Population Standard Deviation & Sample Variance Formula

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Population Standard Deviation, Standard Error, Sample Variance Formula – Statistic

The key terms in these statistic chapters include Estimated Standard Error, Population Standard Deviation, Rejecting the Null Hypothesis,  Sample Variance Formula & Sample Variance.


A sample of n=4 scores has SS=48. What is the estimated standard error for the sample mean?

2 (find the sample variance SS/n-1
estimated standard error= under square root (sample variance/ n) = 2)


A sample of n=4 scores has SS=60 what is the variance for this sample?

20 sample variance = SS/n-1


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.


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 and estimated standard error= under square root (sample variance/ n)


A repeated-measures experiment and a matched-subjects experiment each produce a t statistic with df = 10. How many individuals participated in each study?

11 for repeated, and 22 for matched


A researcher plans to conduct a research study comparing two treatment conditions with a total of 20 scores in each treatment. Which of the following designs would require only 20 participants?

A repeated-measures design

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?

2 (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?

The t statistic could be considered as an estimated z statistic
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


A sample of n=4 scores has SS=48. What is the estimated standard error for the sample mean?

2 (find the sample variance SS/n-1
estimated standard error= under square root (sample variance/ n) = 2)


A sample of n=4 scores has SS=60 what is the variance for this sample?

20 sample variance = SS/n-1


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.


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 and estimated standard error= under square root (sample variance/ n)


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

34
n-1= 34


What is the variance for the sample for n=31 that has an SS of 120

Sample for variance= SS/n-1 = 4


A repeated measures study using a sample of n = 20 participants would produce a t statistic with df = ____.

19


For which of the following situations would a repeated-measures research design be appropriate?

Comparing pain tolerance with and without acupuncture needles


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

s^2= SS/n-1


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)

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


t statistic:

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

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 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

Large variance is bad for inferential statistics. It means that the scores 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.

Estimated d

When the population values 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. This 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.


S

Sample variance


Population variance:

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

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.


t statistic

A statistic used to summarize sample data in situations where the population standard deviation 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.


A sample of n=4 scores has SS=48. What is the estimated standard error for the sample mean?

2 (find the sample variance SS/n-1
estimated standard error= under square root (sample variance/ n) = 2)


A sample of n=4 scores has SS=60 what is the variance for this sample?

20 sample variance = SS/n-1

Why are t statistics more variable than z scores?

The extra variability 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.


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 and estimated standard error= under square root (sample variance/ n)


t distribution

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.


Confidence interval

An interval estimate that described in terms of the level (percentage) of confidence in the accuracy of the estimation.


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