pptMarch 21st , 2016
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- Allows the researcher to make generalizations from sample data to the population from which the sample was drawn.
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Systematic Variation
Differences in performance created by a specific experimental manipulation.
Unsystematic Variation
Differences in performance created by unknown factors.
Age, Gender, IQ, Time of day, Measurement error etc.
Randomization
Minimizes unsystematic variation.
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- Unbiased random sample contains errors.
- Eliminate bias reducing sampling error.
- Larger samples yield smaller sampling errors.
- Precision: same results will be obtained if another random sample were drawn from the population.
Increase sample size
- Smaller and the anticipated difference
Larger sample size
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- Small samples can identify very large differences
Consider the variables in your study
Treatment studies (new medication to reduce a virus)
- Populations with very limited variability
Small samples precise results
- More variable the population, the larger the sample size.
Achievement Scores
- In most cases, 10-15 participants per variable/construct
- Large samples do not correct for bias.
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- Difference between population parameter and a sample statistic
What is unaccounted for
- Even with random sampling, there will be some error
- Can estimate the expected error and include in statistical analysis
- Sampling error decreases with increases in sample size
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- Take several samples from the population
- Samples will differ slightly
- Each sample will have its own mean
- We can calculate the sample mean, the average of the sample means
- Samples will vary because they contain different members of the population: sample variation
- Sampling distribution: diagram that shows the frequency distribution of sample means from the same population
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- Means: Sampling Distribution of means
- Central Limit Theorem: sampling distributions of means will resemble the normal curve
- Standard deviation of the sampling distribution: Standard error of the mean SEM
M = 9
= 10
M = 8
M = 10
M = 11
M = 12
M = 11
M = 9
M = 10
M = 10
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- Central Limit theorem
Large samples will take the shape of a normal distribution regardless of the shape of the population from which it is drawn
Equal to the standard deviation of the sample (s) divided by the square root of the sample size.
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- Standard deviation of the sample mean
Remember that standard deviation a representation of how close the observations are to the mean
- A large standard error means that there is a lot of variability between the means of different samples
The sample might not be representative of the population
- A small standard error indicates that most sample means are similar to the population mean
Sample is likely to be an accurate reflection of the population
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- Larger the sample, the smaller the standard error of the mean.
Larger samples have greater precision
- Less variability in a population, the smaller the standard error the mean.
Less variability more precision.
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- Z-Score
Standardizing a score with respect to the other scores in the group.
Expresses a score in terms of how many standard deviations it is away from the mean.
The distribution of z-scores has a mean of 0 and SD = 1.
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- We can assess the accuracy of the sample mean as an estimate of the mean in the population by calculating boundaries within which the true value of the mean fall.
95% and 99% are the percent range for most of the confidence intervals calculated.
A confidence interval for the mean is a range of scores constructed such that the population mean will fall within this range in 95% of samples
If we collect 100 samples, calculated the mean and then calculated the confidence interval for that mean, then for 95 of these samples the confidence interval would contain the true value of the population mean
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- Sample means will be different form the population means due to sampling variation.
- We use Z scores to calculate confidence intervals.
Properties of Z score
As such, 95% of z-scores lie between −1.96 and 1.96.
Calculate a 95% confidence interval
99% of z-scores lie between −2.58 and 2.58
Calculate a 99% confidence interval
99.9% of them lie between −3.29 and 3.29.
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- Lower boundary of confidence interval
- (1.96 X SE)
- Upper boundary of confidence interval
+ + (1.96 X SE)
- Value of the mean might be influenced by sampling errors.
- m=75, s=16, n=64 Sem =2.00
- Actual mean is called a point estimate
95% or 99% confidence interval
95% confidence interval: We have 95% confidence that the true population mean is between 71-79
99% confidence interval:
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- Research Hypothesis H1
Statement of the relationship the researcher seeks to find as a result of the study
Directional or Non-directional
- Null Hypothesis H0
States that there is no relationship between the variables under study.
Assesses the probability that the results of the study were due to chance.
Ex. There is no relationship between mathematical intervention and mathematic achievement.
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- States the expected relationship between variables.
- Must be testable
- Should be consistent with the existing body of knowledge
- Statement should be concise.
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- State in operational terms, the relationships that should be observed if the research hypothesis is true
- State the null hypothesis.
- Gather the data
- Determine if evidence is sufficient to accept or reject the null hypothesis
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- Rejection of research hypothesis after empirical testing does not mean that the study was failure.
- Unconfirmed hypothesis are apart of scientific research and still add to the body of knowledge
- Hypothesis are never proved or disproved
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- A researcher finds the following differences in Self-Esteem scores between boys and girls.
- Girls m=70.00 Boys m=80.00
- Do boys have higher self-esteem or are the results due to chance factors?
We use inferential tests to answer the question.
- Null Hypothesis:
The true difference between the means (in the population is zero)
H0: u1-u2=0
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- Symbolized by a Greek lowercase level alpha (α).
- Rejection of a true null hypothesis.
States that the results of the study were not due to chance.
- Researcher incorrectly concludes that significant differences WERE found or that a relationship between the variables exists.
- Considered to be more serious than Type II error.
Inferential statistics allow the researcher to estimate population parameters based on what is known about the sample, and sampling error
- IF YOU REPLICATE the DATA 100 TIMES……
- On 5 occasions we would have a see an effect., probability of error, we would have a test statistic large enough to tell us that there is an effect in the population.
- If we lower the probability~ make alpha more stringent we increase our risk of making a type II error
- If we take 100 samples of data from a population in which an effect exists, we would fail to detect that effect in 20 of these samples.
Use tests of significance
Null hypothesis is TOOL for significance testing; because you are testing whether your results are due to chance alone, or a real relationships among variables ( so what we do is RULE OUT that is happened by chance alone) – THE CHANCE EXPLANATION IS THE NULL HYPOTHESIS
You always begin statistical tests with the assumption that the null hypothesis is true – so you reject the null or fail to reject the null
TYPE I – reject the null, but it IS true; you say you have a significant finding when you don’t
SAY THE NULL IS FALSE, BUT IT IS TRUE; GENERALLY THIS IS THE MORE SERIOUS
(say guilty, but innocent)
TYPE II – accept the null but it is NOT true; say there is no difference when really there IS a difference
SAY THE NULL IS TRUE, BUT IT IS FALSE
(say innocent, but they’re guilty)
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- Symbolized by a Greek lowercase level beta (β).
- Investigator retains a false null hypothesis
- Researcher incorrectly concludes that significant differences were NOT found or that a relationship between the variables does not exist.
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- Null Hypothesis
- Type I error:
reject true null hypothesis - Type II error:
accept false null hypothesis
| Innocent | Guilty | |
| Not guilty verdict | Justice | Type II Error β |
| Guilty verdict | Type I Error α | Justice |
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- Investigators must weigh the consequences of a Type I or Type II error before the conducting the experiment.
- Level of significance: is the level at which the null hypothesis will be rejected.
Probability that the investigator is willing to risk in rejecting a null hypothesis.
Most common used levels are .05 and .01
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- Type I errors can lead to changes in the educational system that are not necessary such as:
Best practices
Curriculum modifications
Teacher training programs
- Type II errors can lead to the lack necessary changes being made. Maintenance of the status quo.
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Sample Mean
6
7
8
9
10
11
12
13
14
Frequency
0
1
2
3
4
Mean = 10
SD = 1.22
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