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why are t statistics more variable than z scores?
The extra variability is caused by variations in the sample variance
what is a fundamental difference between the t statistic and z-score
The t statistic uses the sample variance in place of the population variance.
A sample of n=4 scores has SS=60 what is the variance for this sample?
sample variance = SS/n-1
what is the sample variance and the estimated standard error for a sample of n=9 scores with SS=72
s^2 = 9 and Sm =1
sample variance= SS/n-1
estimated standard error= under square root (sample variance/ n)
A sample of n=4 scores has SS=48. what is the estimated standard error for the sample mean?
find the sample variance SS/n-1
estimated standard error= under square root (sample variance/ n) = 2
A sample of n=25 scores has a mean of M=40 and variance of s^2 =100 what is the estimated standard error for the sample mean?
estimated standard error= under square root (sample variance/ n) = 2.041
Which set of characteristics will produce the smallest value for the estimated standard error?
A large sample size and small sample variance
which of the following statements are true of the t statistic?
1. the t statistic could be considered as an estimated z statistic
2. The t statistic provides a relatively poor estimate of z with small sample size
when the population standard deviation is unknown you can use the t statistic, assuming all relevant assumptions are satisfied.
what is formula for for t statistic
t = (M-u) /Sm
with the exception of whether the population standard deviation is known the necessary assumptions for hypothesis tests with the t statistic and with the z statistic are
essentially the same
what is degree of freedom n=35
what is the variance for the sample for n=31 that has an SS of 120
sample for variance= SS/n-1 = 4
what is estimated standard error for the sample for n=16 that has a sample variance of 400
under the square root (sample variance/sample size)= 5
sample Variance formula
estimated standard error for sample formula
estimated standard error= square root (sample variance/sample size)
Sm (big S and small M)
estimated standard error
formula= Sm= s/square root n
t statistic formula
t = (M-u) / Sm
estimated standard error (Sm)
is used as an estimate of the real standard error, σM, when the value of σ is unknown. It is computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean, M, and the population mean, μ.
two reasons for making this shift from standard deviation to variance:
1. the sample variance is an unbiased statistic; on average, the sample variance (s2) provides an accurate and unbiased estimate of the population variance (σ2). Therefore, the most accurate way to estimate the standard error is to use the sample variance to estimate the population variance.
2. the t statistic that require variance (instead of standard deviation) in the formulas for estimated standard error. To maximize the similarity from one version to another, we use variance in the formula for all of the different t statistics. Thus, whenever we present a t statistic, the estimated standard error is computed as
is used to test hypotheses about an unknown population mean, μ, when the value of σ is unknown. The formula for the t statistic has the same structure as the z-score formula, except that the t statistic uses the estimated standard error in the denominator.
-A t statistic is used instead of a z-score when the population standard deviation and variance are not known.
Degrees of freedom:
describe the number of scores in a sample that are independent and free to vary. Because the sample mean places a restriction on the value of one score in the sample, there are n − 1 degrees of freedom for a sample with n scores.
The t Distribution
is the complete set of t values computed for every possible random sample for a specific sample size (n) or a specific degrees of freedom (df). The t distribution approximates the shape of a normal distribution.
Like the normal distribution, t distributions are bell-shaped and symmetrical and have a mean of zero. However, t distributions have more variability, indicated by the flatter and more spread-out shape. The larger the value of df is, the more closely the t distribution approximates a normal distribution.
The Unknown Population
The mean value is known before the the treatment. The question is whether the treatment influences the scores and causes the mean to change. In this cause, the unknown population is the one that exists after the treatment is administered, and null hypothesis simply states that the value of the mean is not changed by the treatment.
Influence of Sample Size and Sample Variance
- Because the estimated standard error, sM, appears in the denominator of the formula, a larger value for sM produces a smaller value (closer to zero) for t. Thus, any factor that influences the standard error also affects the likelihood of rejecting the null hypothesis and finding a significant treatment effect
large variance is bad for inferential statistics. Large variance means that the scores are widely scattered, which makes it difficult to see any consistent patterns or trends in the data. In general, high variance reduces the likelihood of rejecting the null hypothesis.
When the population values are not known and you must substitute the population values are not know and you must substitute the corresponding sample values in their place. the population mean with treatment and the standard deviation are both unknown. Therefore, we use the mean for the treated sample and the standard deviation for the sample after treatment as estimates of the unknown parameters.
Standard Error of M:
The standard deviation of the distribution of sample means, σM, is called the standard error of M. The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ). The standard error serves the same two purposes for the distribution of the sample means.
The Alpha Level:
The alpha (α) value is a small probability that is used to identify the low-probability samples. By convention, commonly used alpha levels are α = .05 (5%), α = .01 (1%), and α = .001 (0.1%). For example, with α = .05, we separate the most unlikely 5% of the sample means (the extreme values) from the most likely 95% of the sample means (the central values).......The alpha level, or the level of significance, is a probability value that is used to define the concept of "very unlikely" in a hypothesis test.
is the square root of the variance and provides a measure of the standard, or average, distance from the mean.
The t statistic
allows researchers to use sample data to test hypotheses about an unknown population mean and not require any knowledge of the population.
t distribution small vs. large values
For large values of df, the t distribution will be nearly normal, but with small values for df, the t distribution will be flatter and more spread out than a normal distribution.
degrees of freedom and shape
The exact shape of the t distribution changes with degrees of freedom
df gets very large, t distribution gets closer to normal z-score distribution
Under what circumstances is a t statistic used instead of a z-score for a hypothesis test?
A t statistic is used instead of a z-score when the population standard deviation and variance are not known.
In general, a distribution of t statistics is flatter and more spread out than the standard normal distribution. (True or false?)
For df = 15, find the value(s) of t associated with each of the following:
a . The top 5% of the distribution.
b. The middle 95% of the distribution.
c . The middle 99% of the distribution.
A. +1.753 B. ±2.131 C. ±2.947
estimated standard error
An estimate of the standard error that uses the sample variance (or standard deviation) in place of the corresponding population value.
A statistic used to summarize sample data in situations where the population standard deviation is not known. The t statistic is similar to a z-score for a sample mean, but the t statistic uses an estimate of the standard error.
The distribution of t statistics is symmetrical and centered at zero like a normal distribution. A t distribution is flatter and more spread out than the normal distribution, but approaches a normal shape as df increases.
percentage of variance accounted for by the treatment (r2)
A measure of effect size that determines what portion of the variability in the scores can be accounted for by the treatment effect.
An interval estimate that is described in terms of the level (percentage) of confidence in the accuracy of the estimation.
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