Chapters 6 & 7 Flashcards
Statistical Inference
A procedure by which we use information from a sample to reach conclusions about a population. 2 general areas: Estimation and Hypothesis Testing
Estimation
Uses sample data to calculate a statistic.
Sampled Population
The population from which you draw your sample
Target Population
The population you wish to make an inference about; the population you wish to generalize your results to
Point Estimate
A single numerical value used to estimate the corresponding population parameter
Interval Estimate
A range of values (with a lower and upper bound) constructed to have a specific probability (or confidence) of including the population parameter
Estimator
The rule that tells us how to compute the estimate
Estimate
A single computed value
Precision of the Estimate (Margin of Error)
Reliability Coefficient times Standard Error
Reliability Coefficient
Z or T value when finding CI
Probabilistic Interpretation
In repeated sampling of a normal distribution population with a known s.d., 100(1-alpha) percent of all intervals will in the long run include the population mean.
Practical Interpretation
When sampling a normally distributed population with a known s.d., we are 100(1-alpha) percent confident that the single computed interval will contain the population mean
Z Values for 90, 95, and 99% CIs
90 = 1.645, 95 = 1.96, 99 = 2.58
T-Statistic
the result of using s instead of sigma is a distribution with a s.d. > 1, so it is not normal. The resulting distribution is the t-distribution
Degrees of Freedom (df)
the number of independent pieces of information that go into the estimate.
t vs. z distribution
- both means = 0, 2. both symmetrical about the mean, 3. both have values from negative to positive infinity, 4. t is more variable than z, 5. The shape of t changes with sample size, when the sample size is infinite the two distributions are identical. 6. t distribution is less peaked in the center and has higher tails.
Rule of Thumb for Equal Variance
If the larger sample variance is more than 2x as large as the smaller variance the variances are unequal
Hypothesis Testing
Procedure for testing a claim about a population
Research Hypothesis
The question or idea that motivates the research
Statistical hypothesis
Hypotheses that are stated in such a way that they may be evaluated by statistical techniques
- Data
Determine the type of data and scale of measurement
- Assumptions
Normal Distribution, equal population variances, independence of the samples, random samples, etc
- Hypotheses
Two statistical hypothesis: Ho and Ha
Null Hypothesis
Statement of agreement with the conditions presumed true in the population. The complement of the conclusion the researcher is seeking. “The hypothesis of no difference. “ Always a statement of equality.
Alternative Hypothesis
What the researcher hopes to conclude. By rejecting Ho you support Ha. States there is a difference.
Two-Tailed Hypothesis
Does not specify the direction in which the null is incorrect
One-Tailed Hypothesis
Specifies the direction in which the Ho is incorrect
- Test Statistic
A statistic that is computed from the sample statistic. The decision to reject or fail to reject the Ho is based on the magnitude of the test statistic.
(Statistic - hypothesis/standard error)
- Distribution of the Test Statistic
We must determine and specify the probability distribution. ex: the t-distribution with n-1 df. Use z-test if normally distributed and the population variance is known. Use t-test if normally distributed and the population variance is unknown.
- Decision Rule
Find critical values to determine the rejection and non-rejection regions.
Level of significance (Alpha)
The probability of rejecting a true null hypothesis. This would be an error, so alpha is typically small (.01, .05 or .10)
Type I Error (Alpha)
False positive. When we reject a true null hypothesis. Probability of a type I error is alpha
Type II Error (Beta)
False negative. When we fail to reject a false Ho. Beta is not controlled by the investigator. It is effected by sample size, alpha, hypothesized value, and the true value of the parameter. Typically beta > alpha
Power
The chance of making a correct decision of rejecting the Ho when the Ho is false. As sample size increases power increases. (1-beta)
- Calculate the Test Statistic
Compare this value to the rejection and non-rejection region
- Statistical Decision
Reject or Fail to Reject Ho
- Conclusion
If the null is rejected, there is evidence in favor of the Ha
If the null is not rejected, there is not evidence in favor of the Ha
- P-values
Tell us the probability associated with obtaining the computed test statistic or more extreme, given that the null hypothesis is true. Smaller values are better for justifying doubting the truth of Ho. If the p-value of the calculated test statistic is less than or equal to alpha, we can conclude the groups are different or there is an association (statistical significance)
Statistical Significance
if a p-value is less than or equal to alpha, it implies statistical significance. Ability to state that the observed difference is not due to chance alone. Necessary for clinical significance but does not imply magnitude of effect.
