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Introduction to the t Statistic - Coggle Diagram
Introduction to the t Statistic
z-scores usually require more information than is available - we need the population standard deviation to compute the standard error.
the solution is to use a corresponding sample value in place of the standard deviation
sample variance = s²= SS/(n-1) or SS/df
sample standard deviation = s = √SS/(n-1) or √SS/df
estimated standard error
= s/√n or √s²/n
estimated standard error is used to estimate the actual standard error of the population when he standard deviation of the population is unknown
requires a shift to focus on variance rather than standard deviation
estimated standard error = √sample variance/sample size
t = (M-µ)/estimated standard error
as sample size increases, the sample variance more closely approximates population variance
for the t-statistic, sample size is expressed in degrees of freedom (df)
basic difference between z-score and t-statistic is that the z-score uses population variance and the t-statistic uses sample variance
if you calculate the t-statistics for every sample the entire set of values will form a t distribution the same way that an entire set of z-scores forms a z-score distribution
how well a t distribution approximates a normal distribution is determined by degrees of freedom
typically, the larger the sample size, the larger the degrees of freedoms and the closer a t distribution is to normal
the shape of a t distribution changes with degrees of freedom
distributions of t are bell-shaped, symmetrical, and have a mean of zero
has more variability than a z distribution - especially when df values are small - tends to be flatter and more spread out than a z distribution
for a z-score, only the numerator of the formula can vary
for a t-statistic, both the numerator and the denominator can vary
sample size and df increase = variability decrease = closer to a normal distribution
"family of distributions": there is a different sampling distribution of t for each possible number of degrees of freedom
t distribution tables are used to find proportions for t-statistics
as df increases, t values decrease in the .05 region until they reach 1.96 +/- in a z distribution, the z-score of 1.96 +/- is connected to the critical region of .05
this demonstrates that the larger the degrees of freedom, the more closely the t distribution approximates a normal distribution
hypothesis test with the t-statistic
the null hypothesis provides a specific value for the unknown population mean
variance and estimated standard error are computed from the sample data
t = (sample mean [from the data] - population mean [null hypothesis])/estimated standard error (computed from the sample data)
the unknown population is the one that exists after treatment is administered
t test can be used when the null hypothesis is gained from a theory, logical prediction, or even wishful thinking
assumptions of the t test:
1) the values in the sample must consist of independent observations
2) the population sampled must be normal
influence of sample size and sample variance:
1) any factor that influences estimated standard error also affects the likelihood of addressing the null hypothesis (sample variance and sample size)
2) larger the variance = larger error = less likely to receive a significant treatment effect
3) estimated standard error is inversely related to the number of scores in a sample = the larger the sample, the smaller the error
4) if all other factors are held constant, large samples produce bigger t statistics and are more likely to produce significant results
measuring effect size for the t-statistic
estimated d: (M-µ)/s
estimated d of 1.00 indicates that the treatment effect is equivalent to 1 standard deviation
measuring the percentage of variance (r²)
treatment is causing scores to increase or decrease which means it is causing the scores to vary
can measure effect size by determining how much of the variability is explained by treatment effect
to measure variability if treatment effect is removed = compute SS for each set of scores
the sum of squares represents the variability that is NOT explained by the treatment effect
total variability - variability not explained by treatment = variability explained by treatment
r² = variability accounted for by treatment effect/total variability, expressed a decimal or percentage
r² = percentage of variance accounted for by treatment
r² = can also be found by a single formula using the t-statistic:
r² = t² / (t²+df)
r² = 0.01 = small effect
r² = 0.09 = medium effect
r² = 0.25 = large effect
r² is not impacted by sample size
another way to measure the size of treatment effect is to compute an estimate of the population mean after treatment
confidence interval: an interval or range of results centered around a sample statistic
relies on the idea that a sample statistic should be relatively close to the population parameter
µ = M+/-t(estimated standard error)
have to pick a level of confidence:
95% (most common)
90%
99%
"the 95% confidence interval for µ"
find the t values bounded for the level of confidence
factors affecting the width of the confidence interval:
1) to gain more confidence in your estimate, you must increase the width of the interval which also produces a larger t value - there is trade off between precision and the confidence you can have that the interval contains the population mean
2) the larger the sample size, the smaller the interval because you are able to make a more precise estimate
directional hypothesis and one-tailed tests using the t-statistic:
1) state hypotheses and select and alpha level
2) locate the critical region (you must use the t distribution table)
3) calculate the t statistic
4) make a decision - is the t statistic in the critical region? either reject or accept the null hypothesis