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Math
Statistics
Hypothesis Testing
Statistics Lecture 4: The t-Test Lecture
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Terms in this set (15)
What is the problem with a single z test?
requires a population parameter or SD, which in the real world we usually dont have
Because the sample variance (s2) is an unbiased estimator of σ2, we can use s2 to estimate what?
The standard error!!! σM
Sampling distribution of s2:
the distribution of variance estimates obtained from all possible random samples of a particular size that could be drawn from a population
What are the consequences of Sampling distribution of s2 having a positive skew, particularly with small N?
-The expected value of the sampling distribution is equal to σ2 (and thus unbiased), but any single, randomly selected sample value is likely an underestimate of σ2
-Underestimation of σ2 results in systematic inflation of test statistic (i.e., z-value)
What is Student's t?
specifies the sampling distribution of the test statistic t which uses s2 as an estimate of σ2
t distribution actually represents a family of curves with a different shape at each df
t distribution approaches normal curve as df approach ∞
Equation: m-u/ SE
Critical values become more extreme as df decrease
Greater threshold for significance at lower N
Result is a systematic correction for inflation of the test statistic computed with s2 as an estimate of σ2
Recap: So what do we do with z test versus a t- test
z-test is structured to determine whether a random sample mean is likely to have occurred in a population with a known μ and σ2
Rejection of H0 interpreted as evidence that data are unlikely to have been drawn from a population with mean μ
Single-sample t test is used when comparing a sample mean to μ when σ2 is unknown
t distribution and corresponding df correct for the underestimation of σ2 observed under the sampling distribution of s2 (particularly at small N)
Threshold for interpretation becomes greater for smaller vs. larger N
Critical values for single-sample test indexed in t table
Factors influencing t?
Difference between M and μ
Sample variance
Sample size
Degrees of freedom (df):
refer to the number of values in the calculation of a statistic that are free to vary
In the case of a single sample t test, df = N - 1. What does this mean?
Loss of 1 df a result of using sample mean in the calculation of s2
1. s2 calculated as a function of the deviation of each score from the mean
2. Because ∑(X - M) MUST equal to 0, the final X value in the distribution is fixed (i.e., its not free to vary)
3. Because only one variance value is estimated in a single-sample t, df = N - 1
What is an effect size?
an estimation of the magnitude of the effect on a population, independent of sample size.
What does an effect size provide us?
ES provides an estiamte of the population value
Although some conversion possible, different analyses have different indices of effect
Help augment the interpretation of inferential stats
Effect size for singles samples t?
Raw difference of M-u/s
OR
d=t/sqaure root of n
What are the interpretative standards of an effect size?
Small: d=.20
Medium: d= .50
Large: d=.80
What is an s-value?
Test provides roughly the same evidence AGAINST Ho given A as observing s successive heads on tosses on a fair coin
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